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PacificJournal ofMathematics
IN THIS ISSUE—Richard Horace Battin, Note on the “Evaluation of an integral
occurring in servomechanism theory” . . . . . . . . . . . . . . . . . . . . . . . . 481Frank Herbert Brownell, III, An extension of Weyl’s asymptotic law
for eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483Wilbur Eugene Deskins, On the homomorphisms of an algebra onto
Frobenius algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501James Michael Gardner Fell, The measure ring for a cube of
arbitrary dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513Harley M. Flanders, The norm function of an algebraic field
extension. II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519Dieter Gaier, On the change of index for summable series . . . . . . . . . . . 529Marshall Hall and Lowell J. Paige, Complete mappings of finite
groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541Moses Richardson, Relativization and extension of solutions of
irreflexive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551Peter Scherk, An inequality for sets of integers . . . . . . . . . . . . . . . . . . . . . 585W. R. Scott, On infinite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589A. Seidenberg, On homogeneous linear differential equations with
arbitrary constant coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599Victor Lenard Shapiro, Cantor-type uniqueness of multiple
trigonometric integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607Leonard Tornheim, Minimal basis and inessential discriminant
divisors for a cubic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623Helmut Wielandt, On eigenvalues of sums of normal matrices . . . . . . . 633
Vol. 5, No. 4 December, 1955
PACIFIC JOURNAL OF MATHEMATICS
EDITORS
H.L. ROYDEN
Stanford UniversityStanford, California
E. HEWITT
University of WashingtonSeattle 5, Washington
R . P . DILWORTH
California Institute of TechnologyPasadena 4, California
* Alfred Horn
University of CaliforniaLos Angeles 24, California
ASSOCIATE EDITORS
H. BUSEMANN
HERBERT FEDERER
MARSHALL HALL
P.R. HALMOS
HEINZ HOPF
ALFRED HORN
R.D. JAMES
BφRGE JESSEN
PAUL LEVY
GEORGE POLYA
L J . STOKER
KOSAKU YOSIDA
SPONSORSUNIVERSITY OF BRITISH COLUMBIA
CALIFORNIA INSTITUTE OF TECHNOLOGY
UNIVERSITY OF CALIFORNIA, BERKELEY
UNIVERSITY OF CALIFORNIA, DAVIS
UNIVERSITY OF CALIFORNIA, LOS ANGELES
UNIVERSITY OF CALIFORNIA, SANTA BARBARA
MONTANA STATE UNIVERSITY
UNIVERSITY OF NEVADA
OREGON STATE COLLEGE
UNIVERSITY OF OREGON
UNIVERSITY OF SOUTHERN CALIFORNIASTANFORD UNIVERSITYUNIVERSITY OF UTAHWASHINGTON STATE COLLEGEUNIVERSITY OF WASHINGTON
• * *
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During the absence of E.G. Straus.
UNIVERSITY OF CALIFORNIA PRESS BERKELEY AND LOS ANGELES
COPYRIGHT 1955 BY PACIFIC JOURNAL OF MATHEMATICS
NOTE ON THE "EVALUATION OF AN INTEGRAL OCCURRINGIN SERVOMACHANISM THEORY"
R. H. B A T T I N
In a recent paper [1] W. A. Mersman considers the evaluation of the integral
>=— Γ -2πi J-ooi hix
dx
where gix) and hix) are polynomials in x of order 2n — 2 and rc, respectively.
Because of the importance of Mersman's result the present writer wishes to call
attention to an alternate and somewhat more direct evaluation of this integral.
We shall utilize Mersman's notation in the main and begin with his equation
(3). By division it is clear that
where it is important to observe that each of the quantities Bj^ will, in general,
depend upon k except the first which is simply B^^ - α o Then
X-Xk -x ~xk
T T.s=0 7=1
n 2r
Σ, Σ2(n-r)
In the above expression it is understood that as = 0 for s < 0 or s > n and
Bjfr ~ 0 for j > n. Mersman's equation (3) then becomes
Mn-r)
r = i
n 2r
/c=i
Received September 20, 1953.
Pacific J. Math. 5 (1955), 481-482
481
482 R. H. BATTIN
For simplicity we define
Ak Bjk J = If 2, , n
so that Fι -I. There results the following set of n linear algebraic equations:
2Γ σ
/=2 2 α °
Using Cramer's rule we may now solve directly for / to obtain Mersman's result
as expressed by his equation (6) .
R E F E R E N C E
1. W.A. Mersman, Evaluation of an integral occurring in servomechanism theory,Pacific J. Math. 2 (1952), 627-632.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
AN EXTENSION OF WEYL'S ASYMPTOTIC LAW FOR EIGENVALUES
F. H. B R O W N E L L
1. Introduction. Let D be a bounded, open, connected subset of the plane
E2 whose boundary B = D — D is a simple closed curve whose curvature exists
everywhere and is continuous with respect to arc length; consider the eigen-
values λ - λn > 0 of the problem
(1.1) V 2
α + λ u = 0 on D, u = 0 on B,
where u(x) is to be continuous over ϋ and have continuous second partials
over D, ^ being the Laplacian. It has long been known (see [7, bibli-
ography]) that in this situation, with 0 < λ^ < λ^+i repeated according to
multiplicity, the asymptotic distribution of λn is given by Weyl's law
μ2(D)( 1 . 2 ) / V U ) = T 1 = — t + o ( t ) , t—» + oo,
where μ 2 ( D ) is the two dimensional Lebesgue measure of D. This can be ob-
tained by Tauberian theorems from the estimate
~ 1 μ a ( D ) lnω C
d 3 ) Σ TTT ,=~Λ + - + O ( ω - 5 / 4 ) , ω - ^ + co,n=ι An(λn + ω) 4 77 ω ω
(see Carleman [ 2 ] for the E3 analogue). By domain comparison methods [3,
p. 386 ] Courant has shown that o(t) in (1.2) can be replaced by O(\Jt l n ί ) .
In a recent paper [6, p. 177, equation 16] Pleijel replaces the estimate (1.3)
by the very much stronger
(14) y l n ω , c nB) 1 ! o l 1 )
over ω > 1 in case the curve B is very smooth (that is, it has an infinitely
Received October 19, 1953.Pacific h Math. 5 (1955), 483-499
483
484 F.H. BROWNELL
dίfferentiable parametric representation), where C is an unknown real con-
stant and KB) is the total length of S. Pleijel's estimate (1.4) follows easily
from a deep investigation jointly with M. T. Ganelius on the compensating part
of the Green function, as yet unpublished. This investigation uses integral
equations over the boundary B, while estimates like (1.3) come from a simple
application of the maximum principle over D to the modified Green function.
Pleijel suggests it should be possible to sharpen (1.2) by using his methods to
investigate the analogue of (1.4) over complex ω.
It is the purpose of this paper to show that from (1.4) alone we can replace
(1.2) by
(1.5) ΛK)4 77
in a certain sense. Precisely our result (2.13) is that with
KB)1 4ιr 4π J
have
over all real u >_ e and all p > 0 for some M < + oo. Moreover, if Nit) has an
ordinary asymptotic series in powers of ί, it must be consistent with (1.5). We
discuss briefly the possibility of sharpening (1.5) by replacing averaged 0
estimates by ordinary ones. We also note the utility of our consistency result
in proving false a conjecture of Minakshisundaram [5, p. 331, no. 2] about the
asymptotic behavior of Nit).
Clearly our theorems will apply to give results like (1.5) for a wide variety
of more general problems than (1.1) for which estimates like (1.4) obtain; in
particular such results hold for (1.1) in 3-space £3 .
2. Results and proofs. The difficulty arising in trying to get an asymptotic
series like (1.5), with 0 replaced by an ordinary 0 or o, is that Tauberian
theorems yielding such results seem to require essential nonnegative condi-
tions after subtracting all but the last term of the series. It is quite clear that
Nit) does not satisfy such a condition. For this reason we use an indirect
AN EXTENSION OF WEYL'S ASYMPTOTIC LAW FOR EIGENVALUES 4 8 5
Abelian type argument [4, p, 224] to get averaged error estimates of the 0 type.
The two first theorems here establish the significance of these averaged
error estimates, which despite the resemblance to Gaussian summability seem
to be little used for asymptotic series. Crame'r [ 1 , p. 819 and p. 823, ( 3 ) ] has
used Caesaro-1 type averaged error estimates on lattice point problems, but
such processes do not appear strong enough for use here.
Throughout the paper all integrals are to be understood in the Lebesgue or
Lebesgue-Stieltjes sense, and for the following two theorems it is understood
that Fit) is to be real valued of bounded variation over every finite interval
of [0, oo), with positive b a continuity point of Fit). Also \dF it) | stands for
dVpit) where Vpit) is the total variation of F over [bf t].
T H E O R E M 1. //
[°° i r ° \ d F i t ) \ < + o oJ b
for some r0 > 0, if
φ(s)= f° fs dF(t),
which must exist and be analytic in s over H[s] > ΓQ, also has an analytic
continuation without singularities throughout R [s ] > 0, and if
over 0 < r <^ro and all real v for some Mi < + oo and h > 0, then over all real
u >_ e and p > 0 we have
(2.1) I Γ β-<pV»> (In <•/«»» dF{t) < l2Mt exp ( l + Plh* + —\Jb \ \ 2 2p
In v i e w of ( 2 . 1 ) i t b e c o m e s c o n v e n i e n t t o de f ine Fit) = 0 if it)) over
t >^ b for s o m e n o n n e g a t i v e f it) d e f i n e d over t >^ k > 0 if for e a c h p > 0 t h e r e
e x i s t s s o m e Mp < + oo s u c h t h a t the left s i d e of ( 2 . 1 ) e x i s t s and i s <_Up f iu)
for a l l u >_k. With t h i s d e f i n i t i o n we c a n r e s t a t e t h e c o n c l u s i o n of T h e o r e m 1 a s
Fit) =
486 F. H. BROWNELL
over t >_ b with k = e. Note that in (2.1) Up —> + 00 as either p —» 0 + or
p—»+oo, so that (2.1) becomes meaningless then. The significance of the
result (2.1) is greatly increased by the following consistency theorem.
THEOREM 2. //
f t'r°\dF(t)\ < + o ob
for some r0 > 0, if
over t >_ b, and if
as t —> + ex) for some rλ > 0, then C\ — 0.
Proof of Theorem 1. Let
4 ( y ) = exp \~ — y2 ~zy]
for p > 0 and any complex z; thus
vNow
M> [Tt'r° \dF(t)\ >r'Γ°VF(r)
Jb
shows
VF(t)=O(tr°);
thus
Jy=ln b z
AN EXTENSION OF WEYL'S ASYMPTOTIC LAW FOR EIGENVALUES 4 8 7
exists as an entire function of z over all real ω and all complex z The Fubini
theorem also shows g(z9 ω) G Lt(-00,00) over ω with
(2.2) f°° ( ( \= -exV( (v + - \ )ψ(z+iυ)P \ 2p2 \ i l l2πJ'°° P \ 2p
over H[z+iv]-H[z]>_ro9 v being real. But the right side of (2 .2) is in
L]i— oo, oo) over v since
fr \dF(t)\,b
and thus the Fourier transform inverse yields
( 2 . 3 ) [ ° ° f ( ω ~J=\ b[ f (y=\n b
1 Γ~ I (v + z / ί ) 2 \ , , ,„,, dv
= exp \φ (2 +iv)eιvω —
v^J-~ \ 2p2 j P2P
for R [ z ] >, r 0 . T h e g i v e n e s t i m a t e on φ{s) a c t u a l l y m a k e s t h e far r i g h t s i d e
of ( 2 . 3 ) e x i s t a n d be a n a l y t i c in z t h r o u g h o u t R [ z ] > 0, and t h u s by a n a l y t i c
c o n t i n u a t i o n ( 2 . 3 ) h o l d s t h e r e a l s o . T h u s wi th z = r w e h a v e for e v e r y p o s i t i v e
r and p a n d for e v e r y r e a l ω the e s t i m a t e
I Γ°° / P2
(2.4) exp - — ( ω - -
I Jr=ln6 \ 2
2p2
Multiplying (2.4) by e r α )and letting r = 1/ω > 0 we note for ω > 1 that
488 F. H. BROWNELL
r2 1 1rω H — In r = 1 + + In ω < 1 + + In ω,
2p2 2p2ω2 ~ 2p2
thus with y = In t and ω — \nu >^ 1 we get the estimate (2.1) as desired.
Proof of Theorem 2. As before we have
\ F ( t ) \ < \ F ( b ) \ + VF(t) = O ( t r ° ) ,
so that we can integrate by parts in the left side of (2.1) and obtain from
F ( t) = 0 (In t) over t > b the estimate
(2.5) ply'lnb
over ω >_ k > 0. Now we are given
C l ί Γ l + / ( ί ) ί Γ l
over ί > b with l i m ^ + o o / ( ί ) = 0 Thus multiplying (2.5) by e ι , letting
y = ω — Λ;, and taking ω —> + oo we get
f Γω-lnfe / p2 \0 = lim j cι I Λ expj-ΓiA; — x \
α>—+ oo I J-°° \ 2 /
2 \\dx
Defining f ( t) = 0 for t < b we obtain
(2.6) 0 = C l /"°° * e x p ί - Γ i * x2\dxJ \ 2 /
lim J Γ " /•(βω"a:) ίKe+ Ii
fit) being bounded over all real t since lim^_>+Oo / ( ί ) = 0 , and thus also
limω_» + o o / ( eω"x) = 0 , dominated convergence applied to (2.6) yields
AN EXTENSION OF WEYL'S ASYMPTOTIC LAW FOR EIGENVALUES 4 8 9
= Ci I x expl ~rγ% x )dx.J -βo \ 2 /
But
< 0
for rι > 0, so that c^ = 0 follows.
To apply these two theorems we use a standard contour integral transforma-
tion on Pleijel's estimate (1.4). The contour Cp, p >_ 0, in the z plane is de-
fined to be first along the negative real axis from — oc to — p, then around the
circle z = peι from θ = - π to θ = 77, then back along the axis to -en. On this
contour we define
with θ ~ — π9 — π < θ < π, θ = π on the three parts respectively. The well known
results are formulated in the following two lemmas (Carleman [2]) , and we
sketch the proofs for the sake of completeness.
LEMMA 3. IfO< λn < λ n + l f an real, if
n
and if
λ 2
then
converges absolutely and is analytic in all complex z except for simple poles
at each λn. Moreover, for 0 < p < λι the function
490 F. H. BROWNELL
l r dz_ / h(z)2τ7i Jcp (z)Sml
exists and is analytic in s over ft [s ] > 2,
2 * an An
converges absolutely and uniformly over ft[s] ^ 2, and over R [ s ] > 2 we ob-
tain
(2.7) r - / h(z)(z) S - l
LEMMA 4. //ίAe assumptions of Lemma 3 are satisfied and if
(2.8) Λ(~ω)= ς w p
ω V . ' P
l n ω + Q ( ί )over ω > 1 with 0 < r^ < rjc.ι < < rγ < 2, then
an analytic extension into ft [s ] > 0 e#cepί /or poZes at rp,
(2.9)in ίr(rD — 1) 1s i n
s i n+ ton
(s-rp)2
cos 7r(r p -l)
analytic in s throughout ft [s ] > 0, aτιcf
Λ )\ 77^
over 0 < r < 2 ancί all real v for some M2 < + ω .
AN EXTENSION OF WEYL'S ASYMPTOTIC LAW FOR EIGENVALUES 4 9 1
We remark that TΊ < 2 is no real restriction in (2.8), since the assumptions
of Lemma 3 imply \Ίmω _ + oo h ( - ω ) = 0 . In demonstrating Lemma 3, the stated
analyticity of h (z) is clear as well as
; l/ \ n = 1
SO
g ( s ) = _ L / h(z) J t2 π ί CP (z)
exists and is analytic in s over R [s ] > 2. To show
there for (2.7), let Cm be the vertical line contour from xm — ΪOO to xm + ico for
%m with λnml < xm < λnf so that using the estimate on h ( z ) to shift from
Cp to Cm we obtain
(2.10) g(s)~ T a, λ : s = — f h{z) —
for R [ s ] > 2, h(z) having the residue
at λ.
To pass from (2 .10) to ( 2 . 7 ) , note that
t lim sup λ2
n ( λn - λnm i )f] = + <
since otherwise
492 F. H. BROWNELL
would be bounded by
which contradicts λ π —» + oo and therefore contradicts
n ι n
Thus there exists a sequence nm such that
nm < Λm +19 λn — λn_ i > 0, and λ^ ( λn - λn. t ) —> + oo as m —» + oo for
We choose
for n = nm, so that
K 1_ ^ 1 + > i
and
< Λm + — = *m + 2 * J
n
With z = xm + it and s = r + ίvf r > 2, clearly
and
I 1 I e x p U | t ; | / 2 )
with L (v) = M exp ( π \ v |/2) make
AN EXTENSION OF WEYL'S ASYMPTOTIC LAW FOR EIGENVALUES 4 9 3
k(z).s-1
\ L(v)over \t\ < λ n - xm ,
\h(z) < (χm)2'r L(v)o v e r λ n - x m <\t\ <xm9
and
\h(z)s-i
K \ L(v)I K \ L(v)over χm < \t I.
Thus integrating over these respective parts of Cm, and using
the right side of (2 .10) —> 0 as m —> + oo and ( 2 . 7 ) follows.
P a s s i n g to Lemma 4, from the estimate ( 2 . 8 ) it is clear that
dzh(z)
e x t e n d s a n a l y t i c a l l y f r o m R [ s ] > 2 t o R [ s ] > r i A l s o f o r rι < R [ s ] < 2 ,
C ^ c a n b e s h i f t e d t o C o y i e l d i n g
(2.11) g U ) =
Now here
sin 77 ( s - 1)
77
h(z)dz sin π(s - 1) /*oo dcύ
/ h(-ω) .Jo „ s-i
ω
ω s i n τ 7 ( s - l )=
s-ι _ τ r ( s - 2 )j A ( - l ) + / h (-ω) ,
which is analytic in s over R [ s ] < 3, having a removable singularity at s - 2.
Also
in 77 ( s — 1 )s i n Γ-2in 77 ( s - 1 )sin
ωs -1 77 ( s - r )
and
s i n 77(s — 1 )
/
oo c?ω s i n 77 ( s •ω In ω =
ωs-ι π { s - r
c?ω s i n π(s - 1 )
494 F. H. BROWNELL
with principal parts
sin 77 (r - 1)
π(s - r )
s i n 7 7 - ( r - l ) c o s τ r ( r - l )and +
π(s-r)2 s-r
respectively at s =r. Thus (2.9) clearly follows from (2.8) and (2.11). Also
from
i n 7 7 ( 5 - 1 ) 1 < 2eπ\v\ a n ds in/
oo 1 day 1
2 r-1 Γω ω
the stated estimate for g^is ) follows.
We combine Lemma 4 with our two previous theorems to obtain the following
result.
THEOREM 5. If the assumptions of Lemma 4 are satisfied with
lp sin (πrp) = 0
in (2.8), then
- Σ an
satisfies
( 2 . 1 2 ) H(t) = \ •' — lp c o s π rp) J + O ( l n ί ) ,
over t >_b where 0 < b < λ 1 # Furthermore, if H(t) has an ordinary asymptotic
series in powers of t as t—» + oo, such a series must coincide term for term
as far as it goes with the terms o / ( 2 . 1 2 ) .
Proof. Let
F{t)=H(t)~ -= l ' P
p lp COS TΓΓp) I ,77 / I
and note that
AN EXTENSION OF WEYL'S ASYMPTOTIC LAW FOR EIGENVALUES 4 9 5
ir-s
t'-'-i dt = —s ~r
for R [s ] > r and b > 0. Also with 0 < b < λu we have
Γ7 1 = 1
for K [s ] >_2. Thus from Lemma 4 we see that
t's dF(t)b
has an analytic continuation without singularities into R [ s ] > 0 by the can-
cellation of principal parts at each rp = s. Also the conditions of Theorem 1
are satisfied with r0 = 2 and h = π; thus (2.1) yields (2.12). Theorem 2 gives
the consistency statement obviously.
To apply this theorem to our problem (1.1), we remark that the desired
condition
follows from Green's function being in L2(D x D), and thus a Hilbert-Schmidt
kernel. Thus Pleijel's estimate (1.4) yields (2.12) with
2
k = 2, r! = 1, mi = C, lχ = , sin ( πr±) - 0, cos ( nr±) = - 1,4 π
1 Z ( B ) . .ro — — 9 rrio — , lo — 0> s i n v ^ Γ 2 ) = 1 >
2 8
and we can state the following.
COROLLARY 6. Let the open, bounded, connected set D in the plane E2
have its boundary B an infinitely differentiable Jordan curve so that Pleijel's
estimate (1.4) holds for the problem (1.1). Then over t >_ λj2 we have
2
(2.13) /VU)= Y, 1 = tλn<t *π
496 F. H. BROWNELL
and as in Theorem 5 any ordinary asymptotic series for N ( t ) must be consistent
joith (2.13).
If we consider the real valued eigenfunction un(x) of problem (1.1), in
place of (1.4) Pleijel gets [6, equation 6 and second equation of p. 177] over
X G D and ω > 1
- \un(x)\2 1 lnω C(x) 1(2.14) V — — = + ' +
Tί λπ(λπ + ω) 4π ω ω 2π
/ - 2 Λ r ( x ) V u Γ \
•°h?H/4 > 0, r ( x ) the distance from x G D to β, the 0 symbol being uniform over
x G D as well as ω > 1, Now K 0 ( Γ )> t n e modified Bessel function of the second
kind and zero order, has
as r — » +00 [ 8 , p . 374] . Thus for each fixed x G D , with r ( x ) > 0, we have
over ω >^ 1
, r N M ) | 2 1 lnω C(x)(2.15) ^ ΓΠ "ϊ β 1 + + K
^ ί λ n (λ π + ω) 4τr ω ω \ω
1\
ω
2}
where the symbol 0χ now depends on x G D. It is also easy to see that at each
x ^ y with x, y G D we have over ω >_ 1
~ ^(xWy) C(x,y)
(2.16) Σ T T T x- — — +0 x ,/ 1 \
, y _ ,\ω2/
and indeed much better estimates than O(l/ω 2 ) hold in (2.15) and (2.16).
Also
+ c o
π=ι λ2
n
is known at each x G D; thus Theorem 5 yields the following.
AN EXTENSION OF WEYL'S ASYMPTOTIC LAW FOR EIGENVALUES 497
COROLLARY 7. Let D be as in Corollary 6, so that (2.15) and (2.16) hold
at each x ^ y with x, y £ Z). Then over t >_ λχ/2
( 2 . 1 7 ) £ K ( x ) | 2 = — ί + O ( l n ί ) , Σ, unix)uniy)=Oilnt),4
λn<t -*π λn<t
with consistency of these series with ordinary asymptotic series, if any, as in
Theorem 5.
3. Discussion of results. It is quite clear that O ( l n ί ) in (2.17) can be
replaced by much stronger estimates in the 0 sense, say O ( l / ί ) , since much
more than Oil/ω2) holds in (2.15) and (2.16). In (2.13) additional terms
enter if a stronger 0 type error estimate is required. These are due to additional
terms entering PleijeΓs equation (1.4), one of them involving the mean square
curvature of B, if 0 ( 1/ω2) is replaced by a stronger estimate.
A much more difficult and interesting question is the extent to which the
averaged 0 estimates in our results may be replaced by ordinary 0 estimates
for the problem (1.1). It is clear that by improving the Oieπ\v\) estimate on
the analytic continuation of
oo
Λ^ , s = r + ιv,
we can replace the Gauss kernel
I p2( A
in our definition of 0 by less well behaved ones. We could get ordinary 0 esti-
mates if we could use the characteristic function kernel X[_t ^ ( ω - y ) , but
since its Fourier transform is essentially v~l sin v, the analogue of the proof of
Theorem 1 would then seem to require stronger conditions on
£-^ n
than can be expected to hold.
It is known from the refined results of geometric number theory [1, p. 823]
that Mix), defined as the number of integer lattice points im,n) in the plane
satisfying m2 + n2 <, x, satisfies
498 F. H. BROWNELL
πx + O(xι/3).
Since
λ = U 2 + m 2 ) — , n > 0, m > 0
for the eigenvalues of (1.1) with D a square of side 6, the eigenfunctions being
products of sine functions, we clearly see that
/ V ( t ) = - IM — -b2 4b r- , 1 / 3 χ
= t y/T + O ( ί ι / 3 )4π 4 7r
for square D, 4[byt/π] + l being the number of lattice points on the axes.
This asymptotic result for Nit) agrees with (2.13), although the corners of a
square prevent it from satisfying the smooth boundary conditions required in
Corollary 6. By carelessly dropping the y ί term in going from Mix) to Nit)9
Minakshisundaram [5, p. 331, no. 2] is led to the conjecture that domain com-
parison methods [ 3 , p. 386] should yield
t + Oitί/3)4/7
for general domains D. Clearly the consistency statement of Corollary 6 makes
such asymptotic behavior impossible for Nit).
REFERENCES
1. H. Bohr and H. Cramer, Die neuere Entwicklung der analytischen Zahlentheorie,Encykl. der Math. Wiss., 2, part 3, no. 8.
2. T. Carleman, Proprietes asymptotics des functions fondamentales des mem-branes vibrantes, Forhand. 8th Skand. Mat. Kongress, (1934), 34-44.
3. R. Courant and D. Hubert, Methoden der mathematischen Physik, vol. 1, Springer,Berlin, 1931.
4. G. Doetsch, Laplace trans forms, Springer, Berlin.
5. S. Minakshisundaram, Lattice point and eigenvalue problems, Symposium onSpectral Theory, Stillwater, Okla., (1951), 325-332.
6. A. Pleijel, Sur les valeurs et les functions propres des membranes vibrantes,Comm. Sem. Math. Univ. Lund (Medd. Lunds Univ. Mat. Sem.), suppl. (Riesz) vol.,(1952), 173-179.
AN EXTENSION OF WEYL'S ASYMPTOTIC LAW FOR EIGENVALUES 499
7. H. Weyl, Ramifications of the eigenvalue problem, Bull. Amer. Math. Soc, 56(1950), 115-139.
8. E.T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge, 1946.
UNIVERSITY OF WASHINGTON AND
INSTITUTE FOR ADVANCED STUDY
ON THE HOMOMORPHISMS OF AN ALGEBRA
ONTO FROBENIUS ALGEBRAS
W. E. D E S K Ϊ N S
1. Introduction. A linear associative algebra possessing a nonsingular
parastrophic matrix is known as a Frobenius algebra after the mathematician
who first investigated the properties of such an algebra [3] . In more recent
years the properties of this class of algebras have been studied in papers by
a number of mathematicians, notably R. Brauer, C. C. MacDuffee, T. Nakayama,
and C. Nesbitt (see References).
Since Frobenius algebras are defined in terms of the parastrophic matrices,
a natural question to ask is the following: Does a parastrophic matrix of rank
m of an algebra U of order n determine in some manner a homomorphism of Cl
onto a Frobenius algebra of order m? As the answer to this query is, in general,
negative, it is the purpose of this paper to investigate the question: When does
a parastrophic matrix of rank m determine in some manner a homomorphism of Cl
onto a Frobenius algebra of order m? First a "manner of determination" is
selected. Since the parastrophic matrices of G form a double CUmodule, various
ideals of & of annihilating elements correspond to each parastrophic matrix.
These are studied and conditions are developed (Theorem 9) which insure the
determination from these annihilators an ideal B such that the difference algebra
d - B is a Frobenius algebra of order m. These requirements are shown to be
necessary, also, in the sense that any homomorphism of Cl onto a Frobenius
algebra of order m implies the existence of a parastrophic matrix Q of rank m
which satisfies these conditions. Furthermore, the kernel of the homomorphism
will be the ideal B determined from among those elements which annihilate Q
as an element of a double CUmodule.
Basic terminology is introduced in v 2, parastrophic modules are defined,
and the order of such a module is discussed. In § 3 one-sided ideals determined
by the parastrophic matrices are considered, while § 4 is devoted to a study of
two-sided ideals determined by certain parastrophic matrices and of the homo-
morphisms of an algebra onto Frobenius algebras. Certain of the ideals
Received February 10, 1954. Presented to the American Mathematical Society May1, 1954.
Pacific J. Math. 5 (1955), 501-511501
502 W. E. DESKINS
introduced in § 4 have radical-like properties, and these ideals are considered
in § 5. A supplementary re 'c on the order of the radical of a Frobenius algebra
is given in §6.
The author wishes to express his gratitude to Professor C. C. MacDuffee for
his counsel during the preparation of this paper.
2. Preliminary remarks. Let U be a linear associative algebra of order n
over the field 3, and let βi, « f e n be an 3-basis for G. Multiplication in &
follows from the multiplication of the basis elements,
ei ej = Σ cijk ek i» ί = !> > n,k
where the Cijk are elements of 3, the constants of multiplication.
The associativity condition, written in terms of these constants of multi-
plication, is equivalent to each of the following sets of n2 matric equations:
(1) Q/Ri-Σ, cikjQk,k
(2) SiQj = Σ, ckijQk i f / * 1 . ' • • » Λ ,
k
where the matrices R(9 S(9 and Qι are defined as (c ; S Γ ) , ( c Γ ι s ) and (c S Γ ^), re-
spectively, where r denotes the row and s the column index.
Let a G G; then
a = a ί e i + ••• + a n e n ,
where the a( are field elements. Let
S(a) =atSi + + α n S π ,
+ ••• + anQn.
R{a) is called the first matrix, S(a) the second matrix, and Q(a) the para-
strophic matrix, of α. (Note that Q(a) as defined here is the transpose of the
parastrophic matrix as defined by other authors). The set R(®) of all the first
(second) matrices of 0* form an algebra which is a homomorphic image of U. The
HOMOMORPHISMS OF AN ALGEBRA 5 0 3
set o2 of all the parastrophic matrices of CL does not in general have this proper-
ty, but if the following definitions are made,
then 12 is a double Ci-module, the parastrophic module of G.
If a change of basis is made for U, the elements of R and & undergo similar-
ity transformations, while the elements of 2 undergo congruency transformations
[ 8 ] . Hence rank and symmetry are invariant set properties of 2 .
MacDuffee has obtained [7] necessary and sufficient conditions that R and
& be algebras isomorphic with U. A corresponding result for 2 is given by
THEOREM 1. 2 is of order m (as an Cί-module) if and only if the following
conditions are satisfied:
( i ) U contains an ideal w of order m such that the difference algebra
(1 — w is a zero algebra.
( i i) & contains no ideal of lower order with this property.
The proof of this theorem is a standard reversible procedure involving a
change of basis for U. Let there be n — m linearly independent linear relations
among the Q^; then there exist n - m linearly independent row vectors
Ti = (ti i, , tin ) i = m + 1, , n ,
such that
23 HkQk =° i> m.k
If B is a nonsingular n by n matrix with the Γj as its last n - m rows, and if
u i, , un form a new basis for Q,
Σ Pkiekk
where P = (pΓ ) = S"1, then the Qι are transformed into
i > m.
504 W. E. DESKINS
Thus if the new constants of multiplication are cf.^ , then
c.' k = 0 i% j = 1, . . , n k > m,
and uχ9 9um span an ideal ID of order m such that U — IΛ is a zero algebra.
The process is clearly reversible.
COROLLARY. // & has either a left or right identity element, then 2 is of
order n.
u is said to be a Frobenius algebra if °2 contains a nonsingular element.
THEOREM 2. U is a Frobenius algebra if and only if 2 is a cyclic module
of order n.
If 2 c o n t a i n s a n o n s i n g u l a r e l e m e n t Q, t h e n ( 1 ) and ( 2 ) imply t h a t
and 2 is of order ra since a Frobenius algebra possesses an identity element.
Conversely, if 2 is generated by an element Q and is of order n, then (1),
(2), and Theorem 1 imply that Q is nonsingular.
3. Ideals of CL Let 6 be a right ideal of Cl of order n - m. If a basis is
selected for Cl such that the last n - m elements of the basis span 6, then the
m matrices Q\, ,Qm have all zeros in their last n - m columns. The task of
determining a right ideal by a process involving reduction of certain elements of
2 through changes of bases of d seems formidable, if possible. However, a
somewhat similar process is given by the following theorem.
THEOREM 3. A parastrophic matrix Q of rank m determines a right ideal
6 of order greater than or equal to n - m.
Let B be the set of all elements b G Cl such that Q * b = 0. Clearly 6 is a
right ideal. That its order is at least n - m will follow from the next theorem.
That B may actually be of order greater than n - m is proved by the follow-
ing example. Let G have basis elements e\ and β2, ^\-^\^2 = e 2 e l = 0>
e£ = βi Then the corresponding (^ has rank 1 but B = CL
A more desirable result is contained in the following.
THEOREM 4. A parastrophic matrix of rank m determines a right ideal of
order n - m.
HOMOMORPHISMS OF AN ALGEBRA 5 0 5
Let Q be a parastrophic matrix of rank m. Then there exists a nonsingular n
by n matrix P such that the elements of the last n — m columns of QP are all
zeros. Let P effect a change of basis for A; that is, if P = (prs \ l e t
ui =Pkiek
be a new basis for U. If Q — Q(a), then Q'(a)9 with respect to the new basis,
is P QP and hence has nothing but zeros in its last n - m columns.
Now assume Q is of this form. Then
= Σ aiQi =ί Σ, aicsri\i * i '
so that
(3) Σ, ai cjki = ° 7 > m* k = 1, *, n.i
From (1) and (3) it follows that
(4) QRi= Σ
Hence Q * e{ ~ 0 for i > m, and B = (e m + j_, , en) is the right ideal determined
hyQ.
A right ideal of fl which may be determined in this way will be called a
parastrophic right ideal.
THEOREM 5. A sufficient condition that the ideal of Theorem 3 be a para-
strophic right ideal is that 12 be of order n.
Suppose Id = ( e m + 1 , , en) is determined from Q as above, and considereit i <. 77i. If @ * βj =s 0, then if =2 is of order nf (4) implies
which is impossible since Q is assumed to be of rank m.
Let Q = Σ,aιQι be in the reduced form described above.
506 W. E. DESKINS
THEOREM 6. If U has a right identity element, then aι = 0 for i > m.
Since U has a right identity element there are field elements f. such that
Then ( 3 ) implies
j > m.
The results of this section are obviously valid if the word "r ight" is re-
placed by "left".
Since the existence of ideals in an algebra & has been shown to be equiva-
lent to the existence of singular elements in 2, the following theorem is immedi-
ate.
THEOREM 7. Q, is a division algebra if and only if 12 contains no singular
elements.
4. Homomorphisms of G. The following result is an immediate consequence
of Theorem 4 and its analogue for left ideals.
THEOREM 8. If Q is congruent to a matrix of the form
iτ °l\ o o | ,
where T is a nonsingular m by m matrix, then the right paras trophic ideal B is
also a left paras trophic ideal. Conversely, if B is a right paras trophic ideal
determined by Q9 then if B is also a two-sided ideal, Q satisfies the above
condition.
Such an ideal will be called a parastrophic ideal, and Q will be said to have
P-rank m. While P-rank is not defined for every matrix, it is a property of every
symmetric matrix. Thus, if the characteristic of 9 is greater than n, the radical
of β is a parastrophic ideal. (It will be apparent shortly that this is true re-
gardless of the field characteristic since a semisimple algebra is a Frobenius
algebra.)
It does not follow that a matrix of 2 of P-rank m determines a homomorphism
HOMOMORPHISMS OF AN ALGEBRA 507
of G onto a Frobenius algebra of order m, for any commutative nilpotent non-
zero algebra contains proper parastrophic ideals. The following indicates a
necessary criterion.
LEMMA 1. If π is a homomorphism of U onto C, an algebra with an identity
element 1, then U contains an idempotent element e such that πe = 1. Further-
more, the set of left annihilators of e is contained in the kernel of the homo-
morphism.
This follows simply from the structure theory for algebras.
Now suppose the last n — m basis elements of U form a parastrophic ideal
13, and suppose that d has an idempotent element u such that du u 6 = 0,.
Then 13 will be called a regular parastrophic ideal,
THEOREM 9. A homomorphism of d onto a Frobenius algebra of order m has
as its kernel a regular parastrophic ideal of order n — m, and conversely if ID
is a regular parastrophic ideal of Cl of order n — m, then U — ID is a Frobenius
algebra of order m.
Suppose 0/ is a Frobenius image of U, with basis βι, , e m and kernel
θ spanned by em + ι, . . , en.
Then U possesses a nonsingular m by m parastrophic matrix Q$
and
is an element of 12 with Pπrank m. Hence 13 is a parastrophic ideal. By Lemma
1, IS is a regular parastrophic ideal.
The converse follows from the regularity of 6 and Theorem 6.
Thus, if Q is a parastrophic matrix of rank m9 if Q can be reduced to a
corner matrix by a change of basis of U, and if Q is associated with a linear
combination of the first m of the new basis elements, then Q determines a
homomorphism of U onto a Frobenius algebra. Furthermore, each homomorphism
of 0/ onto a Frobenius algebra may be determined in this fashion.
508 W. E. DESKINS
5. Radical-like ideals. A function f of & into the s e t of all ideals of & i s
called a radical function of Cl if the contraction of f to the difference algebra
C = U —/"(Q/) maps C onto the zero ideal. The ideal f(Q) is called a radical-
like ideal of CL
Let l be the set of all regular parastrophic ideals of U and let ΓL be an
element of Γ of minimal order, with the agreement that Γl is the zero ideal if
U is a Frobenius algebra and Q, if & is nilpotent. Then define / ( &) = ϊl.
T H E O R E M 10. f(&) = U is a nilpotent ideal of CL
If Cί is nilpotent the theorem is trivially true, so assume that & has the
radical K ^ Q,. Suppose K is of order r and that U is of order m. Let
Π n K = C .
Case 1. C = (0) . Let fl have a basis such that the first m basis elements
span \l while the last r span JC. By the definition of H there is an element
Q' of 12 of rank rc — m with its first m rows and columns composed of only zeros.
Now ϊl is isomorphic to a semisimple subalgebra of U — Kt so there is an element
Q" of 12 of rank m with only zeros in its last n — m rows and columns. Then
Q ' + Q " is nonsingular.
Case 2. C ^ ( 0 ) . Then 0/- C is a Frobenius algebra by the above work.
In either case
so that ϊi is contained in & and so is nilpotent.
One important property which Γl may lack is uniqueness. The question of
whether Yί is unique up to an & -isomorphism will now be considered and parti-
ally answered.
The following result indicates a significance of the U-isomorphism of two
minimal elements of ί .
THEOREM 11. Let ϊl and lΐi be minimal elements of r. Then a necessary
condition that U and tfL be &~isomorphic is that ϊi - C and lU - C be zero alge-
bras, where C = ft n ϊtl.
It may be assumed that C = (0) . Then let σ be an G-isomorphism from Ifl onto
ΐίl. If a and b are elements of ϊfl, then ϊi contains an element b ' such that
HOMOMORPHISMS OF AN ALGEBRA 509
ab = a(σb') = σ(a δ ' ) = 0 .
The isomorphism of the minimal elements of ί for certain algebras will
stem from the following lemma.
LEMMA 2. If V and V are n by n matrices of rank m with elements from a
field o which contains at least w + 1 nohzero elements, then o contains an
element t ^ 0 such that U + tV is of rank at least m.
It will be sufficient to prove the result for m = n. Let D and E be nonsingu-
lar n by n matrices such that
DVE = / .
Consider the equation
det(D(U-xV)E)=άet{DUE-xI)**09
which is of degree n in the indeterminate %% Since o contains at least n + 1
nonzero elements, one of them does not satisfy this equation.
Let U and V be n by n matrices, and let
mean that the two matrices do not both have nonzero elements in the same row-
column position.
THEOREM 12. // two minimal elements ϊlχ and Π/2 of I are determined by
symmetric matrices Qι and Q2 of 2 of rank m% if o contains at least m + 1
nonzero elements, and if
<?lΛ<?2=0,
then ft — H ι is isomorphic with u - U 2 .
The cyclic modules
are of order my and the representations of U over these double U-modules yield
Frobenius algebras of order m which are images of U, isomorphic with U - H t
and & - H2 respectively. Let t be a nonzero element of o such that Q{ + tQ2 is
of rank m (since higher rank would contradict the minimality of the order of fli
510 W.E. DESKINS
and Π/2) Since Q^ and Q2 are symmetric
is a cyclic module of order mf and since Qγ
κQ2 ~ 0> (1) a n ( ^ (2) imply that
the mapping
is an (i-isomorphism between (^ * G, and ( ( ^ + ί(?2 ) * Ct. Similarly (? * Cί and
(ζ) t + tQ2 ) * fl are Ci-isomorphic. Hence Qχ * Q, and @2 * Ci are CUisomorphic
which implies that U — Hi and U — \l2 are isomorphic.
6. A remark concerning Frobenius algebras. While Frobenius algebras are
generally regarded as algebras with radicals of sufficiently small order, the
following indicates that their radicals must also be~ of sufficiently large order.
THEOREM 13. Let (λ be a Frobenius algebra bound to its radical K. Then
if & — K is of order m9 K is of order at least m. If K is a zero algebraf then &
is of order m.
By the results of Nakayama [9] the set of all elements of U which annihilate
K from the right is an ideal <C which also annihilates & from the left and has
order n - k = m9 where k is the order of K. Since (X is bound [4] to ίί,
£ cK,
hence m <_k9 and m = k if <C = Jί.
The consideration of bound algebras is, of course, sufficient since an alge-
bra may be written as a direct sum of a semisimple algebra and a bound algebra.
(This result is due to M. Hall [4] ) .
REFERENCES
1. R. Brauer, On hyperkomplex arthmetic and a theorem of Speiser, A SpeiserFestschrift, Zurich, 1945.
2. R. Brauer, and C. Nesbitt, On the regular representations of algebras, Proc.Nat. Acad. Sci., 23 (1937), 236-240.
3. G. Frobenius, Theorie der hyperkomplexen Groessen, S.-B. Preuss. Akad. Wiss.,
Berlin (1903), 504-537.
4. M. Hall, The position of the radical in an algebra, Trans. Amer. Math. S o c ,48 (1940), 391-404.
HOMOMORPHISMS OF AN ALGEBRA 511
5. M. Ikeda, and T. Nakayama, Supplementary remarks on Frobeniusean algebras,Osaka Math. J., 2 (1950), 7- 12.
6. C.C, MacDuffee, Modules and ideals in a Frobenius algebra, Monatsh. Math.
Phys., 48 (1939), 292-313.
7. , On the independence of the first and second matrices of an algebra,
Bull. Amer. Math. Soc , 35 (1929), 344-349.
8. , The theory of Matrices, Chelsea, New York, 1946.
9. T. Nakayama, On Frobeniusean algebras, I, Ann. Math., 40 (1939), 611-633.
10. , On Frobeniusean algebras, II, Ann. Math., 42 (1941), 1-21.
11. , On Frobeniusean algebras, III, Jap. J. Math., 18 (1942),
12. , Orthogonality relations for Frobenius and quasi-Frobenius algebras,Proc. Amer. Math. Soc, 3 (1952), 183-195.
13. C. Nesbitt, On the regular representations of algebras, Ann. Math, 39 (1938),634-658.
14. M. Osima, Some studies on Frobenius algebras, Jap. J. Math., 21 (1951), 179-190.
15. G. Simura, On a certain ideal of the center of a Frobeniusean algebra, Sci. PapersColl. Gen. Ed. Univ. Tokyo, 2 (1952), 117-124.
THE OHIO STATE UNIVERSITY
THE MEASURE RING FOR A CUBE OF ARBITRARY DIMENSION
J.M.G. F E L L
1. Introduction. From Maharam's theorem [2] on the structure of measure
algebras it is very easy to obtain a unique characterization, in terms of cardinal
numbers, of an arbitrary measure ring. An example of a measure ring is the ring
of lίellinger types of (finite) measures on an additive class of sets. In this
note, the cardinal numbers are computed which characterize the ring of Hellinger
types of measures on the Baire subsets of a cube of arbitrary dimension.
2. Definitions. A lattice R is a Boolean σ-rίng if ( a ) the family Rx of all
subelements of any given element x form a Boolean algebra, and (b) any count-
able family of elements has a least upper bound. If, in addition, for each x in
R there exists some countably additive finite-valued real function on Rx which
is 0 only for the zero-element of R, then R is a measure ring. x A Boolean σ-
ring with a largest element is a Boolean σ-algebra. A measure ring with a
largest element is a measure algebra. The measure algebra of a finite measure
μ is the Boolean σ-algebra of μ measurable sets modulo μ null sets.
A subset S of a Boolean σ-algebra R is a σ-basis if the smallest σ-sub~
algebra of R containing S is R itself. R is homogeneous of order Cί if, for every
nonzero element x of R, the smallest cardinal number of a σ-basis of Rx is Cί,
We observe that R cannot be homogeneous of finite nonzero order. If it is homo*
geneous of order 0, it is the two-element Boolean algebra.
Let Cί be an infinite cardinal, / the unit interval [0, 1], and Ia the topological
product of / with itself Cί times (the Cί-dimensional cube). L will denote the
product Lebesgue measure on the Baire subsets of 7α, and M the measure
algebra of Z/ α . Then it is not hard to see that M is homogeneous of order
Cί. Maharam has in fact shown that it is, essentially, the only measure algebra
of order Cί.
1Our use of the terms 'measure ring', 'measure algebra', unlike Maharam's, refersonly to the algebraic structure. In this sense two measure rings are isomorphic if theyare isomorphic as σ-rings.
Received March 3, 1954.Pacific J. Math. 5 (1955), 513-517
513
514 J.M.G. FELL
3. Maharam's theorem and the characterization of measure rings.
THEOREM. (Maharam [2]) . Every measure algebra homogeneous of order
OC is isomorphic to M . Every measure algebra is a direct product of countably
many homogeneous measure algebras.
Consider now a measure ring R. A nonzero element x of R is homogeneous of
order Cί if the principal ideal Rχ is so. The cardinal function p of R will as-
sociate with each cardinal Cί the smallest cardinal number pa of a maximal
disjoint family of elements of R which are homogeneous of order α.
THEOREM, The measure ring R is determined to within isomorphism by its
cardinal function p.
Proof. If, for each r in an index set Kt Nr is a measure algebra, we may de-
fine a σ-ring
rβK
consisting of all functions φ on Kf such that ( a ) φ 6 NΓ for r G K9 (b) φ = 0
for all but a countable number of indices r. Union, intersection and difference
are defined in the obvious manner. N is actually a measure ring. Indeed, suppose
that φ E /V; and that the countable set of indices r for which φr ^ 0 is arranged
in a sequence r 1 ? r 2, . For each i, let m[ be a countably additive nonnegative
finite-valued function on NΓ. having value 0 only for the zero-element of Nr.
Then the function m, defined for subelements ψ of φ by
is countably additive, nonnegative, and finite-valued and has value 0 only for
Now for each cardinal α, let {%ag} (β < pa ) be a maximal disjoint family
of elements of R homogeneous of order Cί; let Raβ be the algebra of subelements
of %aβ By the definition of homogeneity, the %a n are all disjoint, and it is
almost trivial to show that
« s ΓUα/3a,β
THE MEASURE RING FOR A CUBE OF ARBITRARY DIMENSION 515
But, by Maharam's theorem, R o 2i M . Hence
R z ΠtΛί ( α )]
which depends only on the cardinal function p.
4. The measure ring of Hellinger types for the cube. Let F be an additive
class of sets. A finite measure μ on F is absolutely continuous with respect
to a finite measure v on F if all μ null sets are v null sets. If μ and v have the
same null sets, they are of the same Hellinger type. The relation of absolute
continuity furnishes an ordering of the Hellinger types under which, as is well
known, the latter form a measure ring. In this ring, the ideal of all Hellinger
types contained in the type of a measure μ is isomorphic with the measure
algebra of μ
We shall denote by R the measure ring of Hellinger types associated
with the additive class B^a' of Baire subsets of the cube 7α; and shall obtain
the cardinal function characterizing R .
/ it \
LEMMA 1. R^^ R ° for any finite nonzero a.
Proof. This follows from Kuratowski's result [1] that any two complete
metric spaces X and Y having the same cardinal number are connected by a
one-to-one Borel mapping of X onto Y whose inverse is also Borel.
In view of Lemma 1, we restrict ourselves to cardinals OC which are in-
finite.
LEMMA 2. The total number of measures on B a is equal to or less than
ca
9 where c is the power of the continuum.
Proof. Ia is the set of all functions φ on a set H, of cardinal number OC,
to [0,1] . Let P denote a basis for the topology of Ia consisting of all subsets
of la of the form
E(φ£la
9 φβejβ for all β in G)
Φ
where G is a finite subset of H, and, for each β, Jβ is an open interval in /
with rational endpoints. Since CC is infinite, it is easily seen that the cardinal
number of P is OC.
516 J.M. G. FELL
Now suppose μ and v are two measures on β ' α ' which coincide on P. I claim
they must then be equal. If this is so, the lemma is proved, since the number
of real functions on P is c α .
Since P is closed under finite intersection, it is easily seen that μ and v
must coincide on finite unions of sets in P. Now let A be a closed Baire set
in Ia; A must then be a countable intersection of open sets. But, between A and
any open set containing it, we may by the compactness of /α, place a finite
union of sets in P. Passing to the limit, we find that μ and v have the same
value for A, Now by the regularity of μ and v, their coincidence is assured for
all Baire sets.
LEMMA 3. For each cardinal γ which is either 0 or infinite and for which
γ <_ α, there are at least ca disjoint measures on Z? whose measure algebras
are homogeneous of order γ.
Proof. With each point φ in Ia associate the measure μ , of a unit point
mass at φ. Any two such points φ and φ can be separated by Baire sets, hence
the Hellinger types of μ, and μ, are disjoint. Since there are c α points in 7α,
the lemma is true for γ = 0.
For any infinite cardinal y, divide //, the index set of Ia (see proof of Lemma
2), into two disjoint parts M and /V, of cardinal number γ and Cί respectively.
Let / and K be the γ- and CX-dimensional cubes with index sets M and N res-
pectively. Since M u N = H$ two points φ and φ, in / and K respectively, define
a point of Ia which we call φ u φ, for which ( φ u φ)r = φr or φr according as
r G M or r E N. We fix φ in K; and for each Baire subset A of /α, let Tφ{A) be
the subset of / consisting of those φ for which φ u φ £ A.. I claim that Tψ is
a σ-homomorphism of B^a' onto all Baire sets in /.
That Tψ is a σ-homomorphism is evident. Now B^a* is the smallest additive
class of subsets of Ia which contains P (defined in the proof of Lemma 2).
That every Baire set A in Ia maps into a Baire set in / will follow if sets in
P go into Baire sets; but the latter is evident. A corresponding argument in /
shows that all Baire sets in / are maps of Baire sets in /α. Hence the claim
made for Tψ is correct.
Now let I / 7 ' denote the product Lebesgue measure defined on the Baire
sets of /. For each φ in Kf and each Baire set A in /α, put
Evidently the measure algebras of μ, and Lry' are isomorphic; the former is
THE MEASURE RING FOR A CUBE OF ARBITRARY DIMENSION 517
t h e r e f o r e h o m o g e n e o u s of order y. F u r t h e r , if φί9 ψ2 £ K9 r E /V, ψ^ir) j^ ψ2(r),
and A i s t h e B a i r e s e t of a l l φ in / α w i th φ(r) = ψιir), we s e e t h a t
It follows that μxpι and μψ2 are disjoint whenever ΦlfΦ2 ^ ^ a n <^ Φι ^ Φ2
The fact that there are ca elements in K completes the proof of the lemma.
THEOREM. If Cί is an infinite cardinal^ the cardinal function p characteriz-
ing the measure ring R of Bellinger types of measures on the Baire subsets
of the Cί-dimensional cube is given by:
ca if γ is a 0 or infinite cardinal with γ <_ Cί;
0 otherwise.
Proof. Since P, a σ-basis of B (see proof of Lemma 2), is of cardinal
number Cί, we must have p = 0 for y > Cί. If y is 0 or infinite, and y <_ Cί,
Lemmas 1 and 2 prove the existence of a maximal disjoint family of elements
of R homogeneous of order y, of cardinal number exactly ca. That ca is the
smallest possible cardinal number of such a family, follows easily from the fact
that no element of R intersects more than a countable number of disjoint
elements of R{a\
R E F E R E N C E S
1. C. Kuratowski, Sur une generalisation de la notion d'homeomorphie, Fund. Math.,22 (1934), 206-220.
2. D. Maharam, On homogeneous measure algebras, Proc. Nat. Acad. Sci., 28 (1942),108-111.
CALIFORNIA INSTITUTE OF TECHNOLOGY
THE NORM FUNCTION OF AN ALGEBRAIC FIELD EXTENSION, II
H A R L E Y F L A N D E R S
1. Introduction. In our previous paper [3], we consider the general norm
of a finite extension K of an algebraic field k. We proved that this form is the
(n/m)th power of an irreducible polynomial in λ[λΠ, where m is the maximum
of the degrees of the simple subfields k(θ) of K over k. The proof of this result
used a considerable amount of the heavy machinery of the theory of algebraic
extensions: the maximal separable subfield, conjugates, transitivity of the norm,
etc. Using only the fact that the general norm is a power of an irreducible, we
obtained a characterization of the norm function NK/k ι n terms of inner proper-
ties.
In the present paper we shall approach these matters from a different point
of view. We shall give an entirely different proof that the general norm is a
prime power—this one based on very little field theory and completely rational.
From this, as noted above, the intrinsic characterization of the norm function
follows. We shall then use this to derive certain theorems in field theory, such
as the transitivity of the norm.
Section 2 contains some preliminary results on polynomials and their norms
and the details of proof for certain results used in [3] , In § 3 we prove the main
result and in § 4 we give some applications.
2. Tool theorems. We shall be dealing with polynomial rings &[X]in inde-
terminates X = (Xί9 ,Xr) and shall take for granted the fundamental fact that
such rings are unique factorization domains [ l , p . 39]. The following is well
known, but we include it—as we do several of the results of this section—for
completeness.
LEMMA 1. Let f (X), g(X) £ k[X] and suppose f and g are relatively prime.
Let k <^K so that k[X] < K[X], Then f and g are still relatively prime when
considered as elements of the extended ring K[X].
Received October 19, 1953.
Pacific J. Math. 5 (1955), 519-528
519
520 HARLEY FLANDERS
For the case r = 1 of one variable, this is so because of the Euclidean
greatest common divisor algorithm.
In general, we suppose H{X) is a common factor of f and gs H(X) EK[X\
Without loss of generality, we may assume that H has positive degree in Xr,
We form the fields of rational functions,
K == K (x i, , xr_ i), k = k (x i, , xτ. i),
and the p o l y n o m i a l s
/ ( T) = / ( % i , •• , x r - i > T), g~( 71) = g ( % i , •• , * Γ . i , 71)
of t h e r ing k[T]. T h e s e p o l y n o m i a l s h a v e a non-tr iv ia l factor H(x9 T) in K[T],
h e n c e by the c a s e r = 1, they h a v e a n o n - c o n s t a n t factor hι(x$ T) £ k[T]:
J(T) = hί(x,T)fι(T), ^(T) = hί(χ,T)gι(T).
Here hi, fy, gjβre polynomials with coefficients rational functions over k in
x = {xι, ' , xr.i ). Multiplying by a suitable denominator q (x ), we obtain
q(x)f(x, T)=h(x, T)f2(x, T), q(x)g(x, T) = h(x, T)g2(χ, T),
where all terms are polynomials. This implies
q(Xi,...,Xr.i)f{X)-h{X)f2(X), q(Xι,' ',Xr.ι)g(X)"h{X)g2(X).
Since h(X) actually involves Xr, it follows from unique factorization that some
irreducible factor of h must divide both f and g,
LEMMA 2. Let k be a field, JO an integral domain such that k < D, and such
that if JO is considered as a linear space over k$ then £ is finite dimensional.
Then c is a field.
Proof. Cf. [2, p. 75 ]. If α £ JO and a 0, then the mapping b—> ab is a
one-one linear transformation on JO into JO. Since JO is finite dimensional and
rank plus nullity equals dimension, it must map JO onto £>. Thus 1 = ab for some
b then a has an inverse.
LEMMA 3. Let [K:k] = n and ωί9 ,ωn be a basis of K over k. Then
[K(X):k(X)] = nand(ω) is a basis of K(X) over k(X).
Proof. Let
THE NORM FUNCTION OF AN ALGEBRAIC FIELD EXTENSION, II 5 2 1
o = k(X)ωt + . . . +k{X)ωn.
Then o is a finite dimensional integral domain over k(X) and
k(X) <a
By Lemma 2, JO is a field; since
K = kωt + + kωn < o ,
we have
K(X) = K-k(X) < a ,
hence JO = K(X). It follows that (ω) spans K(X) over k(X). But it is clear
(by equating coefficients) that (ω) is linearly independent over the rational
function field k(X).
We introduce the norm in this way. If [K:k] = n and A G K9 then Nj^/^A is
the determinant of the linear transformation B —* AB on K over k Specifically,
if cύij * , ωn is any basis of K over /c, and
then
We similarly define the irace
Sκ/k A = Σ α ί t
for later purposes. The rules
= (NK/kA){NK/kB)9 Sκ/k(A+B)=Sκ/kA+Sκ/kB,
= αn. Sκ/k(a) =n - a,
follow immediately.
We form the fields K(X), k(X) so that also [K(X): ft (Z) ] = n and we
may discuss
522 HARLEY FLANDERS
for R(X) G K(X). We shall use the abbreviation
as we did in [3] since this can hardly lead to confusion.
LEMMA 4. Let
F(X)eK[X] and f (X) = Nκ/kF(X).
Then
f(X)£k[X]
and F (X) divides f (X) in the ring K [ X ].
Proof. We write
ΣA(a) A'(α)
where A G K and X(a) is a monomial in X — (Xγ, , XΓ). We have
hence
I = Z^o A!(α) ω, = έ^f..\X)cύj,l>] J If J
f(X)~Nκ/kF=\f..\ek[X]t
where f . . G ί ; [ ^ ] . Thus
which settles the first point. We may also write
which implies
On expanding the determinant we soon see that F(X) does indeed divide f(X).
LEMMA 5. //
THE NORM FUNCTION OF AN ALGEBRAIC FIELD EXTENSION, II 5 2 3
F(X),G(X)eK[X], h(X)ek[X],
and
F(X) = G(X) (moάh(X)),
then
Nκ/kF(X)^Nκ/kG(X) (modh(X)).
Proof. We may write
F(X) = G(X) + h(X)Q(X)
with Q(X) E K[X\ As above, we have
ωj, G(X)ωi=Σgij(X)ωj,
i t h fij 8ij> Ίij G * [ ^ l T h u s
and therefore
7V(F) == 1/ .| = \gij + hq.j\ = l g ι 7 l = ^ V ( G ) U o d A U ) ) .
LEMMA 6. Let F(X) be an irreducible polynomial in K[X]. Let f (X) =
/Vχ/^F(Z) and suppose that g{X) is any non-constant divisor of f(X) in
k[Xl Then F{X) divides g(X).
The case r = 1 is given in [4, p. 19].
Proof. If r = 1 and F(X) does not divide g(X), then we can find polyno-
mials U(X), V(X) eK[X] such that
Thus
U(X)F(X) = l (moάg(X)).
By Lemma 5 we obtain
524 HARLEY FLANDERS
u(X)f{X) = l ( m o d g U ) )
which is clearly impossible.
In the general case we may suppose that the degree of F in Xr is positive
and pass to the rational function fields k — k(x)y K = K(x)y where x = (x ι ? ,
xΓmi). The usual unique factorization argument shows that F( T) = F(x i9 •• ,χr-ι* T) is irreducible in K[T]. For the norm we have
The polynomial ~g{T) = g{xι, ,xrm\$T) divides f (T) in k[T]; it follows
from the case r — 1 that F(T) divides g(T):
We multiply by the denominator of // to arrive at a relation of the form
Since F(X) is irreducible, this implies that F(X) divides g(X),
THEOREM 1. Let F(X) be irreducible in K[XI Then {(X) = Nκ/k F{X)
is a power of an irreducible polynomial in k\_X\
Proof. If p(X) and q(X) are irreducible factors όί f(X) in k[X]9 then by
Lemma 6, F(X) divides both p{X) and q(X). This implies, by Lemma 1, that
p(X) -q{X). Hence f {X) has only one distinct irreducible factor.
NOTE 1. In the proofs of both Lemma 1 and Lemma 6, the reduction of the
case of general r to the case r = 1 could have been effected by the Kronecker
device of substituting suitable powers of a new variable T for the X(9 since in
these statements we dealt with only a finite number of fixed polynomials and
their divisors, all of bounded degree.
N O T E 2. Lemma 1, for the case in which [K:k] ~n9 is an immediate con-
sequence of Lemma 4. For if H{X) E K{X) and H{X) is a non-constant common
divisor of f and g, then we have f=HFί, g = HGί9 and thus
fn = N(f) = N(H)N(Fι\ gn = N{H)N{Gι).
But H divides Nκ/kH, hence N{H) is non-constant. This is clearly impossible
when f and g are relatively prime.
THE NORM FUNCTION OF AN ALGEBRAIC FIELD EXTENSION, II 5 2 5
Once Lemma 1 is proved for finite extensions, it can be proved for arbitrary
extensions by the use of a transcendence basis.
3. The general norm. Let [K:k]-n and let ω 1 ? , ω n be a basis of K
over k. As in [3] , we form the general element
Ξ= o)ιXι + + ωn Xn eK[X]
and the general norm
NN/k(~)£k[X]
which is a form of degree n
THEOREM 2. The general norm is a power of an irreducible polynomial in
k[X].
Proof. The general element E is a linear form in K[X], hence irreducible;
Theorem 1 now applies.
From this now follow the results of § 3 of [3]; we state the following in-
stance.
THEOREM 3. Let [K:k] -n and let φ be a function on K into k with the
following properties:
(1) φ(AB) = φ(A)φ(B).
(2) φ(a)=an.
( 3 ) φ ( Σ αj CU( ) = / (a i, , an),
where f is a polynomial of degree at most n. Then φ(A) -Nj^/kA for all A in
K.
4. Applications. Let k <.L < K, where K is a finite extension of k, and
consider the function
A—>NL/k[Nκ/LA]
on K into k. Evidently this satisfies the properties (1,2,3) of the theorem
above, so we obtain
526 HARLEY FLANDERS
Next, let [K: k] = n and let A G X. The /"je/rf polynomial of 4 is
It is clear that / ^ ( Ό = 0 and that fA(T) is the minimum polynomial of A in
case X = k{A )—since 1, /4, , /471"1 is a basis in that case. If K >_ L > k9
then
fA,κ/k{TUNκ/k(T-A)=NL/k[Nκ/L(T-A)]
Especially HA GL, then
Here is another consequence; if K > L >_k and A £ K, we have
For if [K:fc] = r, then
Our statement follows at once from this and the following lemma.
L E MM A 7 . L e t [ K : k ] = n and
f ( T ) = Γ + A x Tr'1 + + A r e K I T ] .
T h e n
This is proved by slightly modifying the proof of Lemma 4.
Finally we derive the familiar expressions for the norm and trace in terms
of conjugates. Let [K: k] = n and let K < ί/. Suppose σ1 ? , σw are n not
necessarily distinct isomorphisms over k on K into ί/ with the property that
whenever h (A^, , Xn) is a symmetric polynomial in k [A] then λ ( OΊ (/4 ), ,
σn(A )) E & for all A £ K. We consider the mapping
THE NORM FUNCTION OF AN ALGEBRAIC FIELD EXTENSION, II 527
on K into k. This sat is f ies properties ( 1 ) and ( 2 ) of the las t theorem. To show
that it also sat is f ies the third property, we let ω t , , ωn be a bas is of K over
k and let A - ΣL α; ω^ be an element of K, α t £ k. Then
ί=l J
where f is a form of degree n in aγ9 9an whose coefficients are, until we
say more, in U If k is infinite, one finds that these coefficients are in k from
the fact that f (aχ9 9an) G k for all vectors ( α l 5 , an); when & is finite,
then K — k(B) is simple over k9 and we may use 1, Bf •• 9Bn~ for a bas i s .
Then the coefficients of f are symmetric in σx ( B ) , , σn( B ), and hence are
in &. At any rate we obtain
If F(Γ) = Σ/4 t Γ, we set
and make the obvious extension to rational functions. A similar argument to
that above implies that
h(Rσι(T),.->,Rσn(T))ek(T)
when h (X) is symmetric in X = (XΛ , , Xn ), h (X) £ k[X], and R(T) E K(T).
It follows that the formula for the norm as a product (of conjugates) is also
valid in K(T) over k(T), hence in particular
and by comparing the second coefficients,
REFERENCES
1. A. A. Albert, Modern higher algebra, Chicago (1937).
2. N. Bourbaki, Algebre, Chapitre V, Corps commutatifs, Paris (1950).
3. H. Flanders, Norm function of an algebraic field extension, Pacific J. Math.3 (1953), 103-113.
528 HARLEY FLANDERS
4. H. Weyl, Algebraic theory of numbers, Princeton (1940).
THE UNIVERSITY OF CALIFORNIA, BERKELEY
ON THE CHANGE OF INDEX FOR SUMMABLE SERIES
D I E T E R GAIER
1. Introduction. Assume we have given a series
(1.1) α 0 + a l + a2 + + an +
and consider
(1.2) b 0 + b i + b 2 + + b n + w i t h b 0 = 0 a n d b n = a n . ι ( n > _ l ) ;
denote the partial sums by sn and tn, respectively. Since sn = ί Λ + i , the con-
vergence of (1.1) is equivalent to that of (1.2). However, if a method of sum-
mability V is applied to both series, the statements
(1.3) (a) V-Σan=s (b) V-Σbn=s1
need not be equivalent (for example, if F i s the Borel method; see [4, p. 183]).
If V(x;sv) and Vix tp) denote the F-transforms of the sequences { sn \ and
{tn \, respectively, it is therefore interesting to investigate, for which methods
V and under what restrictions on { an \ the relations
(1.4) ( a ) V(x;sv)^ K . x* ( b ) V(x; tv) ~ K . χ<?
(x—> XQ , K c o n s t a n t ; q >_ 0, f i x e d ) 2
are equ iva lent .
The c a s e s V^C^ ( C e s a r o ) and V~A ( A b e l ) are quickly d i s p o s e d of
( § 2 ) , while V~E ( g e n e r a l E u l e r t ransform) and V~B ( B o r e l ) p r e s e n t some
i n t e r e s t ( § § 3 - 5 ) .
2. THEOREM 1. The statements (1.4.a) and (1.4.b) are equivalent for
XWe shall always let L ° ° s o n = Σ, an.
x—y XQ through values depending on the method V,
Received December 1, 1953. This work has been sponsored, in part, by the Officeof Naval Research under contract N5ori-07634.
Pacific J. Math. 5 (1955), 529-539529
530 DIETER GAIER
V = Ck(k > - 1 ) and V ~ A. 3
Proof. If
s < f c ) - < * ( * . „ ) . ( " +
and
we have by definition of the Cesaro means
(2.1) (l-x)k+ι Σ.τlk)xn
the series being convergent for | x \ < 1. The proof of Theorem 1 now follows
from the inner equality in (2.1) and the relation
γ(k) o(k) S^^n τ ι - 1 " - 1 . .
\n —> oo .ίn + k\ jn + k\ /n-l + k\
\ n I \ n I \ n - 1 /
3. Let g(w) = Σ,γnwn be regular and schlicht in | w \ £ 1, and assume
g ( 0 ) = 0 , g ( l ) = l. Then the ^-transforms of Σ,an and 22b n are obtained
by the formal relations [ 5 ]
Σ,anzn = Σ,an[g(w)]n = Σ,0Lnw
n; E(n;sp)= ^ av
(3.1) (τι = 0 , l , . . . ) .
Σbnzn=Σbn[g(w)]n=Σβnw"; E(n;tv)=Σ β*
THEOREM 2. The statements (1.4.a) and (1.4.b) are equivalent for V - E.
Proof. First we note that if either
E{n;sv)^0(n^) or E U; tv) = O(n^) (n—*ω),
}For q = Osee [4, p. 102].
ON THE CHANGE OF INDEX FOR SUMMABLE SERIES 5 3 1
then the formal relations (3.1) are actually valid for \w \ < 1 and also
( 3 . 2 ) Σ,βnwn= Σbn[g(w)]n=g{w). Σan[g(w)]n = g(w). Σ Clnw
n
(\w\ < 1) .
Denote by Anf Bn9 Cn the partial sums of Σdn, Σβn, Σγn, respectively.
We assume first
E (n; sv) - An ~ K n^ (n —> oo) .
Then, since by (3.2) Σ,βn is the Cauchy product of Σ,(Xn and Σ y π , we have
E(n;tv) = Bn=γnA0 +γnmlAί + ... + γιAn_ι
and for ^ >. 1
( 3 . 3 ) — = — A 0 + γ _ . _ + ..,. + γ(u-l)q
For the matrix cnV in this transformation of the convergent sequence \Ann"^
we have clearly
lim cnV = 0 ( v = 0, 1, •)«
n —• oo
Furthermore
n-1 vq \γn\ n oo
finally we prove
lim
For ςr = 0 this follows from
532 DIETER GAIER
for q > 0
nH
£ ί/n v\? fn-v-l\1
and the last term is a positive regular transformation of the sequence { Cn\
tending to g ( l ) = 1, whence
^2. cnv —> 1 ( n —>oo) .
v
Therefore the transformation ( 3 . 3 ) of \ An n"^ \ converges to K, which proves
Bn - K-n? ( Λ — > ( » ) .
Assume on the other hand Bn ^ Knq in —» oo). Putting w = 0 in ( 3 . 2 ) , one
obtains βQ = 0, so that
Zanwn = [g{w)}-1 Σβnw
n=w[g(w)Yι
is regular in \w\ < 1. Furthermore the expansion of the function w[g{w)]~ι
for w = 1 converges absolutely to 1, s ince w = 0 is the only zero of g{w) in
Iu; I £ 1. An argument similar to the one above shows then that Bn^.ί gί Knq
(n—> QQ) implies An ^ Kn^ (n —>oo), which completes the proof of Theorem
2.
We add a few remarks about the assumptions on the function z = g(w) by
which the E-method is defined.
a. Theorem 2 becomes false if only regularity of g{w) in \w\ < 1, and con-
tinuity and schl ichtness in \w\ <_1 are assumed. For there exist such functions
g(w) whose power ser ies do not converge absolutely on | w \ = 1 (cf. [ 2 ] ) .
Therefore in ( 3 . 2 ) one could find a convergent Σ α n whose transform Σ,βn
diverges.
b. All that was used about the function g(w) in the proof of Theorem 2
was that the power ser ies of g(w) and of w[g{w)Yι converge absolutely to
the value 1 for w = 1. This can be guaranteed by the weaker assumption that
g(w) with g ( l ) = l and g ( 0 ) = 0 is regular in \w \ < 1, continuous and schlicht
ON THE CHANGE OF INDEX FOR SUMMABLE SERIES 533
in \w\ <. 1, and that the image of \ w \ = 1 under the mapping g(w) is a recti-
fiable Jordan curve. Because then
< ooΪ27T \g\e^Jo
and hence ϋL \γ \ < oo [8, p. 158]; on the other hand also
/ \G'{eιφ)\dφ < oo,Jo
where
I' giw) -wg'(w)£'(„,)_[-£_] =ίg(w)]2
so that also the power series of G(w) converges absolutely to the value 1 for
c. If
g(w) ~ w[(p + 1 ) - pw]~ι (P >_ 0> fixed )
one has E - Ep as the familiar Euler method of order p, for which Theorem 2 is
known in the case q — 0 [4, p. 180],
d. The function
g(w) = (2 - « , ) - 2(1 -wΫΛ ( g ( 0 ) = 0 )
leads to the method of Mersman [6] , as Scott and Wall showed [7, p. 270 ].
Here Theorem 2 is also applicable, since the more general conditions about
g{w) in remark (b) are satisfied, as is readily seen.
4. The Borel method is defined by the transformation
svxv
-χΣ ( x > o ) ,e
where the power series is assumed to define an entire function. It is known
that B(x sp)—>K [x—»oo) implies B{x;tv) —>K (x —»co), but not con-
versely [4, p. 183]. We now prove more generally
534 DIETER GAIER
THEOREM 3. The relation
B(x;sv) ~ Kx^ {x—> oo)
implies
B{x\tv) Ξί Kx^ (x—>oo).
Proof. We have for x > 0 [4, p. 196]
v\ {v+ 1)1
B (t; sv )<; f v /*r , * B(t:sv)-^- dt-χ-1 I e-^-'h* —dt.v\ Jo t^
This transformation of the convergent function B{t;sv)t"^ (t—»oo) by means
of the 'matrix
, (0<t<x)
is regular, since
I \c(x9t)\dt—*0 (%—> oo t u t 2 > 0, fixed)Jtγ
and
Γx foe / t \ q
\c(x,t)\dt= I c(x, t)dt =e"x e Ί - l dt—>1 (x—>oo).Jo Jo \x I
Therefore B(x;tv) ^ Kx^ (x—> oo).
We discuss now the converse of Theorem 3.
THEOREM 4. The relation
B(x;tv) z Kx* U—» oo)
implies
ON THE CHANGE OF INDEX FOR SUMMABLE SERIES 5 3 5
B(x;sv) 21 Kx^ (x—>oo),
if
( 4 . 2 ) l im s u p \an \ί/n < oo,
that is, if the series 2Lanzn has a positive radius of convergence.
Proof. Using (4 .1) we have for % > 0
Fix) ^x'^B(x;tv)=x^e"x [% etB (t; sv)dt.
J o
C o n s i d e r n o w F ( x ) a s f u n c t i o n o f t h e c o m p l e x v a r i a b l e x f o r K { x ) >^ 1 . T h e n
( 4 . 2 ) i m p l i e s \tn\ <_ Mn f o r s o m e c o n s t a n t M > 0 a n d h e n c e i n H ( x ) >_ 1
and also
(4.3) \F(x)\ < aeP\x\ H(x)
for positive constants α and β. Hence one knows that
F(x)—>K (x—>+
implies
F'(x)—>0 (x—>+ω
t h a t i s ,
[XB(t;sv)dt ί - 1 - !
from which the result follows.
5. We now show that Theorem 4 is best possible in a certain sense.
4 I f F{χ) i s r e g u l a r in K(x) > 1 a n d ( 4 . 3 ) h o l d s , t h e n Fix)—>A ix — > + o o ) im-p l i e s F'(x)—>0 ix—>-f CXJ). T h i s lemma w a s u s e d a l s o in [ 3 ] , w h e r e T h e o r e m 4 w a sp r o v e d for q = 0.
536 DIETER GAIER
THEOREM 5. In Theorem 4 the Condition (4.2) cannot be replaced by
(5.1) l i m s u p ne \an\Wn < oo ( e > 0 ) .
For the proof we need the following
L E M M A . For every β > 1 , there exists a n entire f u n c t i o n f ( z ) of order
β satisfying
(5.2) / U ) _ > 0 U ^ + α>)f/'(*)-/-> 0 U — > + α>) U = *
Proof. Put α = /3"ι and consider the Mittag-Leffler function
which is an entire function of order Cί" = β. Let m be the integer with
α α< m < + 1.
1 - α 1 - α
We first study the derivatives of Ea( z) of order 1, 2, , m on the line arg z -
OLπ/2 for large | z |. For these z (assume for definiteness | z | > 2) one has
[ 1 , pp. 272-275]
(5.3) £ . < * > - — / V ' ' - 2 - + i β ' l Λ \2πiCl JL t - z α
the path L being
£ = re I oo > r > 1, Cίπ > φn > — I, t = e ( — ώn < φ < + φn ) ,i — * ' o 9/ ^ — — r u 7
t = reiΦo (1 < r < oo);
ί ι / α is the branch which is positive for t > 0. The A th derivative of the integral
part in (5.3) can then be estimated as follows
Λ/a k\* —dt
k\ Γ , . " a , |Λ
2πa\z\k+i/
A/a\el I
| l - (
ON THE CHANGE OF INDEX FOR SUMMABLE SERIES 537
since for our values of z one has | 1 - (t/z) | >. δ > 0 and on the straight line
segments of L
— e c o s α with cos — < 0 .Cί
Therefore
-j
£ ' ( 2 ) = o ( D + — ez z(X 9
a2
ι/a-1
1a
(5.4)α3
£ > - ι > ( 2 ) = o ( l ) + —
Now we consider the function
which is again an entire function of order CC1. For \z \—» oo on arg z = dπ/ 2
we have by (5.4)
am
however
α m + 1
and herein | e ε l / α | = l and ( ( l / α ) - l ) r o - l > 0, so that F ' ( z ) - / * 0
( I z I—> co on arg z = a n / 2 ) . For the lemma it is therefore sufficient to take
538 DIETER GAIER
Proof of Theorem 5. Define the { an \ of (1.1) by
r< \ fx -t <Γ a v t V i ίx -ί / ^ 7/ ( % ) = / e
Σ L ώ = / e
ι a(t)dt,Jo v\ Jo
w i t h t h e fix) of t h e a b o v e l e m m a a n d /3 = ( l - € ) " 1 . S i n c e fix) i s of o r d e r
β > 1, s o i s o ( ί ) , a n d t h e r e f o r e [ 1, p . 2 3 8 ] 5
anλ / IΛ
suplim sup n ^ — e lim sup n" | α π | < oo ,
that is, (5.1) is fulfilled. Furthermore
/ ( % ) — > 0 (%-^ + oo),
which is equivalent to
B(x;tv)—>0 {x—> + oo).
However, in order that
B(x;sv) —>0 (x—> + oo),
it would be necessary and sufficient to have [4, pp. 182-183]
emχa(x)=f'(x)—*0 (x—>+oo),
which by our lemma is not fulfilled. So we have given an example of a ser ies
Σ o n for which B(x;tv) —» 0 (x —» +oo) does not imply B(x;sv) —> 0
(x —> + oo) and for which ( 5 . 1 ) holds.
Prof. Lδsch (Stuttgart) suggested to me the relation to the coefficient problemfor entire functions.
REFERENCES
1. L. Bieberbach, Lehrbuch der Funktionentheorie, 2. ed., vol. II, Leipzig, 1931.
2. D. Gaier, Schlichte Potenzreihen, die auf \ z \ = 1 gleichmassig, aber nicht absolutkonvergieren, Math. Zeit. 57 (1953), 349-350.
3. , Zur Frage der Indexverschiebung beim Boreί-Verfahren, Math. Zeit.58 (1953), 453-455.
4. G. H. Hardy, Divergent series, Oxford, 1949.
ON THE CHANGE OF INDEX FOR SUMMABLE SERIES 539
5. K. Knopp, Uber Polynomentioicklungen im Mittag-Leffίerschen Stern durch An-
wendung der Eulerschen Reihentransformation, Acta Math. 47 (1926), 313-335.
6. W. A. Mersman, A new summation method for divergent series, Bull. Amer. Math.Soc. 44 (1938), 667-673.
7. W. T. Scott and H.S. Wall, The transformation of series and sequences, Trans.
Amer. Math. Soc. 51 (1942), 255-279.
8. A. Zygmund, Trigonometrical series, Warsaw, 1935.
HARVARD UNIVERSITY
COMPLETE MAPPINGS OF FINITE GROUPS
MARSHALL HALL AND L. J. PAIGE
1. Introduction. A complete mapping of a group G is a biunique mapping
x —>®(x) of G upon G such that x ®(x) =* η(x) is a. biunique mapping of G
upon G. The finite, non-abelian groups of even order are the only groups for
which the question of existence or non-existence of complete mappings is un*
answered. In a previous paper [4] , some progress toward the solution of this
problem has been made. We shall show that a necessary condition for a finite
group of even order to have a complete mapping is that its Sylow 2-subgroup be
non-cyclic, and that this condition is also sufficient for solvable groups. We
shall also prove that all symmetric groups Sn(n > 3) and alternating groups
An possess complete mappings. In the light of these results the following con-
jecture is advanced:
CONJECTURE. A finite group G whose Sγlow 2-subgroup is non-cyclic
possesses α complete mapping.
It is interesting to compare this conjecture with the results of Bruck [2, p.
105].
2. Complete mappings for the symmetric and alternating groups. The follow-
ing theorem is a generalization of Theorem 4, [4] and will be necessary for
considerations of this and other sections.
THEOREM 1. Let G be a group, H a subgroup of finite index {G:H) =k .
Let u\, U29 , w/c be a s e ί °f elements of G that form both a right and left
system of representatives for the coset expansions of G by H. Let S and T be
permutations of the integers 1, 2, , k such that
Ui(us{i)H) =uτ{i)H, i = 1,2, . . . , & .
1ΓΓhe restriction that the index be finite is unnecessary. However, P. Bateman [ l ]has shown that all infinite groups possess complete mappings and so we have chosenthe present restriction for simplicity. In fact, the restriction that G be finite wouldseem appropriate.
Received December 18, 1953. The work of L. J. Paige was supported in part by theOffice of Naval Research.
Pacific J. Math. 5 (1955), 541-549541
542 MARSHALL HALL AND L. J. PAIGE
Then, if there exists a complete mapping for the subgroup H$ there exists a com-
plete mapping of G.
COROLLARY 1. Let G be a factorizable groups that is, G = A B9 where
A and B are subgroups of G with A n B = 1. If complete mappings exist for A
and B9 then there exists a complete mapping for G.
COROLLARY 2. If H is a normal subgroup of G, and both H and G/H pos-
sess complete mappings then G possesses a complete mapping.
Proof. By hypothesis,
(1) G = u\H + u2H + • + u^H = Hui + Hu2 + + Huk
and thus the equation
(2) i t s ( i ) P = p* i t [ s ( j ) t p ] , ( i = 1,2, • • • ! * ) , P€H,
uniquely defines p* and ^fs( ) π as functions of p and ι. Here, wr<j/.\ i = ut for
some 1 <_ t fC k. Moreover, p is uniquely defined by p * and i, for if
usϋ)Pι
then we would have
Since the w's form a system of representatives this would imply
u [ s ( i ) , p ι ] " U [ s ( ) .p a ]
and consequently pί = p 2
We have assumed that there exists a complete mapping for H; hence, there
is a biunique mapping ®ι of H upon H such that the mapping ηγ{p) = p Θ i ( p )
is a biunique mapping of H upon H.
Let us define a mapping of G upon G in the following manner:
O)
where p, p*, w[ s( ι ) 1 a r e defined by ( 2 ) .
COMPLETE MAPPINGS OF FINITE GROUPS 543
In order to show that Θ is biunique, assume that
ΘUίP*) = eu / P*).
Then,
u[sU),Pι] ' ei{Pi] = uίs(j),p2] ' &ι{P2
)I
and t h i s can h a p p e n o n l y w h e n u[sd) ]~u[s(~) ] i m p l y i n g Θ t ( p t ) = Θ L ( p 2 )
or p t = p 2 . Now,
and it would follow from (2) that i =/. If G is finite we may conclude immedi-
ately that Θ is a biunique mapping of G upon G. If G is infinite, we note from
(2) that if p is kept fixed, then as i ranges over 1, 2, . ., k; u[s(^ l ranges
over all coset representatives. Thus for any element ut p', we first find p
from p ' = Θ i ( p ) ; and then holding p fixed we vary i to find the p* such that
ιι . v . p = p* . Ufr For this i and p* we have
ΘUjp*) =Mt ®ι(p) =ut p\
and every element of G is an image of some element of G under the mapping Θ.
Let us now show that Θ is a complete mapping for G. Consider
η(uiP*) =uiP* Θ U p*) = ttip* " [ s (0 ,p] ' Θ ^ P ^ =uiuS(i) ' P Θ l ( p )
First, if ηiuip*) = ηiujp*), we have
(4) ^ ^ ( ί ) P i Θ ι ( P i ) = = w / z x S ( / ) P 2 Θ ι ( P 2 ) ' O Γ uT(i)H = uT(j)H>
and this is impossible unless i = /. Consequently from (4),
P 1 Θ 1 ( P l ) = P 2 θ 1 ( p 2 )
and Θ t being a complete mapping implies pγ - p2. Again the finite case is
completed and if G is infinite we note that there is but one i such that UiUς,ί.\H —
Uj>ί\H and the subsequent solution for p* is straightforward.
Corollary 1 follows from the observation that the elements of A form a sys-
tem of coset representatives satisfying the hypothesis of the theorem.
544 MARSHALL HALL AND L. J. PAIGE
Corollary 2 is proved by noting that if
in G/H, then
We will use Theorem 1, to show that an earlier conjecture [4, p. 115] con-
cerning complete mappings for the symmetric groups Sn(n > 3) was wrong.
THEOREM 2. There exist complete mappings for the symmetric group Sn if
ifn > 3.
COROLLARY. (See conjecture [4, p. 115]). There exist Latin squares
orthogonal to the symmetric group Sn for all n > 3.
Proof. The proof will be by induction and we note first that S3 has no com-
plete mapping [3, p. 420], Thus we must exhibit a complete mapping for S 4 .
We may express S4 = A B9 where
A=\\9 (123), (132) ! ,
B = U , (12), (34), (12M34), (1324), (1423), (14)(23), (13)(24) },
are subgroups of S4 with AAB = 1. Moreover, there exist complete mappings for
A and B given by:
β ( l ) = l, 6(123) = (123), Θ(132) = (132)
for A; and
Θ ( l ) = l, Θ(12) = (34), Θ(34) = (1324), Θ( 12)(34) = (13)(24)
Θ(1324) = (14)(23), ©(1423) = (12)(34),
Θ( 14)(23) = (12), Θ(13)(24) = ( 1 4 ) ( 2 3 ) ,
for B. The fact that S4 has a complete mapping now follows from the corollary
of Theorem 1.
Let us now assume that Sn has a complete mapping with n > 3. Then,
Sn+X =Sn + ( 1 , n + 1)Sn + (2, n + 1)S n + + (n t n + 1)Sn ,
= Sn + Sn (1 , n + 1) + Sn (2, n + 1) + + Sn (n9 n + 1) .
COMPLETE MAPPINGS OF FINITE GROUPS 545
Clearly, two cosets (/, n + l)Sn and (k% n + 1 )Sn (j φ. k) being equal would
imply (/, k, n + 1) G Sn and this is impossible.
Now note that
(/, n + 1) (/ + 1, n + 1) Sn = (/, j + 1, n + 1) Sn = (/ + 1, n + 1) Sn
if 1 < / < re - 1. Also, U , n + l ) ( l , n + l)Sn = ( 1 , n + l)Sn.
We now see that the coset representatives of Sn+ γ by Sn satisfy the con-
ditions of Theorem 1 under the obvious mapping S ( l ) = l , S{i)=i + 1 for
2 <. i <. n and S(n + 1) = 2. Hence, S π + X has a complete mapping and our in-
duction is complete.
The corollary follows from Theorem 7 of [ 4 ] ,
It should be pointed out that the coset representat ives used for Sn+ι in the
argument above do not form a group and hence Theorem 1 is sufficiently stronger
than the corollary to be of decided interest.
THEOREM 3. There exists a complete mapping for the alternating group
Λn$ for all n.
Proof. Aι, Ait a n d A3 (the cyclic group of order 3) possess complete
mappings. Hence assume that there exists a complete mapping for An. Then,
rc+ 1 = An + ( 1 , n, n + 1) An + ( 1 , n + 1, n) An + (2, n + 1) ( 1 , n) An
+ (3, 7i + l ) ( l , n)An + . . . + U - 1 , Λ + 1 ) ( 1 , n)An
and the coset representatives are valid for either a right or left coset decom-
position for An+ι by An,
It is a simple, straightforward verification that the permutation S, given by
S ( l ) = l, $ ( 2 ) = 2 , S(3) = 3, S ( ί ) = i + 1 (4 < i < n), S U + D - 4
satisfies the conditions of our Theorem 1. Here we meet a slight difficulty if
n = 3, but it is known [3, p. 422] that there exists a complete mapping for
/14 and we may take n — 4 as the basis for our induction.
3. Groups of order 2". Although it has been indicated in the literature [4]
that the results of this section are known, it seems desirable (and necessary
for completeness) to include the proofs of these results.
LEMMA 1. Let G be a non-abelian group of order 2n and possess a cyclic
546 MARSHALL HALL AND L. J . PAIGE
subgroup of order 2n~ι. Then a complete mapping exists for G.
Proof. It is known [5, p. 120] that G is one of the following groups:
( I ) Generalized Quaternion Group (n > 3) , A 2n~l = 1, B2 =A2n'\ BAB'ι = A'\
(II) Dihedral Group U > 3), A2n'1 = 1, B2 = 1, BAB'1 = A'\
(III) (n > 4 ) , A2n'1 = 1, β 2 = 1, BAB'1 =A 1 + 2 * ' 2 .
(IV) U > 4) , ^ 2 " " 1 = 1, β 2 = l,
In each case, the elements of the group are of the form
Λ D^ \CL — Ό9 1, , z = 1 ; p = 0, l j .
Let us define a mapping Θ as follows: ( let m ~ 2n~ ),
®(Ak)=Ak; A = 0 , l , . . . , m - l ;
Θ ( / 4 * ) - i 4 * I B β ; A = m, m + 1 , . - . , 2 m - l ;
Clearly, Θ is biunique and we will show that it is a complete mapping for
groups I and II. Thus,
Ak >®(Ak) = Ak .Ak = A2k; A = 0 , l , . . . f m - l .
Ak . Θ(Ak) = Ak - Ak'mB = A2k'mB k = m, m + 1, . . •, 2m - 1 .
>2 i . D 2 _ J2" 2
We see that we have a complete mapping if B = 1 or β = A
A slight calculation in the evaluation of Ak B®(AkB), will show that this
mapping is also a complete mapping for the group IV. It is necessary to use the
fact that n > 4.
COMPLETE MAPPINGS OF FINITE GROUPS 547
In order to obtain a complete mapping for group III, we define:
β(Ak > B)=Ak+m; for /c =
* . β) = 4 * δ ; for A; =
The verification that this mapping is a complete mapping for group III is straight-
forward and will be omitted.
This completes the proof of the lemma.
THEOREM 4. Every non-cyclic % group G has a complete mapping.
Proof. This theorem is known to be true for abelian groups [4] , We may use
induction to prove the theorem if G has a normal subgroup K such that K and
G/K are both non-cyclic Corollary 2, Theorem 1).
In view of Lemma 1, we assume that G is a non-abelian group of order 2n
and does not possess a cyclic subgroup of order 2n~ this implies n > 4. If
G contains only one element of order 2, G would have to be the generalized
quaternion group [5, p. 118] contrary to our assumption. Hence G contains an
element of order 2 in its center and another element of order 2. These elements
together generate a four group V.
If V is contained in two distinct maximal subgroups Mi and M2, then Mγ n M2 —
K D V is a normal subgroup of G such that both G/K and K are non-cyclic. In
this case the theorem would follow by induction.
We now suppose that V is contained in a unique maximal subgroup M\. Gt
being non-cyclic, contains another maximal subgroup M2 and if Mγ n M2 is non-
cyclic our induction again applies. Taking Mί n M2 to be cyclic, we see that
Mi is a group of order 2n" containing a cyclic subgroup of order 2n~ and also
the four group V. Thus Mt is of the type II, III or IV of Lemma 1 or possibly
an abelian group with A2*1'2 = 1, B2 = 1, BAB"1 = A. In all cases, Mx n M2 = {A \.
Now let C be any element of M2 not in \A\. Then by the normality of {A \,
C2 = Ar, where r is even since otherwise C would be of order 2n~l and G has no
cyclic subgroup of order 2n~ι. Also C"1 AC ~ Au with u odd.
Now consider the group H ~ \ A2, B }, which is non-cyclic since n > 4. Here,
548 MARSHALL HALL AND L. J. PAIGE
Λfi = # + HA = # + AH, and
Thus,
G = H +HA+HC + HAG = H + AH + CH + CAH,
where CAH = ACH since
We see that the elements 1, /4, C, AC are two-sided coset representatives for
flinG.
Define
Θ ( l ) = l , ®(A) = C, Θ(C)=AC9&(AC)=A,
and c o m p u t e :
1 . Θ ( l ) / / = 1 .H;
A Θ ( A ) H = A CH
C Θ( C)H = CACH = C C" AC — Ar AUH = /4/Z since r is even, w odd;
4C .ΘUC)//=/lC4tf = C . C ιACH = CAuH = CAH = ACH.
Hence, with these representatives the hypotheses of Theorem 1 are satisfied
and G has a complete mapping.
4. Solvable Groups. The existence of complete mappings for solvable groups
is answered in the following theorems.
THEOREM 5. A finite group G whose Sylow 2-subgroup is cyclic does not
have a complete mapping.
Proof. Let a Sylow 2-subgroup S2 of G be cyclic of order 2m. Then the
automorphisms of S2 are a group of order 2 m - 1 . Hence in G, S2 is in the center
of its normalizer. By a theorem of Burnside [5, p. 139], G has a normal sub-
group K (of odd order) with S2 as its coset representatives. Since G/K = 5 is
cyclic, the derived group G' is contained in K; and clearly,
Π ggee
COMPLETE MAPPINGS OF FINITE GROUPS 549
S is cyclic of order 2m and hence Π s = p, where p is the unique element
of order 2 of S2. Thus, s G S
Π g Ξ p U : ° H P (modK);gGG
and since G'CX, the Corollary of Theorem 1 [4, p. I l l ] is violated and G does
not have a complete mapping.
THEOREM 6. A finite solvable group G whose Sylow 2-subgroup is non-
cyclic has a complete mapping.
Proof. By a theorem of Philip Hall, a solvable group has a p-complement
for every prime p dividing its order. Thus, if S2 is a Sylow 2-subgroup of G
and H is a 2 complement, G — H S and H n S = 1. S has a complete mapping
by Theorem 4 and //, being of odd order, has a complete mapping. By Corollary
1 of Theorem 1, G has a complete mapping.
As further evidence in support of our conjecture we have the following
special theorem.
THEOREM 7. Let G be a finite group whose Sylow 2-subgroup is not cyclic.
If G has (G:S2) Sylow 2-subgroups and the intersection of any two Sylow 2-
subgroups is the identity, G possesses a complete mapping.
Proof. By a well known theorem of Frobenius, G is a factorable group;
that is, G = N S 2 , where N is the normal subgroup consisting of all elements
of odd order. We now apply Corollary 1 of Theorem 1.
REFERENCES
1. P. Bateman, Complete mappings of infinite groups, Amer. Math. Monthly 57 (1950),621-622.
2. R.H. Bruck, Finite Nets, L Numerical Invariants, Can. J. Math. 3 (1951), 94-107.
3. H.B.Mann, The construction of orthogonal latin squares, Ann. Math. Statistics13 (1942), 418-423.
4. L. J. Paige, Complete mappings of finite groups, Pacific J. Math. 1(1951), 111-116.
5. H. Zassenhaus, The theory of groups, Chelsea Publishing Co., New York, NewYork, 1949.
OHIO STATE UNIVERSITY
THE INSTITUTE FOR ADVANCED STUDY AND
UNIVERSITY OF CALIFORNIA, LOS ANGELES
RELATIVIZATION AND EXTENSION OF SOLUTIONS
OF IRREFLEXIVE RELATIONS
MOSES RICHARDSON
1. Introduction. Let >- be an irreflexive binary relation defined over a
domain 2) of elements α, 6, c, . We represent the system (5), >-) by an oriented
graph G by regarding the elements of 3 as vertices of G and inserting an arc
ab of the graph, oriented from a to b, if and only if a >- b. The sentence " α >- b"
is read " α dominates 6". A set V of vertices is termed internally satisfactory1
if and only if x G V and γ E V implies x ^j- y. A set V of vertices is termed ex-
ternally satisfactory if and only if γ E 5) — F implies that there exists an % E F
such that % >- y. A set F of vertices is termed a solution of G, or of ( 3 , >~), if
and only if it is both internally and externally satisfactory. In [4], various suf-
ficient conditions for the existence of solutions were established.
By a subsystem Oo,/*") of the system (§>,>-) is meant a system where
5)0 C 5) and the relation >- for the subsystem is merely the restriction of the
relation >- for the supersystem (5), >-). Let Go be the graph of the subsystem
(^o> >"") a n d l e t ^o he a solution of Go. A solution V of G is termed an extension
of Vo if Fn ®0 = Vo; in this case VQ is also said to be relativized from V. In
this paper, some sufficient conditions for the existence of relativizations and
extensions of solutions are presented. More elegant and more effective extension
theorems, especially with a view toward possible applications to the theory of
ra-person games, remain to be desired. It is hoped that the present paper may
serve to stimulate interest in this apparently difficult problem.
2. A theorem on relativization. If H is a subgraph of the graph G, then the
graph obtained by adding to H all the arcs of G which join pairs of vertices of
H will be termed the juncture of H (relative to G) and will be denoted by //.
In [ 2 ] , internally satisfactory is called satisfactory with respect to non-domination,and in [4] it is called ^/- -satisfactory.
Received March 1, 1954. Part of the work of this paper was done at the Institute forAdvanced Study in 1952-3, and part while the author was consultant to the LogisticsProject sponsored by the Office of Naval Research in the Department of Mathematics atPrinceton University in 1953-4. A statement of many of the results contained hereinappeared without proofs in L 5 J .
Pacific J. Math. 5 (1955), 551-584
551
552 MOSES RICHARDSON
H is termed a conjunct subgraph of G if and only if H = H.
The graph Go of a subsystem (§) O f >-) oί the system (§),>-) having the
graph G is a conjunct closed subgraph of G. If H is any subgraph of G, proper
or not, and x is any vertex of G, then D~ι(χ9H) shall denote the set of all
vertices y oί H such that y >» x. If v¥ is any set of vertices of G, let
D'\X9H)= U D-H%,«),
and let
D-n(X,H)=D-ι(D-n+\X,H), H)
for Λ > 1. Let D°(X,H) = Z by definition.
THEOREM 1. // Go is α conjunct subgraph of G and V is a solution of G,
then a sufficient condition for V n 5)0 to be a solution of Go, where 5)0 is ίAe
set of vertices of Go, is
(1) D - y ,
Proof. We must prove that F n S)o is both internally and externally satis-
factory with respect to G o . That is we must prove that
( a ) χ9 y E V n S)o implies x >/-y relative to Go, and
(b) y G ? ) 0 - F n 5 ) 0 implies that there exists an % € F π S 0 such that
Λ; >- y relative to Go.
But (a) follows immediately from the facts that GQ is a conjunct subgraph of
G and that V is a solution of G. To prove (b) , consider any y E S)o - Fn §)0.
There exists an Λ; E F such that Λ; >- y relative to G since V is a solution of G.
Then Λ; E D"ι(y9 G) C S)o by hypothesis. Thus x$ y E ®0
a n ( l t n e oriented arc
%y C G. Since GQ is a conjunct subgraph of G, arc xy C G o . This completes the
proof.
REMARK. It would suffice to replace Condition (1) by the weaker condition:
y G S o - Fn 5)0 implies that there exists a vertex x E Fn S)o such that % >- y.
3. An extension theorem. If X C 5), let the predecessor-set of Z relative to
G — Go denote the set
RELATIVIZATION AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELATIONS 5 5 3
P(χt G~G0) = U Dmn(X,G-G0).n-i
B y a predecessor-sequence p(xo,G-Go) of x0 E 2)Q r e l a t i v e to G - Go i s
meant a maximal regression2 #o, %i,%2» ' ' •» of finite or infinite length, such
that all its vertices except possibly XQ itself are in G - Go; that is, such that
one vertex xn is chosen from the set D~ι(xn_l9 G - Go) for each n > 0, all
xn's being distinct. Let p* (x0, G — Go ) be the set of all vertices of the pre-
decessor-sequence p (xo9 G — GQ ) other than XQ itself. A predecessor-sequence
is termed trivial if and only if p*(#o» G ~ Go) is empty. We have
for all predecessor-sequences p (x0$ G - Go ) of XQ relative to G - GQ. Note
that the elements of the predecessor-set of x0 or of a predecessor-sequence of
x0 are not necessarily ancestors of XQ, although every ancestor of XQ belongs to
at least one predecessor-sequence of XQ (all relative to G — GQ), If >- is not
asymmetric then a source, which has no ancestor, may have non-trivial pre-
decessor-sequences.
Throughout the sequel we suppose that Go is the graph of a subsystem
(® 0 , >-) of the system (3) f>-) the graph of which is G, that Vo is a given
solution of Go, and that 3
T H E O R E M 2. Suppose that:
(1) All non-trivial predecessor-sequences p (XQ, G — GQ ), XQ E 2>0, are
either infinite or, if finite, of odd length if XQ £$OO and of even length if XQ E F O ;
(2) D( Vo, G)n D- 2 n ( F o, G - Go ) = D ( Vo, G)n D " 2 n + l (f 0 0, G - Go ) = 0 /or
σZZ n > 0;
(3) If h > 0 and k > 0 ore o/ ίAe same parity then
D-h(Vo,G-Go)nD-k(Woo,G-Go) = 0,
and if h > 0 and k > 0 are of different panties then
D-h(V0, G - G0)*D-k{V0, G-Go)= D'h{Woo, G - Go ) n D k(W00, G - GQ) = 0;
2 See [4] for definitions omitted here.3 This is a slight modification of the notation of [ 4 ] .
5 5 4 MOSES RICHARDSON
( 4 ) S _ 3 O C P ( 2 ) O , G - G o ) .
Then a solution V of G which is an extension of VQ exists.
Proof. Let
F = F o u ( u D 2n(V09G-G0)) u ( U D-2n+ι(W009G-G0)),
tf00, G - GoW = Wooυl U D-2n+ι(V0,G-G0)) u( U
We shall show that V is a solution of G. Since Go is a conjunct subgraph of G
and Fo is a solution of Go, it follows that FQ is internally satisfactory relative
to G. By (4), 2) = F u I P . By (3), F n IF = 0; hence W = 3 - F. We have only
to prove:
(a) F n Z ) ( F , G ) = 0 ;
(b) WCD(V9G).
Proof of ( a ) . If Λ; E FQ, y G Fo, then x yf- γ since FQ is internally satis-
factory relative to G.
If x e Fo, y G D " 2 n ( F o, G - Go ), then % )f y by (2) .
If x e Vo, y e D'2n+ι (IFoo, G - Go ), then Λ y. γ by ( 2).
If x eD"2n(V0%G - G o ) , y 6 Fo, then jc^-y; for x >- y would imply that
* G D- ι ( Fo, G ~ Go ) contrary to ( 3).
If x £ D"2n ( F o, G - Go ), y G D " 2 m ( FOf G - Go ), then % y; for % >• y would
imply that XeD'2m'ι( Fo, G - Go ) contrary to (3) .
If xeD-2n(V0,G-G0), y e D - 2 m + ι ( i F o o , G - G o ) , then x Jf y; for % >^ y
would imply that Λ; G D " 2 m ( ! F 0 0 , G - Go ) contrary to (3) .
If % E D - 2 m + 1 ( f F o o , G - G o ) , y ^ o , then % >f y; for % >- y would imply
that x e D" ι ( F o, G - Go ), contrary to (3) .
If % e D - 2 m + l ( t F 0 0 , G - G 0 ) , y £ D - 2 * ( F o , G - G o ) , then x^γ; for x ^ ywould imply that x e D"2n~ι (V0$G - GQ) contrary to (3) .
RELATIVIZATION AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELATIONS 555
If x£D-2m+ι(W0Q,G~G0), γeD-2n+ι(W00$G-G0), then x >f y; for
x >- y would imply that x G D"2n (Woθ9 G - Go ) contrary to (3) .
Proof of (b) . If y G IFOO, then there exists an x G Fo such that # >-y.
If y G β - 2 " + 1 ( F o , G - G o ) , then there exists an x G D'2n( VQi G - Go ) such
that Λ; >-y, since y belongs to some predecessor-sequence p(% 0, G - G o ) of
some %0 G Fo and such a predecessor-sequence is infinite or of even length by
(1).
If y eD'2n(W00,G~G0), then there exists an x G D'2n'1 (WOOi G - Go ) such
that x >- y, since y belongs to some predecessor-sequence p(xo9G — Go) of
some %o G $oo* a n c^ such a predecessor-sequence is infinite or of odd length by
(1). This completes the proof.
C O R O L L A R Y . Suppose Conditions ( 1 ) and ( 4 ) of the theorem above, and
that:
( a ) No vertex of any P(XQ9G Λ GO ), XQ G ®O> is adjacent to any vertex of
®o other than XQ; and if XQ and XQ are distinct vertices of ®o then
P(χo,G-Go)nP(xζ,G-Go) = O;
(b) No P(xo9 G — GQ) U ixo), XO G 2>O» contains an odd unoriented cycle.
Then a solution V of G which is an extension of Vo exists.
Proof. We have to show that the hypotheses of the corollary imply those of
the theorem. It will suffice to show that if either ( 2 ) or ( 3 ) are false then
either ( a ) or ( b ) will be violated.
If ( 2 ) were false, there would exist either a vertex
xeD(VOfG)nD'2n(Vθ9G-Go)
or a vertex
yeD(Vo,G)nD'2nU(Woθ9G-.Go).
In either case , the first part of ( a ) or ( b ) is contradicted.
If ( 3 ) were false there would exist either
( i ) a vertex
556 MOSES RICHARDSON
x e D-Hvl, G - Go ) n D-k(wJ00, G - Go)
with h and k of the same parity or
( i i ) a vertex y such that either
Y e D-Hυί, G - Go )n D-Hvi G - Go)
or
y € D - A K 0 , G - Go ) n Z r f e U 0 0 , G - G 0 )
with h and A; of d i f ferent p a n t i e s .
In C a s e ( i ) , Cond i t i on ( a ) would be v i o l a t e d . In C a s e ( i i ) , ( a ) i m p l i e s
i = /. But then P (vι
Q9 G - Go ) u (v^ ) or P (wι
QQ$ G - Go ) u (u)ι
QQ) would contain
an unoriented cycle of odd length h + k contrary to ( b ) .
4. Sinks and inverse bases. We suppose henceforth that 5) - 2)0 C P (5)0»
G - Go ). If H is any conjunct subgraph of G, and % is a vertex of G, let
C - ι ( % , # ) = U D'n(x9H).n-0
That is, C" (x9H) denotes the set of all vertices y of H which chain-dominate
x by means of a chain all the vertices of which, except possibly x, lie in H,
together with x itself; in symbols
If y G C"1 (xfH) and x E C"1 (y, ff), x j/= y9 then Λ; and y are termed cyclically
related relative to //. If y E C"1 (%,//) but x £ C~ι (y,H) then % is termed a
descendant of y relative to H. A sequence #i , %2$ %3> °f vertices of # is
termed a descending sequence of // if xn+\ is a descendant of xn for all τι
(except the last n if the sequence is finite) and if there exists no vertex y
which is a descendant of all xn. If a vertex x of // has no descendant relative
to // then Cml{xtH) is termed an inverse basic set of A/ and x is termed a sink
of this inverse basic set. A subgraph H is termed descendingly finite if every
descending sequence of H is finite. The same inverse basic set may contain
more than one sink; all sinks of the same inverse basic set are cyclically
RELATIVIZATION AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELATIONS 557
related relative to //, and any vertex cyclically related to a sink is a sink of
the same inverse basic set of //.
LEMMA I . 4 If H is descendinglγ finite then every vertex of H belongs to
some inverse basic set of H.
Proof. L e t xι be any v e r t e x of H. E a c h d e s c e n d i n g s e q u e n c e Xι9X29 # 3 , •••
of H b e g i n n i n g with X\ h a s a l a s t e l e m e n t x\. T h e n
- A C V%2» " '9 X 2 ^ V X 3 , Π ) , 9 X\ 1 ^ \ X \ 9 t i )
but
%2 0 C- L ( %i , ), %3 ^ C- l ( %2 , / / ) , . . . , %λ ^ C- [ ( Xλ _ £, / / ) .
Hence
C - 1 ( λ ; 1 , / / ) C C'ι(x2,H) C . C C ^ U ^ t f )
and
λC'ι(xλ,H) = U C'Hxi9H)
is an inverse basic set containing %i of which x\ is a sink.
LEMMA 2. // // is descendinglγ finite, no proper subset B of an inverse
basic set A is an inverse basic set.
Proof. Suppose contrarywise that B were an inverse basic set and a proper
subset of A. Let b be a sink of B and a a sink of A. Then B = C"ι{b9H) and
A - C"1 (α, H). Since B is a proper subset of /4, 6 a and 6 G C'1 (a9H). Since
the sink 6 can have no descendant relative to H9 we have a G C"ι{b9 H)9 other-
wise a would be a descendant of b. Then C"1 (α, //) C C"1 (b9 H), or A C B.
Therefore A = β contrary to hypothesis.
By an inverse basis of // is meant a set S of vertices of H such that (a)
x G S, y G S, x ^ y9 implies that x is not chain-dominated by y relative to H,
4Lemmas 1-5 are duals, in an obvious sense, of Lemmas 1-5 of [4] which are in turngeneralizations of theorems of Kδnig [ 1 , pp. 88-90], for finite graphs. Lemma 2 of [4,p.58l] should be corrected by adding to its statement "if B has a source", and de-leting from the proof all mention of Case (c); this change does not affect the rest of [ 4 ] ,
558 MOSES RICHARDSON
and (b) y G//n S — S implies that there exists a vertex x of S such that x is
chain-dominated by y relative to H (that is, y G C ' 1 (x$ //)).
LEMMA 3. Every descendinglγ finite subgraph H has an inverse basis.
Proof, Let the distinct inverse basic sets of H be Bί9 B2$ , where βχ ^ βy
for i 5^/. (The range of i and / is any lower segment of ordinal numbers, finite
or not.) By Lemma 1, every vertex of H belongs to at least one β t . Let 6; be a
sink of 5 j . Then no b{ chain-dominates bj9 i ^ j . For, if so, i , G C"1 (bj9 //).
Then bi has 6y as a descendant unless bj G C~ι (bi,H); that is, unless 6j and
6y are cyclically related relative to //. In this case,
C'ί(bhH)cC'ι(bJ9H) and C-ι(bJ9H)CC'ι(bhH);
that is, βj = βy, a contradiction. Let S be the set of b^s just chosen, con-
sisting of one sink from each inverse basic set β;. It has just been shown that
Condition (a) of the definition of inverse basis is satisfied by S. That Con-
dition (b) is satisfied follows immediately from Lemma 1.
LEMMA 4. If H has an inverse basis S and b{ G S9 then C" (b(9H) is an
inverse basic set of which b( is a sink.
Proof. If not, 6t has a descendant p in H. That is,
bieC'Hp.H) but P£C'ι(bi9H).
Since p G H n §>, there exists a vertex bj of S such that p G C" (bj9H). Now,
bj φi b[ since p f. C~ (bι9H) Hence b{ chain-dominates p which chain-dominates
bjf so that bi chain-dominates bj since chain-domination is transitive. This
contradicts the fact that b{ and bj both belong to the inverse basis S.
LEMMA 5. Every inverse basis S of a descendinglγ finite subgraph H con-
sists of one sink from each inverse basic set of H.
Proof. By Lemma 4, each vertex of S is a sink of some inverse basic set.
Two distinct vertices of S cannot both be sinks of the same inverse basic set
since, if so, they would be chain-dominated by each other. There remains only
to show that every inverse basic set has a sink in the given basis S, Suppose
β were an inverse basic set none of the sinks of which were in S. Let b be a
sink of β. Since b is not in S, there exists a vertex b' of S such that b chain-
dominates b\ Hence C"l(b9H) C C~ι{b'9H). But b has no descendant relative
RELATIVIZATION AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELATIONS 559
to H s ince b is a sink. Therefore 6 ' and b must be cyclically related relative
to // s ince, if not, b/ would be a descendant of 6. Therefore C" {b'9H)C
C'ι(b9H\ so that C'\b'%h)=C'\b9H) = B. Then b' is a sink of B which does
lie in 5.
5. Progressively finite graphs. A graph H is termed completely descend-
ingly finite if and only if all its closed subgraphs are descendingly finite. A
sequence \xn\ of vertices of H is termed a progression of // if and only if
xn >-#ft + ι, and Cl (xnxn + ι) C H for all n (except the last if the sequence is
finite ). // is termed progressively finite if and only if all the progressions of
// are finite.
LEMMA 6. A necessary and sufficient condition that H be completely de-
scendingly finite is that H be progressively finite.
Proof. If H is progressively finite then it is descendingly finite. If H is
progressively finite then every closed subgraph of H is progressively finite.
Hence if H is progressively finite then it is completely descendingly finite.
If H is completely descendingly finite, there can exist no infinite progression
#ι >~ ^2 >~ y^χn >"•••• For, if so, the subgraph consisting of the vertices
%i and the oriented arcs x^x^x (i - 1, 2, 3, ) would constitute a closed sub-
graph which would not be descendingly finite. This completes the proof.
For example, the graph G of Figure 1 is descendingly finite but not com-
pletely descendingly finite since G — St(y) is an infinite progression.
We suppose henceforth that Cl (G — Go ) is progressively finite^ where Go is
a conjunct closed subgraph of G having the solution Vo. Let5
Woo = D(V0,G0) and Wo = D( Vo, G) υ D'1 ( Vo, G - Go).
Let
G.t = G - S t ( F 0 u f 0 ) .
Let V.i be an inverse basis of G. ι which exists by Lemma 3. For each finite
ordinal number k >_ 1, let
W.k=D(V.k,G.k)uD-HV_k,G_k),
^This is a slight modification of the notation of [4],
560 MOSES RICHARDSON
ad inf.
Figure 1
RELATIVIZATION AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELATIONS 5 6 1
and
- St [ U ( Vmi u Wmi ) I = G Λ - S t ( V.-k u »LΛ
and let F_ _ j be an inverse basis of C^-i
LEMMA 7. G.^-i is α conjunct subgraph of G for all k >, 0.
Proof. Any arc of G not in G.^.i lies in
St f U (F. u IF.,.)]
and hence has at least one endpoint in this star. Thus if x and y are vertices
of G_£_i and x >- y relative to G then x >- y relative to G.£. i since arc %y
cannot lie in the star while both endpoints are in G./ . ι
LEMMA 8. For α/Z A; > 0,
u vmi
o < j X /c+i
is internally satisfactory.
Proof. We prove the lemma by mathematical induction.
For k - 0, we must prove that
( F o u F . 1 ) π D ( F o u H l f G ) = 0.
(1) #, y G Fo implies Λ; > - y relative to G; for % >/- y relative to Go since
Vo is a solution of Go and Go is a conjunct subgraph of G.
(2) x E Vo, y € H i implies Λ; )f y relative to G; for Z) ( VOf G - Go ) n G. t = 0
by definition of G. t while V_χ C G. j .
(3) x E F . i j G F o implies % >f y relative to G; for ZTι ( Vo, G - Go) nG.! = 0
by definition of G. i while F. t C G. ι
(4) Λ;, y G F . i implies x >/- y relative to G; for F. t is an inverse basis of
G.i which implies x >/- y relative to G_ t while G_ i is a conjunct subgraph of G
by Lemma 7.
Assuming that U.^ , Vm( is internally satisfactory, we complete the proof by
562 MOSES RICHARDSON
showing:
( b ) Vmkmln D ( F Φ 1 , G ) = 0 ;
(c ) / U VmλnD(Vmkml9G)=0.\i<k J
If
x e U Vmim y G K.L..1 ,
then % >ί- y; for if % >- y then
y e U
and y jέ G.^ . i , while F-^-i C G.^. i This proves ( a ) . Since G.£. i is a conjunct
subgraph of G, x >- y relative to G, where x, y E Vm/Cmχf would imply x >-y
relative to G./j . i, contrary to the definition of inverse bas i s . This proves ( b ) .
If
then Λ; >/- y; for if x >- y then
%e U D " ι ( H / t G . ^ c U Wmi
so that Λ: G.^.. t , a contradiction. This completes the proof*
It may happen that G_n = 0 for no finite ordinal n, in which case we may let
V = any inverse basis of G , and
W.ω = D( V.ω, G.ω) uD-ι(V.ω,G.ω),
and so on. Transfinite induction shows that if β is an ordinal number for which
RELATIVIZATION AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELATIONS 5 6 3
PLα is nonempty for all α < β then
U V.aa<β
is internally satisfactory. Let the cardinal number of the set 2) be K . Let λ be
the next largest ordinal after those of 3 ( K ) where 3 ( K ) is the set of all
ordinal numbers of well-ordered sets having cardinal number jr Then no
matter how we well-order the elements of ®, its ordinal number is < λ. Well-
order them as follows:
•• Xa»Xa+l$ '• xβ,Xβ + 1, * * Xy 9Xy +\9
V W
Then every vertex of 2) is in some F.ζ or some ίf.ζ with ζ < λ. Let K be the
lowest ordinal for which G.κ = 0. Then every vertex of G is ultimately used up
in some fiζ or ULζ, ζ < K. We have then the following theorems in which we
let
V= U V.a:0 < α<κ
THEOREM 3. If Vo is a solution of the subsystem (®o>^~) °f t n e system
(2), >•), and if the graph C\(G~G0) is progressively finite, and every vertex
of G - Go is in the predecessor-set P (®o» G - Go), then V is a maximally in-
ternally satisfactory set.
THEOREM 4. //, in addition to the hypotheses of Theorem 3, there exist
inverse bases V.afor each CC with 1 <_ Cί < K such that
D-l{V0,G-G0)CD(V,G) and D" ι ( V.a, G.a) C D( V, G),
then V is a solution of G and an extension of VQ.
THEOREM 5. //, in addition to the hypotheses of Theorem 3, >- is sym-
metric, then V is a solution of G and an extension of VQ.
The proofs of Theorems 4 and 5 are immediate. 6
6 As to Theorem 5, the fact that if >~ is symmetric then every maximally internallysatisfactory set is a solution is established in [2],
564 MOSES RICHARDSON
THEOREM 6. If the hypotheses of Theorem 2 are satisfied, then so are the
hypotheses of Theorem 4.
Proof. Let
V.i = D-2( Vo, G - Go ) u D - ι ( IF,,,,, G-Go),
V.i = D-3(V0, G - Go ) u D - 2 ( I F 0 0 , G - Go ) = D ' ι ( V.u G - Go) u D( F ^ , G - G o ) ,
F.j =* Z)"4( F o , G - Go ) u D-'dT'oo. G - G o ) ,
and so on. Then
= U F.α
0< α
and
H7= U !F.α = IF0 0uUD-2"+ 1(F0,G-Go)uUD-2 n(IF0 0,G-G0),0 £ α
so that V is a solution.
There remains to show that V.a is an inverse basis of G. α . Clearly, neither
of two distinct vertices x9 y^V.a chain-dominates the other by virtue of the
parity restrictions (2), (3) of Theorem 2. We must show now that every vertex
γ oί G.a chain-dominates some x of K α . This is obvious since by (4) every y
belongs to P ( $ 0 , G - G 0 ) , t h a t i s> t o s o m e D'n(Vθ9G—GQ) or to some D'n(W00,
G - Go), that is, to some H α or W_a. By (1) it is clear that every D"l ( H α , G.α)C
D(VSG). This completes the proof.
The example of Figure 2 shows that Theorem 4 is less restrictive than
Theorem 2. For
but an extension exists and the hypotheses of Theorem 4 are satisfied.
6. Some extension theorems. If H is a subgraph of G, let
K(x,H)=D(x,H) u Z ) - 1 ( % , # ) ,
RELATIVIZAΉON AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELAΉONS 565
Figure 2
566 MOSES RICHARDSON
let
K(X,H)= U Kix.H), X C S ;x£X
let
Kn(X,H)=K(Kn ι(X9H),H) lorn > 1.
That is, Kn(X,H) denotes the set of vertices of H connected to vertices of X
by unoriented one-dimensional chains of length n.
LEMMA 9. If 2) — S)o C P ( Vo$ G - Go ), then every inverse basic set B of
G-j-i has a sink in K2 ( F.j , G./.i), i >_ 0.
Proof, Suppose i = 0. Each sink y of B chain-dominates some vertex of
Vo since 5) - ®0 C P ( Vo, G - Go )• Consider the chains of minimum length m by
which y chain-dominates vertices of FQ. Then m >_ 2 since /£( PQ, G — GQ ) n
G_ι = 0 . Suppose the lemma were false, so that m > 2, and let yQ be a sink of
B for which this minimum length is attained. Then there exist distinct verticesχl$ χ2$ 9 xm- i of G ~ GQ such that
ϊo>- xmΊ >~xm-2 >- •*• >- x l >• v{
for some t>0 G FQ. Then either
(1) Λm-iJfG.i,
or ( 2) x m . i G G. i and is a descendant of yQ,
or (3) %m-i G G β l and is cyclically related to y0 relative to G.χ.
In Case (1), xm.\ £ Vo v WQ so that
xm-i£Kl(V0,Gmi) and yQ eKHV^G.,)
contrary to the supposition that the lemma is false. In Case (2) yQ is not a
sink of B since a sink can have no descendant. In Case (3), m is not the mini-
mum length since xmm\ would be a sink of B which chain-dominates vJ
Q by means
of a chain of length m — 1.
Now suppose i > 0. Let B be an inverse basic set of G_, _i Each sink y of
B chain-dominates some vertex of Vm( since Vmj is an inverse basis of G.t 3 G.t . χ
RELATIVIZAΉON AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELATIONS 567
Consider the chains of minimum length m by which y chain-dominates vertices
of Pίj. Then m >_ 2 since K{ V-i, G.j) n G-i-ι ~ 0. Suppose the lemma were false,
so that m > 2, and let yQ be a sink of B for which this minimum length is at-
tained. Then there exist distinct vertices χu x2, 9 xm. i of G_; such that
y o >- %m.i >- Λm.2 > >- xι >- ^ for some i Λ G K.j .
Then either
(1) Xn.iϊG.i.i,
or ( 2) χmm j G G. t _ L and is a descendant of y 0,
or (3) xm-ι G G. t _ i and is cyclically related to yQ relative to G. . 1#
In Case (1),
and hence %m.\ G Kj u W.i and hence %m. i G X l ( PI;, G.j) so that yQ G A 2 ( Kj,
G.j . i ) contrary to our supposition that the lemma is false. In Case (2), y is
not a sink of B since a sink has no descendant. In Case (3), m is not minimal
since xm.\ would be a sink of B which chain-dominates v { by means of a chain
of length m — 1. This completes the proof.
The example of Figure 3 shows that we must take Kn in the unorίented
sense; for here v]γ G P ( F o , G - Go ), in fact υ\ G D'4 (Vo , G - Go ) but υ\<£
D'2 ( Vo, G ~ Go ) although ^ G K2 ( Vo, G - Go )."
A subgraph // of G is termed progressively bounded at the vertex y if all
progressions of H beginning with γ have lengths forming a bounded set of
natural numbers. H is termed progressively bounded if it is progressively bound-
ed at each of its vertices.
LEMMA 10. // S - ® 0 C P ( F o , G - G o ) and if Cl (G-Go) is progressively
bounded then every vertex y of S• - S o i\s arc element of F. t or $ l t for some finite
ordinal i.
Proof. Every vertex y of 2) — 2)0 i s a n element of C" (v^Qi Cl (G — Go )) for
some t;^ G Vo by hypothesis. Consider all progressions of Cl (G - Go ) beginning
with y and ending with elements of Vo. Their lengths have a least upper bound
M(y) by hypothesis. By Lemma 9, we may select inverse bases
V.i.ιCK2(V.i,G-G.i.ι), »>0.
568 MOSES RICHARDSON
-1
Figure 3
RELATIVIZATION AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELATIONS 569
Since V.i is an inverse basis of G_ u there exist progressions starting with y
and ending with elements of F. t unless y is in Vmjc or W^ with k <^ 1. All such
progressions have lengths < M(y) - 2. For if there existed a progression from
y to some v t 6 F_ ι of length > M(y) - 2, there would be a progression from y
to some element of Vo of length > M(γ) since there exists some progression
from v t to some element of Fo and its length must be > 2 because Vm i CG.i
Similarly the lengths of all progressions from y to elements of V_ι must be
<_ M(y) -2i. But this can be >_ 0 for only a finite number of values of i. Hence
there exists a value of i for which y chain-dominates some element of Vm( by
means of a progression of length 0 or 1; that is, y is in either Vmj or ίF.j .
By a relative cycle (of Cl (G - Go ) mod Fo with modulo 2 coefficients) shall
be meant an unoriented one-dimensional chain lying in Cl (G — GQ) except for
its set of boundary vertices (possibly empty; that is, absolute cycles are in-
cluded among the relative cycles ) which lies in Vo.
THEOREM 7. Suppose that Vo is a solution of Go such that'
(1) Cl (G — GQ) is progressively bounded^
(2) each vertex of every K n" (VQ, G — Go ) is dominated by some element
(3 ) Cl ( G — GQ ) contains no relative cycle of odd length;
( 4 ) 3 ) ~ 5 ) o C P ( F o , G - G o ) .
Then there exists a solution V of G which is an extension of Vo.
Proof. Choose V.i as in Lemma 9. To show that V = U Q < i V_ι is a solution
of G we have, by Theorem 4, only to show that
D - ι d / 0 , G - G 0 ) c D ( F , G ) a n d D - 1 ( F . i , G . i ) C D ( F , G ) f o r i > 1.
L e t
Then
£K2n-ι(viQ,G-G0)
570 MOSES RICHARDSON
for some / and n by virtue of the way in which the F.^ were chosen. By ( 2 ) , w is
dominated by some vertex x of 5) — 2)0. If x G F, there is no more to prove. If
x e S - F , then
for some k and m. Hence there exists a relative cycle of odd length, contrary to
(3). This completes the proof.
THEOREM 8. Let V be any maximally internally satisfactory set containing
Fo such that:
(1) every v G V belongs to K2n(V0,G - Go) for some n > 0;
(2) each element of K m" (VQ9 G — GQ)9 for every m > 0, is dominated by
some element of 5) - S o
(3) Cl ( G - Go ) contains no relative cycle of odd length.
Then V is a solution of G.
Proof. Let
γ E ( 5 ) - S 0 ) n ( % - V ) .
We sha l l show that there e x i s t s a n % 6 F such that x >- y. Since V i s maximally
internal ly s a t i s f a c t o r y , F u ( y ) i s not internal ly s a t i s f a c t o r y . Therefore e i ther
( a ) some v >- y, or ( b ) some v -< y. In C a s e ( a ) , there i s no more to prove.
In C a s e ( b ) ,
y eK2n"ι(V0,G-G0) for some τι > 0.
By ( 2 ) , there exists an x G 5) — S o such that a; >- y. If x G F f there is no more
to prove. If not, that is if x G (2) - ® 0 ) n ( S - F ) , then F u (%) is not internally
satisfactory. Therefore there exists a v G F such that either Λ; >- v or x -< t>.
In either case,
xeK2mml(Vθ9G-Go)
for some natural number m But this together with
y£Kanml{V0,G-G0)
RELATIVIZATION AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELATIONS 5 7 1
and x >- y imply that there exists a relative cycle of odd length contrary to
(3). This completes the proof.
COROLLARY. The hypotheses of Theorem 8 imply that
V0,G-G0) and W = §-V =
Proof. We have
Vcυκ2n(V0,G~G0)=E,
and
W = 2> - V C U K 2 m ' ( Vo, G - Go ) = Ω.
Furthermore
K2n ( Vo, G - Go ) n K2m-ι ( Fo, G - Go ) = 0 ,
for, if not, there would exist a relative cycle of odd length. Thus we have
£ n Ω = 0 , E u Ω = S , F c £ , W'Cίl, F u l F = S),
This implies E - V9 Ω = W as follows. Let e E £. Then e E 5) which implies that
either e E F or e E IF. But e E ίP would imply that e E Ω contrary to £ n Ω = 0.
Therefore e 6 F , Hence E C V and therefore E = V. Similarly Ω C IF and hence
Ω = IF. This completes the proof.
Thus Theorem 8 resembles Theorem 2, except that now the parity restric-
tions are on the unoriented chains rather than on the oriented ones, and we do
not restrict the sets Kn(WOOf G -Go).
The examples of Figures 4-6 are covered by Theorem 8 but not by Theorem
2. In Figure 4,
w I e D ( Vo, G) n D -1 (Wo 0, G - G 0 ) £ 0
violating hypothesis 2b of Theorem 2, but the extension exists under Theorem
8. In Figure 5,
v _\ G D'2 (Wo o, G - G o ) n D" ι (Wo 0 , G - G 0 ) ^ 0
violating the second part of the hypothesis 3b of Theorem 2, but the extension
572 MOSES RICHARDSON
Figure 4
RELATIVIZATION AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELATIONS 573
Figure 5
574 MOSES RICHARDSON
W- 1
Figure 6
RELATIVIZATION AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELATIONS 5 7 5
exists under Theorem 8. Note also that an odd relative cycle exists mod Go but
not mod VQ, In Figure 6,
v\ eD-2(V0,G-G0)nD-2(W00,G-GQ) ^ 0
and
j2£D-l(V0,G-G0)nD-l(Woo,G-G0)J-0Wn
both violating hypothesis 3a of Theorem 2, but the extension exists under
Theorem 8.
Let μ (X9 G - Go ) denote the set of vertices of G - Go connected to X by
an unoriented chain of minimal length h, where X C 2). Then
μh(X, G - Go ) n μk (X, G - Go ) = 0 for h £ k.
By μ° (X9 G - Go ) is meant Z.
THEOREM 9. Lei F o be a solution of Go where Go i's α conjunct subgraph
of G. Let WQQ = ®o — o β ^ suppose that every vertex of 3 — 5)0 Ϊ'S connected
to 5)0 ^y some unoriented chain. Let
= U μ 2 " ( J / 0 , G - G 0 ) u Un=o m = i
F = U J u 2 ί l - 1 ( F 0 , G - G 0 ) u U μ 2 m ( I F o o > G - G o ) .
w = l m = o
Suppose that:
(1) every element of W is dominated by some element of V;
(2) μh(V0,G-G0)nμk{W00,G-Go)=0
if h and k have the same parity,
(3) no two elements of the same μ2n"1 (tFOo, G - Go ) are adjacent;
(4) no two elements of the same μ n{ VQ, G — GQ ) are adjacent.
Then V is a solution of G which is an extension of Vo .
576 MOSES RICHARDSON
Prόυf. Clearly 5) = V u Ψ and WCD{V9G). Also (2) implies V n IF = 0.
There remains only to prove that no two elements of V are adjacent.
If *, y E μ2n{V0,G- Go ) then x >f y by (4).
If *, y £ μ2m-1 (iF0o, G - Go ) then x + y by (3).
Let
^ e μ 2 n ( 7 0 , G - G 0 ) , y£μ2m(V0,G-G0), mfn.
Suppose m > n. If x and y were adjacent then y & K2n ι (V0,G — Go). But
2re + 1 < 2m, contradicting the minimal property of μ2m( Vo, G — Go ). A similar
proof is obtained if m < n.
If
* e μ a B - | ( I P o o , G - G o ) , y £ μ a m " ι ( l P o o , G - G o ) , m Φ n ,
then Λ: and y are proved non-adjacent as in the preceding paragraph.
Let
χeμ2n(Vo,G-Go),y£ μ2P ι(W00,G - Go )
and suppose x were adjacent to y. Then
x&K2P(W00,G-G0) or x£K2P-2(W00,G-G0).
Since Λ: is connected to Woo, it is minimally connected to Woo. That is, either
(a) * € μ 2 Λ ( l F O o , G - G o )
or
(b) x e μ
2 h ι{WQ0iG~G0)
for some A. In Case ( a ) , Condition ( 2 ) would be violated. In Case ( b ) , h = p
since either h < p or h > p would violate the minimal property of some μ.
But h =p contradicts Condition ( 3 ) . This completes the proof.
THEOREM 10. Let VQ be a solution of a conjunct subgraph Go of G such
that every vertex of 5) - ®o *5 connected to Vo by some unoriented chain. Let:
RELATIVIZATION AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELATIONS 577
(1) no two elements of the same μ2ι( Vθ9 G — Go), i > 0, be adjacent;
(2) x £ μ ι" (Vo$ G - Go) imply that there exists a j >_0 such that
x -< y for some γ € μ2} ( Vo, G - Go).
Then
V' = U μ2i(V0,G-G0)ϊ = 0
is a solution of G which is an extension of Vo.
Proof. Every element of 5) — 2)0 not in V must be in
IF= U ^ 2 i - l ( F 0 f G - G 0 ) .
Clearly
5) = F u IF and Fn W = 0 .
Also (2) implies W C D (VfG). There remains only to prove that V is internally
satisfactory.
Let
x e μ
2i( Vo, G - Go ), y £ μ2H Vo$ G - G o ) , i ί j .
Suppose i < j . If x were adjacent to y, then y £ K2i ι ( Vo, G - Go ). But 2i + 1 <
2/, contradicting the minimal property of μ2Π VOf G - G o ) . A similar proof
holds if i > j .
Let x9 y G μ ι (Vo$ G - Go). If i > 0, (1) implies that x and y are non-
adjacent. For i = 0, this follows from the facts that Vo is a solution of Go and
that Go is a conjunct subgraph of G. This completes the proof.
The conditions of Theorem 10 do not prohibit entirely the existence in
Cl (G — Go) of adjacent vertices of IF, of odd unoriented cycles, or of transitive
triples. For example, the graph in Figure 7 permits an extension by Theorem 10
and includes the three cited phenomena. Theorems 7-10 may be regarded as
variants of Theorem 2.
7. Dual and alternating procedures. Let G t be a conjunct subgraph of G.
578 MOSES RICHARDSON
-1
Figure 7
RELATIVIZATΊON AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELAΉONS 579
If x £ 5), let D(x$ G — Gι) denote the set of all vertices γ of G - G\ such that
x >- y. If X C 2), let
,G~G 1 )= U D C ^ G - d ) .
Λ G X
For n > 1, let
By the successor-set of /? relative to G - Gι is meant the set
oo
SiX G-G^- U D^ί^G-Gx).
THEOREM 11. Let Gγ be a conjunct subgraph of G, Vγ a solution of G\9
Wί = ®! - Fi where ® t = S) n G t . Suppose that:
( 1 ) for et>ery τι > 0,
F x n Z) 2 n + r ( Vt, G - G r ) = n D2 n (IF ι , G - G % ) = 0
( 2 ) if h > 0 αrac? A; > 0 are of the same parity^ then
Dh(Vι,G-G1)nDk(Wι,G-Gι) = 0;
if h > 0 and k > 0 are of different parities then
Dh(Vι,G-Gι)r\Dk{Vι,G-Gι)=Dh(Wι,G-Gι)nDk(Wι,G-Gι)=O;
(3) S - S 1 C S ( S 1 , G - C ) .
Γλerc there exists a solution V of G which is an extension of V\.
Proof. Let
= F , u U D2n{VltG-Gι)u U D2m ι(Wu G - Gι).n = l m = l
580 MOSES RICHARDSON
We must show:
( a ) F n D ( F , G ) = 0 ;
(b) 5 > - F c D ( F , G ) .
(a) If x G F j , γ G Vx, then x >jί- y since Vι is internally satisfactory rela-
tive to G.
If x£Vl3 y£D2n{VuG-Gι) thens . j f y; for* >- y would imply D ( F,, G) n
D a B ( ί Ί . G - G ι ) ^ 0 contrary to (1) .
If xG Fi , y e D ^ - ' d ί Ί . G - G ! ) then * y- y; for * X y would imply D ( Vι,
G)n D2m ι(WuG -Gι) ^ 0 contrary to (1) .
If xeD2n(VltG-Gι\ y€Vi, then * )/- y; for * >^ y would imply γ£
D^+HVuG-d) contrary t o ( l ) .
If * e D a n ( ^ l f G - G i ) , y e J 9 2 m ( f 1 , G - G l ) then c ^ y ; for * > y would
imply y e D2n+ι (Vlt G - G t ) contrary to (2) .
If x e D 2 " ( Ft, G - d ), y e D ^ ^ (Wt, G - G t ) then x Jf y; for * V y would
imply y e D 2 n + 1 ( F 1 , G - G x ) contrary to (2) .
If xeD2mml(WltG~Gι), y G F t then a; )f y; for * > - y would imply y G
D ^ d f Ί . G - G t ) contrary t o ( l ) .
If x € D2"1"1 (IΓX, G - Gi ), y G Z) 2 n ( Vx, G - G t ) then x ^- y; for * >• y would
imply y G D 2 m ( I F 1 , G - G 1 ) contrary to (2) .
If % G D 2 m - l ( l F 1 , G - G 1 ) , yeD^-^W^G-Gi) then x >/-γ; for x >- y
would implyy G D2m(Wlt G - Gγ) contrary to (2) .
(b) Let
ι U U D 2 n - 1 ( F 1 , G - G 1 ) υ U D 2 m ( I F i , G - G t ) .n=l m = l
By (3) ,
By (1) and (2), V n W = 0. Hence IF = 2) - F.
If y G IFi, then there exists an x G Vι such that x >- y.
If yG D 2 n - 1 ( F 1 , G - G 1 ) then there exists an x G Vγ u D 2 π ( F t , G - Gx
such that x >- y.
RELATIVΪZATION AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELATIONS 5 8 1
If y £D2m(Wl9G~Gι), then there exis t s an x G D2m"1 ( Wx, G - Gι) such
that x >- y. This completes the proof.
COROLLARY. Let Gx be a conjunct subgraph of G, Vι a solution of Gx.
Suppose that:
( a ) no vertex of any S (xι,G - Gχ)9 x\ £ ® i , is adjacent to any other
vertex of 2)χ; and if X\and x^are any two distinct vertices of^i then
( b ) no
S(xlt G - Gι) u (xι), xχ£^>ι,
contains an unoriented cycle of odd length;
( c ) S-^CSI^G-G,).
Then there exists a solution of G which is an extension of V\.
Proof. C o n d i t i o n ( c ) i s i d e n t i c a l with ( 3 ) of the theorem. We h a v e only to
s h o w t h a t ( a ) and ( b ) imply ( 1 ) and ( 2 ) ; t h a t i s , t h a t if e i t h e r ( 1 ) or ( 2 ) were
f a l s e t h e n ( a ) or ( b ) would be v i o l a t e d .
If ( 1 ) were fa l se t h e r e would e x i s t e i t h e r
( i ) a v e r t e x x € D ( F 1 , G ) n D 2 " ( F ι , G - G ι ) ,
or ( i i ) a v e r t e x r G D ( K ι $ G ) n D 2 " - ι ( l F ι , G ~ G ι ),
or ( i i i ) a v e r t e x z G Vx n D 2 n + 1 ( V,, G - Gt),
or ( i v ) a v e r t e x u € Vx n D2n(WuG - Gx).
In C a s e ( i )
xeS{v[9G-Gx)r\S{v{tG-Gx)
and by ( a ) , i - j . But then there e x i s t s an unor ien ted cyc l e of odd leng th in
S(vι
ι<$ G - G\ ) u {v[ ) contrary to ( b ) . In C a s e ( i i ) , the s e c o n d p a r t of ( a )
i s c o n t r a d i c t e d . In C a s e s ( i i i ) and ( i v ) , the first p a r t of ( a ) i s c o n t r a d i c t e d .
If ( 2 ) were f a l s e , t h e r e would e x i s t e i t h e r
582 MOSES RICHARDSON
( i ) a vertex
x £ Dh ( Vx, G - G ι ) n D k (W ί, G - G t )
for some h9 k of the same parity,
or ( i i) a vertex
y EDh(v[9G - Gv)n Dk{υ{9G - Gt)
for some A, A; of different parities,
or (i i i) a vertex
zeDh(w[$G-Gι)nDk(w{fG-Gι)
for some ^, A; of different par i t ies . In Case ( i ) , the second part of ( a ) is con-
tradicted. In C a s e s ( i i ) and ( i i i ) , ( a ) implies i -j and then ( b ) is contradicted.
Now suppose Go is a nonempty conjunct subgraph of Gι and let Vo be a
solution of GQ. For each natural number n, let £271-1 be constructed by adjoining
to G2n-2 the vert ices of P (5)2n-2$ G - ^2^-2)* where S j = S) n G t , and taking
the juncture; that is ,
£2/2-1 " ^2n-2 U P ( >2n-2f G - G 2 Λ - 2 )
Similarly let
G2n = G2n-ι u 5(5)2n-U^ ~ £2^-1).
Then each Gf is a conjunct subgraph of G + i For x% y G 5)j , # >- y relative to
Gj+i implies Λ; >- y relative to Gz since at least one endpoint of every arc in
Gj+ t - Gj is not in G;.
If GQ intersects every component of G, then
3 = U 5)f .
For then every vertex of G is joined to some vertex of Go by a finite unoriented
chain and therefore lies in some G;. In particular, this is true if G is connected.
THEOREM 12. Let Go be a conjunct subgraph of G which intersects every
RELATIVIZATION AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELATIONS 583
£35#~
3 3 3
^ 3 2 ^34 #36
3 2 2
5 23 6 2 l Ί l
^ * "
< f <6 14
' 1 3
Figure 8
584 MOSES RICHARDSON
component of G, let Vo be a solution of G o , and let Gι9 i >_ 1, be defined as
above. Suppose that for every even i, G( satisfies Conditions ( 1 ) , ( 2 ) , ( 3 ) of
Theorem 2 relative7 to Gj+ l 9 and that for every odd i9 G t satisfies Conditions
( 1 ) and ( 2 ) of Theorem 11 relative to G + i. Then there exists a solution of G
which is an extension of Vo.
Proof. The solution Vo of Go can be extended stepwise to a solution Vγ of
£i> ^2 °f ^29 ' ' '» Vi °f £/» by Theorems 2 and 11 applied alternately. Hence
U°t.o Vι i s a solution of G.
For example, in Figure 8, G; has the set of vert ices 2); = [g.j,, g 2 , g 3 , ].
Then
F l = [ £ l 2 , S l 4 » - - ] u F 0 , ^2 = [ g 2 1 , g 2 5 , ••• i § 2 4 ^ 2 8 > # ] U F 1 »
Theorem 11 is a sort of dual to Theorem 2. Theorem 12 merely uses the
procedures of Theorems 2 and 11 in alternation. Similar processess dual to those
of other preceding theorems can be introduced so as to yield extensions in the
direction of successor-sets rather than predecessor-sets, and similar alternating
procedures can then be used.
T h a t i s , w i t h G(+χ in t h e r o l e of G in T h e o r e m 2 .
REFERENCES
1. D. Kδnig, Theorie der endlichen und unendlichen Graphen, Leipzig, 1936.
2. J. von Neumann and O. Morgenstern, Theory of games and economic behavior,Princeton 1944; 2nd edition 1947.
3. M. Richardson, On weakly ordered systems, Bull. Amer. Math. Soc. 52 (1946),113-116.
4. , Solutions of irreflexive relations, Ann. of Math. 58 (1953), 573-590.
5. , Extension theorems for solutions of irreflexive relations, Proc. Nat.Acad. Sci. 39 (1953), 649-655.
BROOKLYN COLLEGE
AN INEQUALITY FOR SETS OF INTEGERS
PETER SCHERK
Small italics denote nonnegative integers. Let A = {a ί, B = { b ! , be sets
of such integers. Define A + B = 1 a + b \ and put
Thus
/4 ( n ) = /I (0, n ) and A(mtn) ** A(n) —A ( m ) if m < n.
The following estimate is well known:
LEMMA. If m < k< n, n fi A + B, then
(1) k-m>_A(n-k-l, n-m-D +B(m9 k).
Proof. If b = rc —α, then n = α + !) E/l + β . Hence the 4 (/z. — Λ; — 1, n — m — 1)
numbers rc — α with m < n —a <^k and the B (m9 k) numbers b satisfying m < b <_ k
are mutually distinct. The right hand term of (1) gives their total number. It is
not greater than the number k — m of all the integers z with m < z <_ k.
The most important result on A + B is due to Mann [2] : Let n fc C = A -f β.
Then there exists an m satisfying 0 < m < n and n — m ft. C such that
C{m9n) ^Ain-m-D + Bίn-m-D.
I wish to prove a less well known inequality which is implicitly contained
in [4] and in a paper by Mann [3], The present proof uses an idea by Besicovitch
and is rather simpler than Mann's method [cf. l ]
THEOREM 1. Let
(2) xeA ( % = 0 , l , 2 , . . .,A; h > 0 ) ,
Received December 29, 1953.Pacific J. Math. 5 (1955), 585-587
585
586 PETER SCHERK
( 3 ) 0 G B or 1 G B9
( 4 ) AΪBCC, n £ C.
Finally let
(5) C(n) < AU-Ό + BU).
Then there is an m satisfying
(6) m £ C9 0 < m < n - h - I
such that
( 7 ) C(m9n) >_A(n-m-l) +B(m,n).
We note that ( 7 ) is trivial but useless without the second half of ( 6 ) .
Obviously, ( 2 ) - ( 4 ) imply m > h if 0 G B and m > h + 1 if 1 G B.
Proof. Instead of ( 3 ) , we merely use the weaker assumption that B is not
empty. Let 60 denote the largest b <^n. Thus B(bo9n) = 0. Since C contains
the integers 60 + a with 0 < a <^ n — bo, we have
(8) C(bo,n) >_A(n-b0) > A (n - bQ - 1) = A U - b0 - 1) + B (bθ9 n).
From ( 5 ) and ( 8 ) , b0 > 0. By ( 2 ) , the numbers bOf b0 + 1, , b0 + h lie in
A + B C C. Hence n jέ C implies 6 0 <. τι - A - 1. Thus
( 9 ) 0 < 60 < / ι - A - l .
By ( 2 ) , b0 £C. Let m denote the greatest z < b0 with z j£ C. If no such
z exists, put m = 0. Applying ( 1 ) with k = &o> w e obtain
(10) C ( m 9 b o ) = b 0 - m > A ( n - b 0 - 1 , n - m - 1 ) + B ( m 9 b 0 ) .
Adding ( 8 ) and (10) , we obtain
C(m9 b0) + C(bθ9n) >_ A (n - b0 -~ I) + A {n - bo - I, n - m - I)
+ B(m9 b0 ) + B(bθ9n)f
that i s (7). By ( 7 ) and ( 5 ) , m > 0. Hence m fi C. Final ly ( 9 ) and m < b0 imply
m < n - h — 1.
The following corollary of Theorem 1 was proved in a different way by Mann.
AN INEQUALITY FOR SETS OF INTEGERS 587
T H E O R E M 2. Suppose the sets A, B, C satisfy the assumptions ( 2 ) - ( 4 ) .
Let 0 < 0Ci < 1 and
(11) Λ{x) > ux{x + 1 ) U = A + 1, A + 2 , . . . , Λ ) .
( 1 2 ) C U )
Proof . By ( 2 ) , OeΛ. F u r t h e r m o r e , ( 1 1 ) and ( 2 ) imply ISA. H e n c e , ( 3 )
i m p l i e s 1 £ C. T h u s our t h e o r e m i s t r u e for n = 1. S u p p o s e i t i s p r o v e d up to
τ ι - 1 > 1.
If C(n) >_A(n-l) + B(n), t h e n ( 1 1 ) with x = n~l y i e l d s ( 1 2 ) . T h u s we
may a s s u m e ( 5 ) . C h o o s e m a c c o r d i n g to T h e o r e m 1. By ( 6 ) , n — m — 1 >_ A + 1.
H e n c e , by ( 7 ) , ( 1 1 ) , a n d our i n d u c t i o n a s s u m p t i o n
C(n) >_C{m)+A{n-m~l)+B(m9n)
>_C(m) + α t (7i - m ) + B(m9n)
The case h = 0 of Theorem 2 is due to Besicovitch [ 1 ]. Obviously, this
theorem can be extended to the case that 0 j£ B$ B in) > 0.
A recent result by Stalley also follows readily from Theorem 1.
REFERENCES
1. A. S. Besicovitch, On the density of the sum of two sequences of integers, J.
London Math. Soc. 10 (1935), 246-248.
2. H. B. Mann, A proof of the fundamental theorem on the density of sums of sets
of positive integers, Ann. of Math. 43 (1942), 523-527.
3. , On the number of integers in the sum of two sets of positive integers,
Pacific J. Math. 1 (1951), 249-253.
4. P. Scherk, Bemerkungen zu einer Note von Besicovitch, J. London Math. Soc.
14 (1939), 185-192.
5. R.D. Stalley, A modified Schnirelmann density, Pacific J. Math. 5(1955), 119-124.
UNIVERSITY OF SASKATCHEWAN
ON INFINITE GROUPS
W. R. S C O T T
1. Introduction. Several disconnected theorems on infinite groups will be
given in this paper. In V 2, a generalization of Poincare"s theorem on the index
of the intersection of two subgroups is proved. Other theorems on indices are
given. In § 3 , the theorem [ 3 , Lemma 1 and Corollary l ] that the layer of ele-
ments of infinite order in a group G has order 0 or o(G) is generalized to the
case where the order is taken with respect to a subgroup. In v 4 , it is shown that
the subgroup K of an infinite group G as defined in [ 3 ] is overcharacterist ic
[ 2 ] . In § 5 , characterizations are obtained for those Abelian groups G, all of
whose subgroups H (factor groups G/H) of order equal to o{G) are isomorphic
to G (in this connection, compare with [ 7 ] ) . Again the Abelian groups, all of
whose order preserving endomorphisms are onto, are found ( s e e [ 6 ] ) .
2. Index theorems. If // is a subgroup of G, let i(H) denote the index of
H in G. The cardinal of a se t S will be denoted by o(S).
THEOREM 1. Let Ha be a subgroup of G, α E S. Then
£(Π//α) <Πi(Ha).
Proof.
gtg'2l£ Γi Ha
if and only if
' e f f α for all α G S.
Thus each coset of Π//α is the intersection of a collection of sets consisting
of one coset of Ha for each (X, and the conclusion follows.
COROLLARY 1. (Poincare) The intersection of a finite number of sub-
groups of finite index is again of finite index.
Received December 23, 1953.
Pacific J. Math. 5 (1955), 589-598
589
590 w. R. SCOTT
C O R O L L A R Y 2. // i(H)=B9 then G has a normal subgroup K such that
<BB.
Proof. Let N(H) denote the normalizer of H, and Cί(H) the conjugate
class of H. Then
HCN(H), o(Cl(H))=i{N{H)) <B.
Thus if K is the intersection of the conjugates of //, Theorem 1 gives i(K) < B .
REMARKS. For every infinite cardinal A9 there is a simple group G of order
A (for example, the *'alternating" group on A symbols). Thus G has no sub-
groups of index less than or equal to B if 2 < A. In particular, if A is such
that B < A implies 2 < A9 then G has no subgroup of index less than its
order A, This is in sharp contrast to the behaviour of Abelian groups, which
have 2A subgroups of index B for Ko <_ B <_ A9 A > Xo [4] , It is an unsolved
problem as to whether there exists a group G of order A with no subgroups of
order A, for A > Ko.
Let U denote the point set union, and + and Σ direct sums ( the latt ice union
of subgroups will not be u s e d ) . If Γ is a nonempty subset of a group G, let
iR( T) = min o (S) such that \JTxa ~ G9 C ί E S .
Define iL(T) similarly, and let i(T) be the smaller of iR(T) and i^i T),
T H E O R E M 2 . If Hι9 ϊ = 1 , * ,n9 are subgroups of G such that i ( / / j ) > ^
A > Ko, then i ( U ^ ) > A.
Proof. The theorem is true for n = 1. Induction on n If, contrary to the
theorem, i(UHi) < A, then, say,
( n
U Hi
- „ i=i
with o ( 5 ) < /4. S i n c e ί(Hι) >_ A9 t h e r e e x i s t s a n % 6 G s u c h t h a t
is empty. Hence
ON INFINITE GROUPS 591
Therefore
n n I n \UfyC U Hi(e υ(ϋaxax'ι)) = U U HΛxβ,
i = i *=2 / 3 G s ' \ i = 2 /
where o ( S ' ) < /4. Hence
= U ( U / / Λ * α = U U U Hixnxa= U U/ αCS / 3 £ s ' j=2 P G "
This contradicts the induction hypothesis. Hence the theorem is true.
REMARK. For every infinite cardinal A, there is a group G of order A9 con-
taining an increasing sequence { Hn } of subgroups, each of index A9 such that
m n «= G.
Let I/A = 0 for A > Ko.
THEOREM 3. // Hi is a proper subgroup of G9 (i - 1, , n) and Σ 1/i (Hi )<
1, then UHt £ G.
Proof. L e t H\, , HΓ have finite index, the others infinite index (if r = 0,
the theorem follows immediately from Theorem 2 ) . L e t
D = Π //; .1
Then D has finite index in G, and it is well known that (UΓ Hi) n Dx is empty
for some x G G. Hence, if U? fff = G, then Dz C U ^ j ffίf whence U ^ + 1 ^ has
finite " index" in contradiction to Theorem 2. Therefore U" Hi ^ G.
3. Layers. Let T be a subset of G, and let n be a positive integer. Let
L i n , T ) = \ g \ g n £ T , g
r £ T f o r 0 < r < n \ ,
Moo, 7') = U | g Λ ( ί 7 ' , n = l , 2 , . . . } .
For T - e, the L(n, Γ) have been called layers. The following theorem general-
izes [3, Lemma 1 ]•
THEOREM 4. Lei G be an infinite group9 H a subgroup9 P a set of primes
and
592 w. R. S C O T T
U U L{λp,H)\\iL(ω,H).,£P λ /
Proof. Deny the theorem. Let x E S. If XeL(λp9H) then xλeL(p9 H).
Hence we may assume that # E L (oo, //) or x E L (p, H), p E P.
Case 1. o (/V(%)) = o ( G ) , where N(%) is the normalizer of %. Then o {N(x)~
S) = o ( G ) . If y E /V (% ) - S, then yΓ E // for some r such that (r, p ) = 1 (if p
e x i s t s ) . If xγ £ S then also (xy)n £ H for some n such that (n 9 p) = 1 (if p
e x i s t s ) . Thus
and # Γ / Ϊ E //. B u t (rn9 p ) = 1 if p e x i s t s , a n d , in any c a s e , we h a v e a c o n t r a -
d i c t i o n . H e n c e xγ E S a n d
o ( S ) > o ( % ( / V U ) - S ) ) = o ( / V U ) - - S ) = o ( G ) ,
a c o n t r a d i c t i o n .
C a s e 2. o ( / V U ) ) < o ( G ) . T h e n o ( C Z ( % ) ) = o ( G ) .
C a s e 2 . 1 . o(H) = o(G). T h e n o ( G ) r i g h t c o s e t s of N(x) i n t e r s e c t H,
T h u s t h e r e are o(G) e l e m e n t s of t h e form h~l xh. But if {h"lxh)n E H t h e n
xn£H, w h e n c e n = λ p and A" 1 xh £ S. T h e r e f o r e o ( 5 ) = o ( G ) , a c o n t r a d i c t i o n .
Case 2.2, o ( / / ) < o ( G ) . We h a v e , s i n c e o{S) < o(G),
(1) o(G)=o(Cl(x))= 21 o ( C Z U ) n L U , t f ) ) .(τι,p) = l
If o ( G ) = Ko, and o{x) = oo, then since H is finite,
C Z U ) C L ( o o , / / ) C S,
a contradiction. If o ( G ) = K0, and o(x)~m, then Cl(x) n L(n, H) is empty
for n > m. Hence, by ( 1 ) , there exis ts , regardless of the s ize of o ( G ) , an n
such that (n9 p ) = 1 and
o(Cl(x)nL(n9 H)) > o(H)o{S).
Let
ON INFINITE GROUPS 593
AU9T) = { g \ g
n e T \ .
T h e n A(n9H) D_L(ntH), h e n c e
o(Cl(x)t>A(n,H))= Σho(CUx)nA(n,h)) > o(H)o(S).
Hence there exists an ho £ H such that
o{Cl{x)nA{n,h0) > o(S).
There i s then a b 6 G such t h a t (b"1 xb)n = h0, whence
xeCl{x)nA(n9 bhob'1).
If
q eCl(x)n A(nibhQb-1),
then
q
n = bhob l =xn.
Hence if qr G H9 then
xnr *qnΓ £H
and p I ΓΪΓJ whence p | r. Thus <7 G S in any case. We have
o(S) >_o(Cl(x)<\A(nsbh0b"l))=o(b{Cl(x)<\ A(n,ho))b-1)
= o(Cl(x)nA(n,h0)) > o(S).
This contradiction shows that the theorem is true.
C O R O L L A R Y . // H is a subgroup of the group G9 then o ( L ( o o , / / ) ) - o{G)
orO.
Proof, In Theorem 4, let P be the empty set.
4. An over-characteristic subgroup. Neumann and Neumann [ 2 ] have defined
a subgroup K of G to be over-characteristic in G if and only if ( i ) K i s normal,
and ( i i ) G/K £ G/H impl ies KCH.
594 w. R. SCOTT
Define ( s e e [ 3 ] ) a subgroup K of an infinite group G a s follows. Let E(x)
be the set of g G G such that x is not in the subgroup generated by g, and let
K be the set of x G G such that o(E(x)) < o ( G ) .
THEOREM 5. //"G is infinite$ and K is defined as above, then K is an over-
characteristic subgroup of G.
Proof, ( i ) K is normal since it is fully characteristic [3, Theorem 6],
(i i) Let G/K ^ G/H.
Case 1. K is finite. Then [3, Corollary 3 to Theorem 8]
K2 =K(G/K) =e.
Hence K(G/H)=e. Now
o(G/H)=o(G/K)=o(G).
If there exis ts a k G K — H, then
o ( E ( k H ) ) < o ( E ( k ) ) < o ( G ) = o ( G / H ) .
Hence kH G K(G/H). This is a contradiction. Hence K C^H, and X is over-
characteristic.
Case 2. X is infinite. Then [3, Theorem 5] A is a p°° group, and [3, Theo-
rem 8] G/K is finite. If there exists a k G X - H then
implies k' £ K - H, and
This contradicts the finiteness of G/H. Therefore X C//, and since G/K is
finite, K ~ H. Hence K is over-characteristic.
5. Abeliaπ groups with special properties.1 If G is an Abelian group such
that 0 C H C G implies G ~ H for subgroups //, then it i s trivial that G i s 0 or
cyclic of prime or infinite order, and conversely. This naturally leads to the
problem of finding those groups which p o s s e s s the following property:
1For the facts used without proof in this section, see [ l ] .
ON INFINITE GROUPS 595
(Pi) G i s Abel ian, and if // i s a subgroup of G s u c h that o(H) = o(G) then
G ~H.
THEOREM 6. G has property (Pi) if and only if (i) G is finite Abelian,
( i i ) G is a p°° group, ( i i i ) G is a direct sum of cyclic groups of order p, p a
fixed prime, ( i v ) G is infinite cyclic, or ( v ) G is the direct sum of a non
denumerable number of infinite cyclic groups.
Proof. If G is of one of the above five types, then it is either trivial or
well-known that G has property ( P i ) .
Conversely, suppose that G is infinite and has property ( P i ) . Let T be the
torsion subgroup of G.
Case 1. o(T) < o(G). Then (see, for example, [3, proof of Theorem 9,
Case 1]) there is a free Abelian subgroup H of G such that o(H) = o ( G ) .
Hence G 21 H. If the rank of G is non-denumerable, we are done. If the rank of
G is countable, then G is countable and contains an infinite cyclic subgroup.
By (Pi ) , G is infinite cyclic.
Case 2. o(T) = o ( G ) . Then G 21 T, that is, G is periodic. If Gp is a non-
zero p-component of G, then G = Gp + Hp, hence G 21 Gp or G 21 //p , a con-
tradiction unless //p = 0. Hence G is a p-group. Thus G = D + R, where D is a
divisible (that is, nD — Ό) and R a reduced (no divisible non-zero subgroups)
p-group. Hence G 21 /? or G 21 D, that is G is reduced or divisible.
Gαse 2.1. G is a divisible p-group. Then G ~ Σ,Ca where Ca is a p°° group.
If there is more than one summand, then there is a subgroup
Cί φ Oio, where G* is a proper subgroup of C α Q . Hence o(H) =o(G)9 but H is
not divisible, a contradiction. Therefore G is a p°° group in this case .
Case 2.2. G is a reduced p-group. Then G has a cyclic direct summand C
of order, say, pn. Zorn's lemma may be applied to s e t s S of cyclic groups
Ca of order pn such that Σ G α , Gα G S, exis ts and is pure in G (that i s , a
servant subgroup of G ) . There is then a maximal such set S*, and if X = C α ,
G α E S*, then X is a pure subgroup of bounded order. Hence K is a direct
summand, G - K + A. It is clear that /I has no cyclic direct summands of order
pn. This implies, by property ( P i ) , that o(A) < o(G)9 hence G 21 K. If, now,
n > 1, there is a subgroup H of K oί order o ( G ) such that H £ K. Therefore
596 w.R. SCOTT
Theorem 6 has a dual.
(P2 ) G is Abelian, and o(G/H) = o ( G ) implies G ~ G/H.
THEOREM 7. G has property {P2) if and only if ( i ) G is finite Abelian,
( i i ) G is infinite cyclic, ( i i i ) G is a direct sum of cyclic groups of order p,
( i v ) G is a p°° group, or ( v ) G is the direct sum of a non-denumerable number
of p groups.
Proof. If G is of one of the above five types, then it is clear that G has
property {P2).
Conversely suppose that G is infinite and has property ( P 2 ) .
Case 1. o(G/T) = o(G). Then, by (P2) G is torsion free. Let C be a cyclic
subgroup of G. Then 2C is cyclic, and G/2C has an element of order 2, hence
o(G/2C) < o{G). Therefore o(G) = K0,-and o(G/C) is finite, hence G is
cyclic.
Case 2. o (G/T) < o(G). Hence o ( T) = o ( G). Let S be a maximal linearly
independent set of elements, B the subgroup generated by S (set β = 0 if S is
empty). Then Γπ β = 0, hence Γ is isomorphic to a subgroup of G/B, and
therefore o(G/B) = o ( G ) . But G/β is periodic, hence G is periodic. It follows,
just as in the proof of Theorem 6, that G is either a divisible or a reduced
p-group.
Case 2.1. G is a divisible p-group. Then G = Σ C α , where C α is a p°° group.
If the number of summands is non-denumerable, we are done. If not, then G is
homomorphic to a p°° group, and o(G) = K0. Therefore by ( P 2 ) , G is a p°°
group.
Case 2.2. G is a reduced p-group. Then, almost exactly as in Case 2.2 of
Theorem 6, it follows that G is the direct sum of cyclic groups of order p.
REMARK. Szelpal [7] has shown that if G is an Abelian group which is
isomorphic to all proper quotient groups, then G is a cyclic group of order p or
a p°° group. Theorem 7 may be considered as a generalization of this theorem.
Szele and SzeΊpal [6] have shown that if G is an Abelian group such that
every non-zero endomorphism is onto, then G is a cyclic group of order p, a
p°° group, or the rationale. The following theorem may be considered as a
generalization.
ON INFINITE GROUPS 597
( P 3 ) G i s A b e l i a n , a n d if σ i s a n e n d o m o r p h i s m o f G s u c h t h a t o(Gσ) = o ( G )
then Go — G.
THEOREM 8. G has property (P3) if and only if ( i ) G is finite Abelian,
( i i ) G is a p°° group, or ( i i i ) G is the group of rationals.
Proof. If G is of one of the above three types, then it is clear that (P3)
is satisfied.
Conversely, suppose that G is an infinite group satisfying ( P 3 ) .
Case 1. G is torsion-free. Then if pG ^ G for some p, the transformation
gσ~pg is an isomorphism of G into itself, so that o (Gσ ) = o (G), Gσ •£ G, a
contradiction. Hence pG - G for all p, and therefore G-ΣLRa9 where Ra is
is isomorphic to the group of rationals. If there is more than one summand, then
there is a projection σ of G onto ΣlRa9 CC ^ Cί0, a contradiction. Hence G is
the group of rationals.
Case 2. G is not torsion-free. Then G = A + B where A is finite (and non-
zero) or a p°° group. Thus the projection σ of G onto the larger of A and B yields
a contradiction unless B = 0. But in this case, since G is infinite, G - A is a
p°° group.
Finally (compare with Szele [5]) consider the following property.
(P4) G is Abelian, and if σ is an endomorphism of G such that o(Gσ ) = o(G)
then σ is an automorphism of G
COROLLARY. G has property ( P 4 ) if and only if ( i ) G is finite Abelian,
or ( i i ) G is the group of rationals.
REFERENCES
1. I. Kaplansky, Infinite Abelian groups, Michigan University Publications in Mathe-matics no. 2, Ann Arbor, 1954.
2. B.H. Neumann and Hanna Neumann, Zwei Klassen charakterischer Untergruppenund ihre Faktorgruppen, Math. Nachr. 4 (1950), 106-125.
3. W. R. Scott, Groups and cardinal numbers, Amer. J. Math. 74 (1952), 187-197.
4. , The number of subgroups of given index in non-denumerable Abeliangroups, Proc. Amer. Math. Soc, 5(1954), 19-22.
5. T. Szele, Die Abels chen Gruppen ohne eigentliche Endomorphismen, Acta. Univ.Szeged. Sect. Sci. Math. 13 (1949), 54-56.
6. T. Szele and I. Szelpal, Uber drei wichtige Gruppen, Acta. Univ. Szeged. Sect.Sci. Math. 13 (1950), 192-194.
598 w. R. SCOTT
7. I. Szelpal, Die Abe Is chen Gruppen ohne eigentliche Homomorphismen, Acta. Univ.Szeged. Sect. Sci Math. 13 (1949), 51-53.
UNIVERSITY OF KANSAS
ON HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS WITH
ARBITRARY CONSTANT COEFFICIENTS
A. S E I D E N B E R G
Let K be an arbitrary ordinary differential field—for our purposes it is suf-
ficient to consider an arbitrary (algebraic ) field K which is converted into a
differential field by setting c ' = 0 for every c G K. Let u be a differential in-
determinate over K and let u — UQ9 uι, represent the successive derivatives
of u. Further, let Co9* 9cm be arbitrary constants over the field K\u)~
K(u0, uι9 •), that is, m + 1 further indeterminates with which we compute in
the usual way, setting cf = 0. In addition to the ring R ^ K{u \ ~ K [UQ$ uχ9 ],
we will also be interested in the rings Rt + m ~ & [UQ9 U\9 • , ι t£+ m ] . Theorems
referring to some one of these rings Rt+m may, if convenient, be regarded as
belonging to ordinary, rather than differential, algebra, but we will still apply
the operation of differentiation to elements of Rt+m (not involving ut+m). This
then amounts to a convenience in writing formulas.
Let IQ = CQ uo + + cmum. This element generates a prime differential
ideal [Zo 1 = ( h9 hi ) in S = K(c)\u\, where /; = c 0 U{ + + cm u; + m . We
a r e i n t e r e s t e d i n h a v i n g e x p l i c i t l y a b a s i s f o r [ l o ] n K \ u \ . I f Δ ( w ) i s t h e d e -
terminant of coefficients of any m + 1 of the Zt regarded as l inear forms in the
c;, then clearly A ( « ) G [ / 0 ] n K U ) and Theorem 2 below a s s e r t s that the
Δ ( u ) obtained from all choices of the Zj form the required bas i s .
Let us confine ourselves to the rings Rt+m
a n ^ $t +m = ^ ( c ) L uo? > ^ ί + m J
I n S ί + m , let p = (Z o , , lt)
LEMMA 1. p = ( l 0 , , lt) is an m-dimensional prime ideal in St + m .
Proof. Let G(UQ, , ut+m) G St+m. Eliminating success ive ly ut+m9
ut+m-W*9Um mod (l0, •• , Z ί ) , we may write G ( uθ9 , z^+ m ) Ξ G I ( M O > * * § >
um.\) mod (Zo» •••>/*), where Gi £St+m is a polynomial in the indicated vari-
ables . Moreover, starting with indeterminate values < . for uι, i = 0, ,wι —l,we
can build up a zero (ζQ, , ζt+m) of p by defining ^ m from the condition
Received December 7, 1953, This paper was written while the author was a Guggen-heim Fellow.
Pacific J. Math. 5(1955), 599-606
599
600 A. SEIDENBERG
/ 0 ( £ ) = 0 , and defining ζ + . successively from the condition Zί ( ^ r ) = 0 . Then
( ξQ, , ζt+m) is clearly a general point of p, whence p is prime and m-dimen-
sional.
LEMMA 2. Let p n Rt+m — P\ and let t >_ m — 1. Then P is a 2m-dimensional
prime ideal in Rt+m
Proof, Consider the equations:
+ + cm ξm = 0
From these we are going to solve successively for the c t , i — 0, , m — l
Since ξQ £ 0, we can solve for c0 and find CQ G K( c i , , cmt ζQ, , ζm)
Suppose in this way, solving success ively for the c t , we find
C O , , C J G K ( C J + I , . . . , c m , ξQ, •• , ί m + / ) , i < TO - 1 .
In fact, assume we have found inductively that
0 2 + 2 + K ( ^ 0 , . , ^ . + m ) cm
Since
dt X ( c 0 , " , c m , ^0,,•• , ^ m + ι . ) / X ( c 0 , > c m ) = m and
dt X ( c o , , c m ) / ί : = m + 1,
we have
dt K ( co, , cm$ ξQ, , ξmH )/K = 2m + 1
= dt
w h e r e d t s t a n d s for " d e g r e e of t r a n s c e n d e n c y " . F r o m t h i s w e s e e t h a t ξQ, •••,
ζm+i a r e a l g e b r a i c a l l y i n d e p e n d e n t o v e r K ( s i n c e t h e s e t c^ + l f , ί m + ι h a s
ON HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS 601
2m + 1 members), in particular they are not zero. The coefficient of c + i in
li + i(ξ) is £2( + 1) plus a term in K( ξQ, • •, £ 2 ί + ι ) arising from c0 ξ^ι + +
since i + 1 < m, we have 2( ί + 1) < m + i + I and ^ ( ^ j ) ί
). Hence <?j+i G X( c t + 2, , fm+ + ι ) ; also ^i + i holds. Con-
tinuing, we have c 0 , , cm. L G K( c m , f0, , £ 2 m - ι )• Hence ίo» " » ^ 2 m . ι
are algebraically independent over K. Thus P is at least 2m-dimensional.
Let Δj ( ^ ) , i >_ m, be the determinant of the coefficients of the forms
IQ( ζ)J •» Zm. ι ( ^ ) , Zi(f) regarded as linear forms in c o , , c m ; that is,
V
Then one finds cyΔj (<f) = O, so that Δj(<f)=O. The coefficient of f + in
this equation is a polynomial in the indeterminates £Q9 , ζ this coef-
ficient contains the term ζQ ζ2 f2m_2
a n ( ^ n e n c e i s n o t z e r o (therefore also
^o( ζ}* * * ' ^m- l ( ί ) a r e linearly independent over X ( f ) ) . Thus P is at most
2m-dimensional, and hence exactly 2τn-dimensional, Q.E.D.
LEMMA 3. Let M = M(u) be the matrix:
α 0
, t >^rn.
Let /4 be the ideal generated in
olM(u). ΎhenA CP.
by the (m + 1) x (m + 1) subdeterminants
Proof. Since lo( ξ), , Zm. i ( £ ) are linearly independent over K(ξ) (and
in fact over any field containing K{ξ)) but Zo( ^ ) , , lm-ι ( ί \ 4 ' ( f ) a1"6
linearly dependent over K(ξ), the matrix M(^) has rank m. Hence A C_ P.
We want to prove A = P, in particular that /4 is prime. Conversely, if we
602 A. SEIDENBERG
knew that A were prime, we could conclude immediately that A - P. In fact,
suppose A is prime and let ηQ, , ηt+m be a general point of A. Since A has
a basis of forms of degree m + 1, no form of degree m vanishes at 77. Hence all
m x m subdeterminants of M(η) differ from zero, and it follows that A is 2m-
dimensional, whence A = P.
In proving A = P, we proceed by induction on m, the assertion being clearly
true for m = 0. For given m, we proceed by induction on t (£ >_ m). For ί = m,
we have to prove the following lemma.
LEMMA 4. Let D be the determinant
uί '
^m ' * ' U 2 m
Then D is different from zero and is irreducible in i?2m
Proof. By induction on m, being trivial for m — 0. D is linear in UQ, the
coefficient δ of UQ> being different from zero and irreducible by induction: in
particular, therefore, D 0. Also D is linear in U2m
a n ( ^ the coefficient δ ' of
U2m i s irreducible. D is reducible if and only if δ is a factor of D - woδ, hence
of D. Similarly for δ'. Now δ and δ ' are not associates, since they are of dif-
ferent degree in UQ SO D is reducible if and only if it is divisible by δδ'. For
m — 1, this means if and only if UQU2 —U^ is divisible by uoii2 This is not
the case. For m > 1, D is reducible only if it is of degree at least 2m, whereas
it is of degree m + 1. Hence for every m, D is irreducible.
DEFINITION. An ideal is called homogeneous if it has a basis of forms.
Similarly we call an ideal isobaric if it has a basis of isobaric polynomials.
LEMMA 5. A and P are homogeneous and isobaric.
Proof. A is clearly homogeneous. Moreover consider one of the (m + 1) x
(m + 1) subdeterminants of M{u)9 say one involving the ith. and th rows, i < j .
Then Mj+&_2 i s t n e element in the ith row and /rth-column and w/ + /.2 is the
element in the /th row and Zth column. Suppose k > L The determinant in ques-
tion has together with a term π wj+/c-2 "/+/-2 a l s o a t e r m ^π ' ui+l-2 ' M/+A>2»
which is of the same weight. Hence if rows i0, ,im are involved, each term
has the weight of the term uiQ z ^ + i ι2+2 ' ' wιm+m> ^ a t is> t n e determinant is
ON HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS 603
i s o b a r i c . T h u s A i s i s o b a r i c . As for P , we know that p i s homogeneous , and
from th i s and the fact that P = p n Rt+m one c o n c l u d e s immediate ly t h a t P a l s o
i s h o m o g e n e o u s . T o s e e t h a t P i s i s o b a r i c , l e t g(u)£P and write g{u) =
gr(u) +gr+ι(u) + •••, where gXu) i s zero or i s o b a r i c of weight /. It i s c lear ly
suff icient to prove gΓ(u) E Pf a s s u m i n g gr £ 0. Since g{u) E P , we have
h{c) g{u) ~ c, u) lί(c9 u),
where h(c) i s a polynomial in the c; a l o n e , and the A^ are polynomia l s in the
c; and uj We a s s i g n to c2- the weight m — i . L e t h{c) = hs(c) + Λ5 +1 ( c ) + •••,
where Ay(c) i s zero or i s o b a r i c of weight / and hs(c)^0. Observe that the
ll{cyu) are i s o b a r i c . Comparing terms of l ike weight on both s i d e s of the above
equat ion we s e e that hs(c) gr(u) = ^ ^ / ( c9u) l(( c9 u). H e n c e griu) G p.
THEOREM 1. A = P. In particular, therefore, for m > 0# A: uo ~ A.
Let
Proof. We proceed by induct ion on m and t, and first show that A:u0 -A.
be the general zero of P introduced above. L e t D(u) be the^.+
determinant occurring in Lemma 4. From D(ζ)=O we see that ζ2m
written as a quotient of two polynomials in the indeterminates ξQ,
with the denominator being
can be
' ^2m-l
d D 2 7 7 1 - 2
which is irreducible by Lemma 4. Hence we see that
ξ, ~LSn+l
(for were it zero, then ξ2m could be written as a quotient of two irreducible
polynomials in ζ^, , ζ , the denominator this time not being an associate
of the other denominator). Hence ξQ is algebraic over K( ζγy , ££+m) Hence
<^i'# * ' ' < +m defines a 2m-dimensional prime ideal Pί in K[uϊ9 , ut+m]; and
Pi is generated by the (m + 1) x (m + 1) subdeterminants of M(u) which do
not involve the first row of M(u). Designating also by Pl9 the extension of
Pi to K [ UQ, , ut+m ], we see that Pγ C_ A. Let now ιiQg(u) G A. We write
604 A. SEIDENBERG
uog(u) = Σ / 4 J ( M ) ΔJ ( W ) , where the ΔJ ( M ) are the (m + l ) x ( τ n + l ) sub-
determinants of M(u), and the /4t are polynomials. We write Aι=A?+uoA."f
where A?does not involve UQ. We then have uo(g(u) - Σ>A?'Δi{u)) = Σ ^ . ' Δ j d ί ) .
The right hand side here i s of degree at most one in UQ, hence gχ = g(u) —
Σ,A?'Δι(u) does not involve u0: g v = g ι ( u i , , ut+m) Now g ( u ) and Δ ; ( u )
vanish at £ Q , . . , ξm+t, hence so does g t that is , g£ vanishes at ^ , , £ m + , .
Hence, g t G P t , whence g € A. Hence A: UQ - A.
As a corollary to the above we get that A : f = A for any polynomial
f £ Rm + t containing a term durQ, d £ K9 d^O (m > 0 ) . For suppose fg £ A: to
prove g G /4. We may suppose / and g isobaric; and also homogeneous. We then
get duΓ
Q g G A$ whence g G /4.
We proceed to prove that /4 is prime. Let Zj = IΪ/UQ = c 0 v + * + cm Vj+m,
where v( = U(/UQ, We p a s s to the rings Rt+m ~ ^ [ ^ ι > * > vt+m J a n ( ^ ^t+m ~
K(c)[v] Observe that t> l y ,vt+m are algebraically independent over K.
Let if be the matrix of the coefficients of the Z, , that i s , the matrix:
V2
v2
V3
and let A be the ideal generated in Rt+m by the (m + l ) x ( m + l ) subdeter-
minants of M(v). Each such subdeterminant is a power of u0 times an (m + 1) x
(m + 1) subdeterminant of M(w); and vice-versa. It would therefore be sufficient
to prove A prime, in fact it would be sufficient to prove that the extension of
A to the quotient ring Q of /?t+ m relative to the ideal ( v Ϊ 9 , vt+m) i s prime.
For suppose this proved and g(u) h(u) G A, where we assume without loss of
generality that g ( w ) , h(u) are homogeneous. Dividing by appropriate powers of
UQ and sett ing
gU)/αJ=i(t;), h(u)/us
0=h(υ),
we get g(v)h(v) G A9 w h e n c e by a s s u m p t i o n f(v)g(v) or f(v)h(v), s a y
fg i s in A for s o m e f(v)ERt+m9 f £ {vi9 , vm). Mult ip ly ing by a power of
^0 w e find uζ f iu) g(u) G A, where f ( u) c o n t a i n s a term duζ. H e n c e g ( u ) G A.
T h e i d e a l /4 in /?£+ m h a s ζ^/ζ^ > ^t+rd^o a s a z e r o » n e n c e i s a t l e a s t
( 2 m — 1 ) - d i m e n s i o n a l . A l so A r e m a i n s at l e a s t ( 2 m — 1 ) -d imens iona l upon ex-
t e n s i o n to Q. In fact , if ξχ/ξ0, •••, ζf^m/^o d e t e r m i n e s P in Rt+m, t h e n
ON HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS 605
P C_{v\ 9 9vt+m), as one s e e s from the fact that £ 0 , » , £ ί + determines a
homogeneous and i sobar ic ideal P and UQ $L P.
Subtracting v{ t imes the first row from the (ί + l ) t h row of M, we get the
matrix
v i v 2
0
Each (TO + 1) x (TO + 1) subdeterminant of this matrix is also an (TO + 1) x
(m . + 1) subdeterminant of /I/. Hence one sees that every m x m subdeterminant
of the matrix
v2
vt+m
is a leading-form of an element in Q A, These m x m subdeterminants generate,
by induction, a 2(m — 1 Vdimensional prime ideal in K\_v2j »vt + m\ a n <^
hence a (2m - 1 )-dimensional prime ideal q in K[vι9 , Vί + m l . The leading
form ideal of A contains or equals ~q. If it contained ~q properly, it would be of
dimension less than 2m — 1. But an ideal and its leading form ideal have the
same dimension [1; Satz 8], Hence q is the leading-form ideal of A and A is
{2m - l)-dimensional.
Moreover A is prime. For quite generally in a local ring, if an ideal A has a
prime ideal ~q as leading form ideal, it must itself be prime. In fact, suppose
gh E A$ g jέ A% h<t A. Then the leading form ideal LFI(A9g) of (A9g) contains
^properly, and likewise for (A,h). But LF1 (A, g) x LF1 (A, h) C LFI ( ( J , g )x
(A$h)) C_ LFIA = q, a contradiction. Hence /4 is prime, and the proof is com-
plete.
The following theorem is an immediate consequence of Theorem 1.
T H E O R E M 2. A basis for ilo]n K\u\ is given by the {m + 1 ) x (m + 1 )
subdeterminants of the oo x (m + 1 ) matrix
um
u2
606 A. SEIDENBERG
REFERENCE
1. W. Krull, Dimensionstheorie in Stellenringen, J. Reine angew. Math. 179 (1938),204-226.
UNIVERSITY OF CALIFORNIA!
BERKELEY, CALIFORNIA
CANTOR-TYPE UNIQUENESS OF MULTIPLE TRIGONOMETRIC INTEGRALS
VICTOR L. SHAPIRO
1. Introduction. It is the purpose of this paper to obtain results in Cantor-
type uniqueness for multiple trigonometric integrals similar to those obtained
previously for multiple trigonometric series ([5, 11, 12]). As might be expected,
the results in the integral case are a bit more difficult to obtain.
Vectorial notation is used for the most part throughout this paper. Thus u
designates the point in ^-dimensional euclidean space, En, with coordinates
( u ί 9 9 u n ) , t h e s c a l a r p r o d u c t ( u 9 x ) = u± x ι + + un x n , w i t h \u\={u9u)^2
and u + 0.x is the point (u ι + axί9 , un + CLxn ).
Previously the author [13], using equisummability between trigonometric
integrals and trigonometric series, has obtained in the special case of double
trigonometric integrals the following result:
Let c{u)9 in L2 on any bounded domain, be 0 ( | u | ), e > 0. Suppose the
double trigonometric integral /„ e ' u c{u) du is circularly summable (C9 1)
to f(x). Furthermore suppose fix) is in Lip Cί, (X > 0, on every bounded
domain id depending on the domain). Then the double trigonometric integral
e-ί{x>u)fix)dx
is spherically summable ( C , 1) to c iu) for almost every u.
Specializing fix) to be the zero function (which is what is meant by Cantor-
type uniqueness, [ 15, p. 274]) and using a more direct attack on the problem,
we are able in this paper both to weaken the hypotheses of the above theorem
as well as to extend the results to ^-dimensional integrals.
2. Definitions and notation. The open π-dimensional sphere with center x
and radius r will be designated by Dnix,r), and the surface of the sphere by
Cn(x$r).
Received November 26, 1953. Presented to the American Mathematical SocietyNovember 27, 1953. This investigation was supported in part by a grant from the RutgersUniversity Research Fund.
Pacific J. Math. 5 (1955), 607-622
607
608 VICTOR L. SHAPIRO
Following Bochner [ l ] , we shall say that the multiple trigonometric integral
fE c{u)eι XfU du is spherically convergent at the point x to the finite value
Lix) if the spherical partial integrals of rank R converge to L(x), that is if
(D /*(*)= ί ei{x>u)cU)du—>L{x) (asR—>oo.)JDn(0,R)
The integral
( 2 ) < 4 α ) U ) = 2 α β - 2 α ίR lr(x){R2-r2)a-ιrdr, α > 0 ,κ Jo
is called the (C, cc)-mean of rank R of the multiple trigonometric integral
fE c(u)e 9 du$ and this integral is said to be spherically summable (C, α)
t o L U ) i f σ < α ) ( z ) — > L ( « ) as/?—>oo.
Given F{x) integrable on Dn{xθ9r), we designate the mean value of F in
this sphere by A (F; xo;r). Given F(x) integrable on CU(XQ; Γ), we designate
the mean value of F on this surface by L{F; xo; t). Thus, designating the
volume of the unit ^-dimensional sphere, 2 πn /nV(n/2), by Ωn and the (n — 1)-
dimensional volume of its surface, 2πn /Γ(n/2), by ωn, we have
A(F;xo;r) = (Qnrn)'1 I F(x)dx
JDn(x0,r)(3)
L(F;xo;r) = ω"1 / F(x0 + rx)dSn. t(x)
where dS^ί is the (n — 1) dimensional volume element of C n (0,1) .
We set
V l ( F ; * o ; r ) = L ( F ; * o ; r - ) - F ( * o ) and V2 ( F;x0; r) = A ( F; xo; r) - F(x0 )
and say that F(x) has a generalized Laplacian of the first or second kind at
the point x0 equal to OCi or 0,2, respectively, if
lim 2n Vι{F;x0;r)/r2 = C^
or
lim 2(n + 2) V2 ( F ; xo; r)/r 2 = a2
CANTOR-TYPE UNIQUENESS OF MULTIPLE TRIGONOMETRIC INTEGRALS 6 0 9
The generalized Laplacian of the first and second kind of F at x0 will be
designated by ΔιF(x0) and Δ 2 F ( % 0 ) , respectively. It is known, [6, p. 261],
that if Fix) is in class C ( 2 ) on Dnix0, r0 ), then ΔF(% 0 ) = ΔιFix0 ) = Δ2F ix0 )
where ΔF(%) is the ordinary Laplacian of F at x.
The closure of the set W is designated by W; and its characteristic function
by \ψix) The set Z is said to be a closed set of vanishing capacity if for
every r§ ZDw(0, r) is a closed set of capacity zero. It is known, [4] , that if
Z is a set of vanishing capacity then Dn(xθ9 r) - ZDniχθ9 r) is a domain.
The trigonometric integral fE e fU c iu)du is said to be of type (U) on
a domain G if
/ ei(x'u)c(u)\u\-2duJEn-Dn(0,\)
converges spherically on G to a function Fix) which is continuous on G.
Throughout this paper En stands for n-dimensional euclidean space where
n > 2, and μ = in - 2)/2.
The function 7j(r) is the Bessel function of the first kind of order i.
3. Statement of main results. We shall prove the following two theorems
concerning Cantor-type uniqueness for multiple trigonometric integrals.
THEOREM 1. Given the multiple trigonometric integral L e fU c iu) du
where ciu) is a complex-valued function which is integrable on every bounded
domain. Let Z be a closed set of vanishing capacity. Suppose that
( i ) The integral is spherically summable ( C, 1) to zero almost everywhere.
( i i ) The (C, 1) spherical mean of rank R, σ^Hx), is such that l i m ^ ^ ^
| ^ l ) U ) | < oo inEn -Z.
( i i i ) c iu) i I u \2 + 1 ) " 1 is in L^ on En.
Then ciu) vanishes almost everywhere.
THEOREM 2. Given the multiple trigonometric integral JE eτ *u ciu)du
where ciu) is a complex-valued function which is integrable on every bounded
domain. Let Z be a closed set of vanishing capacity. Suppose that
( i ) and ( i i ) The same as ( i ) and ( i i ) of Theorem 1.
( i i i ) The integral is of type iU) on En.
( i v ) ciu)i\u\2 + I)"1 is in L2 on En.
Then ciu) vanishes almost everywhere.
610 VICTOR L. SHAPIRO
For the special case of the plane, we prove the following theorem.
THEOREM 3. Given the double trigonometric integral fE e fU c(u)du
where c{u) is a complex-valued function which is integrable on every bounded
domain. Let Z be a closed set of vanishing capacity and W be a closed de-
numerable set such that WZ = 0. Suppose that
( i ) The integral is spherically summable ( C9 1) to zero in E2 — Z,
( i i ) The integral is of type (U) on E 2 — W.
( i i i ) c ( u ) = o ( I u I ) as \u\—» oo
( i v ) c ( u) ( I u I 2 + I ) " 1 is in L2 on En.
Then c(u) vanishes almost everywhere.
4. Fundamental lemmas. Before proving the main theorems of this paper, it
is first necessary to establish a connection between the ( C, 1) spherical sum-
mability of the integral fE e ' u c (u) du and the generalized Laplacians of
the "anti-Laplacian" of this integral. In short, we shall now establish some
lliemann-type, [15, p. 270], results for the multiple trigonometric integrals.
We need prove the following lemma only for the plane, since the conclusion
is hypothesized for Theorems 1 and 2.
LEMMA 1. Let du) be a complex-valued function which is integrable on
every bounded domain in the plane, vanishes in D2{0,ro), r 0 > 0, and is o\\u\)
Suppose that σ ^ ι ) {x0 ) = o {R) where σ^ ι ) (:x;) is the ( C , l ) spherical mean of
rank R of fE e
i(x>u)c(u) du. Then fE ei(x>u) c U ) \u\'2 du is spherically
convergent.
Without loss of generality, we assume x0 to be the origin. Then with //?(%)
given by (1) and σ^Hx) by (2), we have
= 2 ίR r-3lr(0)dr+R-2IR(0)Jo
CANTOR-TYPE UNIQUENESS OF MULTIPLE TRIGONOMETRIC INTEGRALS 6 1 1
Since by assumption σ^ ι ) ( 0 ) = o ( R ) , to prove the lemma it only remains to
show t h a t / Λ ( 0 ) = o ( β 2 ) . But
IR(0) = [(/? + I ) 2 σ
- ( 2 R + I ) ' 1 f c ( u ) [ R + D 2
JD2 (θ,R + l ) -D 2 (θ,R)
and the proof is complete.
LEMMA 2. Let c{u) be a complex-valued function which is integrable on
every bounded domain in En and which vanishes in Z ) π ( 0 , ΓQ ), ΓQ > 0. Suppose
that
( i ) lίm D ^ I On (XQ ) I - d where σ^ (XQ) is the (C9 1 ) spherical mean of
rank R of fF e 'u* c (u)du and d is finite-valued.
( i i ) - f ei{x'u)c(u)\u\-2du
is spherically uniformly convergent in Un(xOf ί0 ), ί0 > 0, to F (x).
Then l i n i ί ^ o I %n^ι (F; xo; t )/t2 \ <_Kd where K is a constant independent of
%o and d.
Observing that for fixed u
(see [1, p. 177]), we have by assumption (ii) for t sufficiently small that
L(F;xo;t)
= -2^T(μ+l) lim / eiix°>u)c(u)\u\-2jΛ\u\t){\u\t)'μduR^ooJDn(o,R)
and consequently that
(4) (2n)Vι(F;x0;t)/t2= l im / e i { x Q ' u ) c(u)η{\u\t)duR^ooJDn(o,R)
where
( r ) r - μ ] / r 2 f o r r > 0, τ ; ( 0 ) = l ,
612 VICTOR L. SHAPIRO
and η(r) i s in C .
Making the following observations:
( a ) By the second mean-value theorem applied to the real and imaginary
parts of IR(XQ) given by ( 1 ) and hypothesis ( i i ) we have IR(XO) = O(R2)9
( b ) F o r f i x e d t9 η(Rt)=O(K'2) a n d η'(Rt) = O(R'5/2) w h e r e τ j ' ( r ) =
dη(r)/dr, we o b t a i n from ( 4 ) t h a t
( 5 ) 2nyι(F;x0;t)/t2 = r ί [°° r2 σ < ι > ( * 0 )t3 Oi(rt) drJ Γo Γ
where OC (r) = dr"177 ' ( r)/dr.
From the fact that Cί(z) is an entire function of the form Σ = 0 b(Z21 l, we
have that there exists a constant Kx such that
(6) | ( χ ( r ) | < «!Γ for r < 1
From the fact that Jμ(r) =0{r'i/2) as r —> 00, and
rfr-μ/μ(r)/rfr = - r " μ / μ + 1 ( r ) f
we obtain that there exists a constant K2 such that
(7) | α ( r ) | < K 2 [ r - ( μ + 7 / 2 ) + r- S ] f o r r > l
From (5), (6) , and (7) , the conclusion of the lemma follows readily. For
given an e > 0, choose RQ so large that \o^Hxo) \ <_ d + 6 for R > Ro. Then
for t < RQ1
9 it is easily seen that
(8) \ 2 r f t ι ( F ; x O t t ) / t 2 \ < K ( d + e ) + O ( ί 4 )
where K is a constant independent of xOi d9 and e. Taking the limit superior
of the left side of ( 8 ) as t —» 0 and then the limit of the right side as e —> 0,
we have the proof of the lemma.
LEMMA 3. Let the hypotheses be the same as in Lemma 2 except that
For if d = 0, the lemma follows immediately from Lemma 2. If d £ 0, choose
cι(u) integrable on En, vanishing for u in [En - Dn(0, 2 ) ] + Dn(0,1), and such
that fE Cι(u)ei{x°'u) = d. Set F ι ( * ) = - / £ Ci(u) \u\'2 ei(x'u)du. Then
CANTOR-TYPE UNIQUENESS OF MULTIPLE TRIGONOMETRIC INTEGRALS 6 1 3
0 = A^Axo)- /iίF^xo) = Δ ^ U o ) - Δ F ^ Λ Q) = ΔιF(xo)-d.
LEMMA 4. Let c(u) be a complex-valued function which is integrable on
every bounded domain in En and which vanishes in Dn(0fro)9 ro> 0. Suppose that
( i ) σ^Hx), the (C, 1) spherical mean of rank R of JP eι XfU c (u)du,
is such that lim/? _» oo | σ^ (Λ O ) I = d
( i i ) c (u ) I u I" is in L2 on En.
( i i i ) - f eί(x°>u)\u\-2c(u)duJ
is spherically convergent to F(XQ). Set
i(x'u)c{u)\u\-2du.fDn(0,R)
Then
ϊhn | 2 ( 7 i + l ) V 2 ( F ; « 0 ; i ) A 2 | < Kd
where K is a constant independent of XQ and d.
Setting
TR(x)=- f ei(x>u)c(u)\u\-2du,JDn{θ,R)
observing that A(F;xo;t) - l i m R _ o o A ( TR xo; t) and that for fixed u,
i { \ nntn Γ rn-ιL(ei{x'u);x0;r)dr
Jo
we obtain
( 9 ) A(F;xo;t)
= - lim (μ)f
JDn(0,R)
and consequently
6 1 4 VICTOR L. SHAPIRO
lίm f ei{x°>u)γ(\u\t)c(u)duR^ocJDn(0,R)
here
U + l ) ( ^ l ) r ) ] / r 2 for r > 0, y(0) = l ,
and γ(r) i s in C ( o ° I
S i n c e γ{r) h a s t h e s a m e form a s η(r) in L e m m a 2 with μ r e p l a c e d by μ + 1,
we c a n p r o c e e d a s in t h a t l emma a n d o b t a i n
2 U + 2 ) V 2 ( F ; * 0 ; ί ) / ί 2 = 2 - 1 [°° t3r2 σ ( 1 ) (x0) β(rt) drJo
where β(r) = dr" γ'(r)/dr. Then we can proceed in a similar manner to obtain
that for e > 0
ϊ ϊ m " \2(n + 2)\72(F;x0;t)/t2\ <K(d + e)
where K is a constant independent of λ'o, d, and e. Since e is arbitrary the con-
clusion of the lemma follows.
LEMMA 5. Let the hypotheses be the same as in Lemma 4 except that
lim/^oo O^HXQ ) = d. Then Δ 2 F(% 0 ) = d.
In the same manner that we obtained Lemma 3 from Lemma 2, we obtain
Lemma 5 from Lemma 4.
LEMMA 6. Let F{x) be real-valued and continuous on Dn(xo,ro ), r0 > 0.
Suppose that
( i ) A 2F(%) = 0 almost everywhere in L)n(x0$r0 )
( i i ) Tϊm \2(n + 2) V2 (F; x; r)/r2 \ < oo for all x in Dn(xθ9ro).r —* oo
77ιeπ F ( # ) is harmonic in Dn(xOiro ).
Following the pattern of proof in [9], we give a proof for n >_ 3.
To prove the lemma, it is sufficient to show that Fix) is subharmonic in
Set
CANTOR-TYPE UNIQUENESS OF MULTIPLE TRIGONOMETRIC INTEGRALS 615
f(x)=2(n + 2)[ ϊ h n ~ V 2 ( F ; % ; r ) / r 2 + lim V2 ( F; x; r)/r2 1/2Γ->0 Γ-»0
for x in 0n(χ0i r0 ). Then / (A; ) = 0 almost everywhere in Dn(x0; r0 ).
By the theorem of Vitali-Caratheodory [10, p. 75] , there exis ts a sequence
of nondecreasing upper semίcontinuous functions ίgm(χ)\ such that gm(x) <
f (x) for a l l x in Dn{xo,ro), gm^x^—> f (x) a l m o s t e v e r y w h e r e in Dn(xOfro),
g ( A : ) i s i n t e g r a b l e on Dn(xoiro)9 and s u c h t h a t
limm
im / £ ( x ) dx — I f (x) dx for r < r0 .-oo JDn(x0>r)*m JDn(xQ,r)
Set
^gJx)=-[ωnU-2)Yιf gmU)\u-x\2-ndu.
Then Δ~ιgm(x) is superharmonic, since gm(u) £ 0 for almost all u in Dn(xQ$ ro)
Furthermore, we observe that for fixed u
A{\x-u \2'n;x0;r) = | x 0 - u \ 2n if |%0 - M | > r
= / ι r " Λ 2 - 1 [ r 2 + |%0 - α | 2 ( 2 - Λ ) τ ι - 1 ] if \x0 ~u\ <r.
Consequently, for xι in Dn(xo,r) with r sufficiently small,
( 1 1 ) V 2 ( Δ - 1 g m ; x ι , r ) = [ ω R ( r a - 2 ) ] - 1 ^ ^ ^gju) \ \u - X ι \2'n
- r a r - " 2 - 1 [ r 2 + |SB t - u | 2 ( 2 - n) n ι ] }rfu.
Suppose g_,(*i ) s finite. Then by the upper semi-continuity of gm(u) at
%i, for e > 0 and r sufficiently small, we have from ( 1 1 ) that
V 2 ( Δ - ι g m ; * i ; r ) < [ g m ( X ι ) + ε ] [ ω n ( n - 2 ) ] - ι [ ω n ( n - 2 ) ] r 2 / 2 { n + 2 ) .
Consequently, we conclude that
( 1 2 ) lim 2 U + 2 ) V 2 ( Δ " l £ xι; r)/r2 < g(xι).Γ->0
S i m i l a r l y , in c a s e g (%i ) = — oo, c o n c l u s i o n ( 1 2 ) r e m a i n s v a l i d .
616 VICTOR L. SHAPIRO
From the fact that Δ" g ix) is superharmonic, we have that F — Δ" g is
upper semi-continuous in Dnix0, r 0 ) . From ( 1 2 ) we conclude that
Tϊm" 2 U + 2 ) V 2 ( F ~ Δ - l g ; % ; r ) / r 2 > 0 for x in D Λ ( * o , r o ) .Γ-»0
Therefore by [8, p. 14], ί F — Δ" g 1 is a nondecreasing sequence of sub-
harmonic functions in Dnixθ9 ro ). But limm_>(X) Δ" g ix) = 0 almost everywhere.
Therefore Fix) is almost everywhere equal to a subharmonic function, Gix), by
[8, p. 22]. But A(F x r) = AiG x r) —> Gix) for all % in Dnixθ9 r0 ). However
from the continuity of F we have A(F x r) —» F(%), and the proof of the lemma
is complete for n >_ 3. For β = 2 a similar proof can be given with the Newtonian
potential replaced by the logarithmic potential.
For the case of the generalized Laplacian of the first kind, we have a similar
lemma with a similar proof, see [9],
LEMMA 7. Let Fix) be real-valued and continuous on Dnixθ9 r0 ), r0 > 0.
Suppose that
( i ) Δ xF ix ) = 0 almost everywhere in Dnixθ9 r0 ).
( i i ) lim \2n^χ (F; x; r)/r2 \ < oo for all x in Dnix0; ro ).
Then Fix) is harmonic in Dnixθ9 ro )•
We now prove s o m e l e m m a s c o n c e r n i n g t h e s p h e r i c a l s u m m a b i l i t y iC9n) of
F o u r i e r t r a n s f o r m s .
LEMMA 8. Let Gix) be a function in L\ on En which vanishes in Dni0,ro),
r0 > 0. Suppose that Fix) = fE eι * Giu)du is in C ( 2 ) on En. Then for u in
Dnio9r0/2)-0
(13) j [e'i(x>u)Fix)~i-e i(x>u)\u\'2AFix))]dx
is spherically summable iC9n) to zero.
For, by Green's second identity, we have
(14) » ( « ) - f [e i(x>u) F (x) - {-e-i(x>u) \u\ 2 AF(x))]dxJD(0R)f
JDn(0,R)
"-1 f F{Rx)i(x,u)e-iR{x'ιι)dSn.ι(x)/cπ(o,i)
CANTOR-TYPE UNIQUENESS OF MULTIPLE TRIGONOMETRIC INTEGRALS 617
+ R n ' 1 f dF(Rx)/dR e'iR(x'u)dSnml(x)\ Ξ \u\'2(AR +BR)JCn(θ,l) i
where dSn_ι(χ) is the (n — 1) dimensional volume element on the unit sphere
CB(O,1).
We shall now show that both AR and BR are ( C, n) summable to zero. For,
by Fubini's theorem, we have
( 1 5 ) {MR2Yι
lifJεn-Dn{o.ro)
=(MR2)
where M =(2π)n/2/2n ι (n-1)1 and
1 for 0 < r < 1
φι(r) =0 for r > 1
( 1 - r 2 ) " - 1 forO < r < 1
0 for r > 1
Since for fixed u £ 0, (x, u) is a homogeneous polynomial which is also a
harmonic function in x, we have by [2, p. 806] and [14, p. 3731 that the right
side of (15) is equal to
(16) G(γ)( y - u9u) \y ~ u \)dy
\y ~u ( Λ l y - u )"
Clearly ( 1 6 ) tends to zero a s R —»oo; so AR is (C,n) summable to zero
for u in Dn(0, r0 / 2 ) - 0.
We also observe after integrating by parts that
17) (MR2)'1 [R rψl-)BrdrJo n\Rl
From the above discussion concerning AR and from [ l , Theorem 1], to show
that BR is (C, n) summable to zero for u in Dn{0,r0/2) - 0 , it is sufficient to
show that
618 VICTOR L. SHAPIRO
(18) (MR2)'1 f F(x)φ ( i l l ) ! l ! _ «>-'•<*.«>,&—»0 as /?—• ooJπ " - 1 \ P / r>2
But by [2, p. 806] and [14, p. 373] the expression in (18) is equal to
En.Dn(θ.ro) l y - u l " " 1 n-μ<- I(R\y-u\)
- K2 dy.
(R\y-u\n-μ-2) \
where Kt and K2 are two constants depending on n.
Clearly (19) tends to zero as R—>oo for u in Dn(0, r o /2) - 0; so BR is
(C, n) summable to zero and the lemma is proved.
LEMMA 9. Let G{x) be a function in L2 on En which vanishes in ^n(0, r υ ) ,
r 0 > 0. Suppose that fE eι *u G(x)dx is spherically convergent to a function
F(χ) which is in C ( 2 ) on En. Then for u in Dn(0,r0/2) - 0
f
is spherically summable (Cfn) to zero.
For (14) also holds in this case, and as in Lemma 8, we have to show that
both AR and BR are (C, n) summable to zero.
Since both F(x) and φ (\x \/R)(x$u) are in L 2 on En, ParsevaΓs formula
gives us both (15) and (16). We therefore conclude as before that AR is ( C, n)
summable to zero for u in Dn(0, ΓQ/2) - 0.
To show that BR is summable (C9n) to zero, we obtain (17) as in Lemma 8.
Then from the fact that AR is ( C, n) summable to zero and from [3, Theorem
55], it is sufficient once again to show that (18) holds.
But by Parseval's formula, we obtain that the expression in (18) is equal
to (19). Observing that for u in Dn{Q, ro/2) - 0 and for y in En -Dn(0,r0) there
exists a constant Kn such that
I W - i ( Λ | y - « P I <κn(R\γ-u\Ti/2 for/? > l
a n d t h a t for s u c h u9 \y -u\2~n i s in L 2 on En - Dn(Q,r0), we c o n c l u d e t h a t
CANTOR-TYPE UNIQUENESS OF MULTIPLE TRIGONOMETRIC INTEGRALS 619
(18) holds and consequently that BR is ( C9 n) summable to zero, which proves
the lemma.
5. Proof of Theorem 1. To prove Theorem 1, it is sufficient to show that
for any r0 > 0, c(u) = 0 almost everywhere in Dn(0y r o /2) . Set
*( \ f [ « ' ( * ' B ) - i - * U»>1 , u
Fι{x) = -I c(u)duJDn(0,rQ) | u | 2
Then, Fχ(x) is in C °° ' on En and
AF t(%)= / eilx'u)c(u)du.JDn(o,ro)
Set
./ \ C ( u )
which is by ( i i i ) continuous in En. Then by Lemma 2 and ( i i ) ,
Tίm | 2 n V ι ( F 2 ; % ; r ) / r 2 | < oo
r-»o
in En - Z and by Lemma 3 and ( i ) , Δ 1 F 2 ( Λ ; ) = - Δ F 1 ( Λ ; ) almost everywhere.
Set F ( Λ ) = Fχ(x) + F2 (x). Take any χ0 in F n and consider Dn{xQir\)>
ri > 0. From the definition of a closed set of vanishing capacity, we see that
there is a closed bounded set of capacity zero Zγ such that
lim \2rNι(F;x;r)/r2\ ^{AF^x)]* lim \2rtfi ( F 2 ; x; r)/r2 \ <
for x in the domain G = Dn(xθ9 ry) — ZιDn(χ9n) Furthermore almost everywhere
in Gf ΔχF(x) = ΔF t (%) + ΔiF 2 (%) = 0. Consequently it follows from Lemma 7,
that F(x) is harmonic in the domain G = Dn(x<)9 r\ ) — ZιDn(xθ9 r ). But F(x)
is continuous in Dn(x 0>ri)« Therefore by [7, p.335], F(%) is harmonic in
Dfji Oi Γ ι ) a n ( i since xQ is arbitrary, F ( Λ ) is harmonic in En
From the fact that F{x) is harmonic in En, we now have that F2(x) =F{χ)
-Fx(x) is in C ( o o ) on F w and that ΔF2 (%) = - ΔFX (%) for all x. Also by [ l ,
Theorem l ] we obtain that
620 VICTOR L. SHAPIRO
[2πYn f e-i(x u)F2(x)dx
is spherically summable (C,n) to zero for u in Dn(0, ro/2) — 0. Therefore by
Lemma 8 for such u,
f e-i{x u)[-ΔF2(x)]dx
is spherically summable (C, n) to zero. But for almost all such u9 we have that
(2πYn f e-i(x'u)AFί(x)dx
i s s p h e r i c a l l y summable ( C , n) to c{u) S ince Δ F ^ Λ ; ) = — Δ F 2 ( % ) , we c o n c l u d e
t h a t for a l m o s t a l l u in Dn(0, ΓQ/2), C(U) = 0, which p r o v e s the theorem.
6. Proof of Theorem 2. T h e proof i s quite s i m i l a r to t h a t of T h e o r e m 1.
O n c e a g a i n it i s su f f ic ient to prove t h a t for any r 0 > 0, c(u) -0 a l m o s t every-
where in Dn(0, ro/2).
Set
F ι ( % ) = - / [ e i ( * tt)-l-ί(*,α)] — du,
and
F 2 ( % ) = - lim ί ei(x>u) ίί^ldu.R-.ocJDn(o,rR)-Dn(o,r0) | α | 2
By (ii i), F2(x) is continuous. Then in a manner exactly analogous to the proof
of Theorem 1 except that Lemmas 4, 5, and 6 are used instead of 2, 3, and 7,
we obtain that F2(x) is in C^°°^ and that ΔF2 {x) = - Δf\ (x). By Lemma 9
and [3, Theorem 55], we obtain that / £ e " ι ^ ' w ^ [ ~ Δ F 2 (x) ]dx is spherically
summable (C$n) to zero for u in Dn(0, ro/2) - 0 . But by [1 , Theorem 1] for
almost all such u, we have that
(2 πyn ί
is spherically summable ( C , n) to C ( M ) . Since ~ Δ F 2 (A;) = Δ F t (x), we con-
clude that c ( u ) = 0 almost everywhere in Dn(0,ΓQ/2) and the theorem is proved.
CANTOR-TYPE UNIQUENESS OF MULTIPLE TRIGONOMETRIC INTEGRALS 6 2 1
7. Proof of Theorem 3. L e t F{ix) be a s in Theorem 2 with n r e p l a c e d by
2, and le t
F 2 U) = - U m /R-oo JD2{θfR)-D2(θ,ro) \u\Z
where r0 > 0. This limit exists for x in Z by ( i i) and for x not in Z by ( i ) ,
(i i i), and Lemma 1. Furthermore by (i i) F2ix) is assumed continuous in E2 - W.
It is clear from the proof of Theorem 2 that to prove this theorem we need only
show that F2ix) is continuous in E2 or what is the same thing that Fix) =
Fγ ix) + F2ix) is continuous in E2.
By (i i) Fix) is continuous in E2 -IF, and by Lemmas 5, Δ2Fix) = 0 in
E2 - Z. Let D2ixOfrι ) be any disc which has a null intersection with W. Then
as in the proof to Theorem 1, Fix) is harmonic in this disc and consequently
in E2 - W. We also observe that now Δ 2 F(%) = 0 in the whole plane and further-
more that Fix) is in L2 on any bounded domain.
Let Wι be the set of discontinuity points of Fix) and let XQ be an isolated
point of Wι» Then there is a closed disc D2ixo,r2) whose intersection with
Wι is xQ. Then by the above discussion we have that Fix) is in L2 on D2ix0% r 2 ),
harmonic in D2ix0i r2) - XQ, and satisfies the further condition that Δ 2 F(% 0 ) = 0.
Consequently by [12, Lemma 4], Fix) is then harmonic in the whole disc and,
a fortiori, continuous at XQ.
Therefore Wl9 has no isolated points and Wι is a perfect set. But W\ C W is
at most denumerable, and by [10, p. 55], Wx is then the empty set. Thus Fix)
is continuous in the whole plane, and, as mentioned above, the proof of this
theorem is reduced to that of Theorem 2.
8. Appendix. In closing we point out that the assumption W and Z have a
null intersection in Theorem 3 is a necessary one. For consider the double
trigonometric integral fE c iu) eι^x'u'du with C ( M ) = 1 . ( i i i) and (iv) of
Theorem 3 are clearly satisfied. Observing that the spherical mean of rank R,
(%), w i th Λ; 5 0 is given by
£ ) ~4πJ2i\x\R)\x\-2=OiR'ι/2),
we see that ( i ) is satisfied with Z equal to the origin. Furthermore, we observe
that for x £ 0
lim / \u\-2ei{x>u)du = 2πfO°i Joir)r'ιdr.R-+OG JD2(O,R)-D2iθ,l) J\χ\
622 VICTOR L. SHAPIRO
Consequently ( i i ) is satisfied with W consisting of the origin. But W and Z do
not have a null intersection, and the conclusion of Theorem 3 does not hold.
REFERENCES
1. S. Bochner, Summation of multiple Fourier series by spherical means, Trans.Amer. Math. Soc. 40 (1936), 175-207.
2. , Theta relations with spherical harmonics, Proc. Nat. Acad. Sci., 37(1951), 804-808.
3. S. Bochner and K. Chandrasekharan, Fourier transforms, Princeton, 1949.
4. M. Brelot, Sur la structure des ensembles de capacite nulle, C. R. Acad. Sci.Paris 192 (1931), 206-208.
5. M. T. Cheng, Uniqueness of multiple trigonometric series, Annals of Math. 52(1950), 403-416.
6. R. Courant and D. Hubert, Methoden der mathematischen Physik, vol. 2 Berlin,1937.
7. O. D. Kellogg, Foundation of potential theory, Berlin, 1929.
8. T. Rado, Subharmonic functions, Ergebnisse der Mathematik, vol. 5, no. 1,Berlin, 1937.
9. W. Rudin, Integral representations of continuous functions, Trans. Amer. Math.Soc. 68 (1950), 278-286.
10. S. Saks, Theory of the integral, 2d. ed., Warsaw, 1937.
11. V. L. Shapiro, An extension of results in the uniqueness theory of double trigo-nometric series, Duke Math. J. 20 (1953), 359-366.
12. , A note on the uniqueness of double trigonometric series, Proc. Amer.Math. Soc. 4 (1953), 692-695.
13. M Summability and uniqueness of double trigonometric integrals, Trans.Amer. Math. Soc, 77(1954), 322-339.
14. G. N. Watson, A treatise on the theory of Bessel functions, Cambridge, 1944.
15. A. Zygmund, Trigonometrical series, Warsaw, 1935.
RUTGERS UNIVERSITY, NEW BRUNSWICK, NEW JERSEY AND
THE INSTITUTE FOR ADVANCED STUDY, PRINCETON, NEW JERSEY
MINIMAL BASIS AND INESSENTIAL DISCRIMINANT
DIVISORS FOR A CUBIC FIELD
L E O N A R D T O R N H E I M
In terms of the coefficients OC, jS, γ of a defining equation
of a cubic field F over the rational number field Q9 Albert [ l ] has given an ex-
plicit formula for a minimal basis, that is, a basis of the integers of Q{θ)
over the rational integers. We solve this same problem with a shorter proof and
a simpler result. This basis is then used to find the maximal inessential dis-
criminant divisor, that is, the square root of the quotient of the g.c.d. of the
discriminants of all integers of Q(θ) by the discriminant of Q(θ). It is known
[3] that the only prime dividing it is 2; we determine the power as 2° or 2 ι .
We first secure a normalized generating quantity,
L E M M A 1. If K is any cubic field, then K = Q(θ) with
( 2 ) 6>3 + aθ2 + 6 = 0 ,
where ( i ) a and b are rational integers, ( i i ) no factor of a has its cube dividing
b9 and ( i i i ) if 3 \\a, then the discriminant Δ = - b ( 4 α 3 + 27 b) of θ is not di-
visible by 3 4 unless 3 | b.
Here gn |1 y means gn | γ and gn l \γ.
Proof. The substitution θ'~ θ+ α/3 is used to obtain an equation of form
(1) with α zero. Follow this by the substitution 0 '= 1/0 to obtain (2) . For
Conditions ( i ) and ( i i) it is obvious that a substitution 0'= hθ will be effective.
If ( i i i) does not hold apply the substitution 0 ' = ab - 3 bθ + a2 θ2; then 0 ' 3 +
cθ'2 +d = 0 where
Received February 10, 1954.Pacific J. Math. 5 (1955), 623-631
623
624 LEONARD TORNHEIM
Now 3 2 \\c since (b9 3) = 1. Also 3 4 \d. If 36|</, then the quantity 0 " = 0 7 9 s
satisfies the conditions of the lemma, where s is the largest integer for which
(s ,3) = 1, s \c, and s3\d. If 3 6 ^ use θ"= 0 7 3 s .
Essentially the following lemma is given by Sommer [2; p. 261],
LEMMA 2. 7^e integers of Q( 0), where 0 is described in Lemma 1, have a
basis over the integers given by
__β + <9 B2 Λ-aB + ( β + α ) 0 + θ2
ωx = 1, ω 2 = — , ω 3 =0 D 2 D
with B9 D9 Dι rational integers satisfying
(3) 3 β - f α =
( 4 ) 3B2 +2aB^0(D2Dι),
( 5 ) B3+aB2 +b^0(D3Df),
(6) -Δ = ό(4α 3 + 276) EO (D6D2),
and D9 D t are maximal subject to these conditions.
Proof. We shall first prove that D = 1. Let p be a prime dividing B and D.
By ( 3 ) , p also divides α. But then by ( 5 ) , p3\b, contradicting the choice of
0. Hence ( B , D) = 1.
From ( 3 ) and ( 4 ) , we have aB = 0 ( D ) . Therefore D \a. But by ( 3 ) , D = 3
or 1.
If D = 3, then 3fi> because from (5) we would get D \B. But then (6) con-
tradicts (i i i) of Lemma 1. Hence D = 1.
Therefore the problem is equivalent to determining the largest Di for which
there is a solution B satisfying (4), (5), (6), when D = 1. It is sufficient to
find solutions of these congruences with Dγ replaced by prime powers pΓ and
then Dγ will be their product. A value of B can be found from solutions modulo
pΓ by using the Chinese remainder theorem.
Thus we wish to determine the maximal value e of r for which there exists
a solution B of the simultaneous congruences
(7) B(3B
CUBIC FIELD 625
( 8 ) B3 + aB2 + b = 0{p2r),
( 9 ) - Δ = 6 ( 4 α 3 + 2 7 6 ) Ξ 0 ( p 2 r ) .
The power p e exists because of ( 9 ) ; in fact if p u | | Δ , then e <_s, where
5 = U / 2 ] .
Case I. ( p , 3 b) ~ 1. Then e - s. For, let B be a solution of
L = 3 β + 2α = 0 ( p s ) ;
hence ( 7 ) is satisfied. By ( 9 )
Now
L 3 - 3 α L 2
Ξ 0 ( p 2 s ) .
T h i s on e x p a n s i o n g i v e s
α ~ 4 α 3 = 0 ( p 2 s ) ,
w h i c h with t h e a b o v e formula s h o w s t h a t ( 8 ) i s s a t i s f i e d . T h u s ( 7 ) , ( 8 ) , ( 9 )
h o l d wi th r - s. H e n c e e >_ s . B u t s i n c e e < s we h a v e e = s .
Case II. p I 3 ό.
H i . ( p , 2 α ) = l . Then e = s . F o r , b y ( 9 ) , pu\\b. S i m p l y t a k e B=0(ps)
t o s e e t h a t ( 7 ) , ( 8 ) , ( 9 ) h o l d w i t h r = s.
Π i i . p \b9 p\a. Then e = 0 if p \\b and e = 1 = s - 1 if p2 \\b. N o t i c e t h a t
p \b by ( i i ) of L e m m a 1. F i r s t , if p \b9 t a k i n g B = = 0 ( p ) p r e s e n t s a s o l u t i o n
of t h e c o n g r u e n c e s wi th r = 1; t h u s e >_ 1. On the o t h e r h a n d , if e >. 1, t h e n
p\B by ( 8 ) ; s o t h a t p2 \b a g a i n by ( 8 ) . F i n a l l y , if e > 1 t h e n p 3 | ό by ( 8 )
s i n c e p | B by t h e p r e c e d i n g s e n t e n c e . T h i s i s a c o n t r a d i c t i o n to ( i i ) of L e m m a
1; h e n c e e < 1. It i s e a s y to s e e t h a t if pφ- 3, t h e n s = 1 w h e n p\\b a n d s — 2
when p 2 11 b. If p = 3, then s - 2 u n l e s s p 11 bf p 2 \ a and t h e n s = 3.
I l i i i . p = 3, p I α, p ^ 6 . N o t i c e t h a t t h e n s = 1 by ( 9 ) a n d ( i i i ) of L e m m a 1.
I Ι i i i ( l ) . 3 2 | α . Then e = 0 unless b = ± 1 ( 3 2 ) in which case e = 1. Now
626 LEONARD TORNHEIM
e < s = 1. Furthermore, the fact that e = 1 if and only if 6 = ± 1 ( 3 ) i s a con-
sequence of (8) since only then does Z > 3 + 6 = 0 ( 3 2 ) have a solution for (3 ,6) = 1,
the solution being given b y β = = - 6 ( 3 ) ; ( 7 ) and ( 9 ) always hold with r = 1.
Iliii ( 2 ) . 3 11 α. Then e = 0, unless b + a = ± 1 ( 3 ), in which case e = 1.
That e < 1 is a consequence of ( 9 ) and ( i i i ) of Lemma 1. If r - 1, then ( 7 ) and
( 9 ) always hold and ( 8 ) has a solution if and only if£> + a s ± l ( 3 ). For,
if B sat is f ies ( 8 ) then 3 | # ; hence £ 2 = 1 ( 3 ) , aB2 + b = a + b ( 3 2 ) . But
B3 = ± 1 ( 3 2 ) so that α + 6 ^ - β 3 = + 1 ( 3 2 ) . Conversely, if a + b = + 1 ( 3 2 ) ,
take J δ Ξ - ( α + 6 ) Ξ ± l Ξ - 6 ( 3 ) ; then B3 + α δ 2 ύ E ί 1 + α + έ E O ( 3 2 ) .
Iliv. p = 2, 2 I 6, 2 | o . Define t and c by 2t \\ b, b = 2*c.
Hiv ( 1 ) . ί odd. From ( 7 ) , 2\B. In the expression on the left in ( 8 ) , there
is only one term, either aB2 or b, containing 2 to the lowest power. Hence
e < [ ί / 2 ] . But B = 0 ( 2 Γ ) with r = [ ί / 2 ] does provide a solution of the three
congruences. Hence e - [ ί / 2 ] Notice that e = 5 — 1 since u — t Λ-\ if ί = l
but u = t + 2 if ί > 1.
Iliv ( 2 ) . t = 2. Let 4 ^ | | ( 4 α 3 -h 27 6) , then u; > 1. Set 4 α 3 + 276 = 4 ^ . By
( 9 ) , e < + 1. Now e >_ w simply by replacing s by w in the solution of Case
I. It remains to determine when e = w + 1. Then from ( 7 ) , 2 | β and from ( 8 ) ,
2 2 | β . Also from ( 7 ) , 3β + 2a EΞ 0 ( 2 " ) ; that is , SB =~2a + 2wS. Now the
product of 27 with the congruence ( 8 ) gives
4 α 3 - 3 . 2 2 M ;
aS2 + 2 3 u ; S 3 + 276 s 0 ( 2 2 u ) + 2 ) .
Hence
2 ^ S 3 + / / ~ 3 α S 2 = 0 ( 2 2 ) .
If S = 0 ( 2 ) f then ff = 0 ( 4 ) f an impossibility. Hence S i s odd, S2 EE 1 ( 4 ) ,
S 3 = 5 ( 4 ) , and
2WS + f f + α = 0 ( 2 2 ) .
But since w > 1, we have 2^S = 2W ( 2 2 ) . Hence
CUBIC FIELD 627
(10) 2M; + / / + α = 0 ( 4 ) .
If w = I, then H = α 3 + 2 7 c = 0 ( 2 ) , a contradiction to (10) . Hence w > 1.
Conversely, if (10) is true, then all the congruences in this paragraph are
satisfied by taking S odd; that is, by taking for B a solution of
3B +2a^2w (2w+ι).
H e n c e e = w + 1 if and only if ( 1 0 ) i s s a t i s f i e d ; t h a t i s , H + a = 0 ( 4 ) . N o t i c e
from the definit ion of w that u = 2 + 2w; h e n c e s - w + 1.
Il iv ( 3 ) . t = 2v(v > 1 ) . From ( 9 ) , u = 2v 4-2; h e n c e e < s = v + 1. Now
β Ξ 0 ( 2 ^ ) y i e l d s a so lut ion of the c o n g r u e n c e s with r — v; h e n c e e >_ v We
determine when e = v + 1. T h e n from ( 7 ) , B i s even. Again from ( 7 ) e i ther
2 | | β or 2V I B. In the first c a s e v <_ 1 by ( 8 ) and t h i s i s a contrad ic t ion to
v > 1; h e n c e β ^ 2 V K. Now ( 7 ) h o l d s while ( 8 ) impl ies
23vK3 + a22vK2 +22vc^0 ( 2 2 t ; + 2 ) ,
which gives, since v > 1,
o χ 2 + C Ξ 0 ( 4 ) .
Thus K is odd and
a + c =0 ( 4 ) .
Conversely^ if this last congruence is satisfied and B is taken as a solution
of B Ξ= 2V ( 2 V + 1 ) , then β is a solution of ( 7 ) , ( 8 ) , and ( 9 ) .
These deductions are summarized in the following theorem.
THEOREM 1. Let θ satisfy the conditions of Lemma 1. A minimal basis
ofQ(θ)is
ω 1 = = l , ω2 = 0, ω 3 = {J3 2 + α β + ( β + a) θ + θ2 \/D,
where D is a product of prime powers ρe determined by the prime powers p
for which ( p 2 ) s | | Δ as described below and B is a common solution of the con-
gruences given below:
(1) If ( p , 3 6 ) = 1 , t h e n e = s a n d SB + 2 a 0 ( p e ) .
( 2 ) If p I a , p 11 b% t h e n e = 0 . , 4 Z s o e = s - 1 ί/ p / 3 cmc? e = s ~ 2 i / p = 3 .
628 LEONARD TORNHEIM
( 3 ) If p\a9 p2\\b$ then e = 1 and B = 0 ( p e ) . Also e = s - 1 unless p = 3
am/ p I α α^ί/ ίλe z e = s - 2.
( 4 ) / / > I 36, ( p , 2 α ) = 1, then e = s and B = 0 ( p e ) .
( 5 ) 7/p = 3, 3 | α , 3^6, ίAerc e < 1 = s; and e = s ι/ cmc? orcZy i / έ + α Ξ + l ( 9 )
ami ίAera B = - 6 ( 3 ) .
( 6 ) / / p = 2, ( 2 , α ) = l , 2 * | | 6 αzzJ
( a ) if t is odd9 then e = s - 1 and B s O ( 2 e ) ;
( b ) if t = 2 ίΛerc e = s - 1 urc/ess H + a = 0 ( 4 ) , wΛere W = -
α/irf then e = s . ,4Zso 3 5 + 2α = 2 s " ι (2s).
( c ) if t > 2 and even, then e = 5 - 1 unless a + c=0 ( 4 ) , where c =6/2*,
and then e = 2. 4Zso B ^ 2 s " 1 ( 2 s ) .
The discriminant of (?(#) is Δ/D 2. It divides the discriminant Δ ( α ) of
every integer α of Q(Θ) and hence their g.c.d. G. The largest inessential
discriminant divisor F is the square root of the quotient G/(Δ/Z)2).
THEOREM 2. TΆe largest inessential discriminant divisor F is 1 except
it is 2 in Case 6b of Theorem 1 when
(11) //-3a + 2 e-U0(23)
αmZ in Case 6c when
(12) a + c + 2 e - 1 ^ 0 ( 2 3 ) .
Proof. The discriminant Δ(α) of an integer Cί = c\ ω\ + c 2 ω 2 + cz ω 3 can
be found from the formula
| α ι 7 | 2 Δ ( 0 ) ,
where the elements of the determinant | α t y | = |α, y (α) | are defined by
α1""ι = α a + ai2 0 + o i 3 02 ( j = 1, 2, 3 ) ,
Since the discriminant of α is unaltered by addition of a rational number, we
have
Δ(α) = Δ ( c 2 ω 2
where
CUBIC FIELD 629
+ C3{B + α ) / D ] # + (,
In computing β2 use the fact that θ3 = - aθ2 - b and θ4 = a2θ - bθ + ab. Also
since the first row of | α t y ( / 3 ) | is 1,0,0, any rational terms can be ignored.
Hence,
c3(B3+aB2+b) c,c2(3B2 + 2aB) c2cAW+a)/ Ί r ) \ | | 3 2 3 2 3 o
(13) | α , μ _ + _ + _ + c ».
Thu s
| α l 7 ( ω 3 ) | =D3
and
M , / x, ( 3 S + 2aS) ( 3 β + o )15 ) I αj.- ( ω 2 + ω 3 ) I - I o ί ; ( ω 3 ) | = + + 1.
D2 D
Now, since GD / Δ is the quotient of the g.c.d G of | α j y | 2 Δ by Δ/D , it
equals the g.c.d of \aη\ 2D2. Hence the inessential discriminant divisor F is
the g.c.d of I aij I D.
To find F we determine for each prime p the highest power p* which remains
in all the denominators of the | α j / ( c θ | expressed in their lowest terms. Then
F is the quotient of D divided by the product of these prime powers and thus F
is the product of all pe"K
In all c a s e s of Theorem 1 except in 5 when a + b = ± 1 ( 3 ), in 6b when
H + a = 0(22), and in 6c when α + c = 0 ( 2 2 ) , B may be chosen to satisfy
either
or
In these cases ( 1 5 ) implies, s ince its first term is then integral, that e = /
when p\a. But if p | a then p | b and s ince we need consider only e > 0 we have
Case 3 of Theorem 1. Then ( 1 4 ) with B E O ( p 3 e ) shows t h a t / = 1= e.
630 LEONARD TORNHEIM
N e x t , in C a s e 5 w h e n b+a=±l ( 3 2 ) , 3 { β . If 3 2 | α , t h e n ( 1 5 ) i m p l i e s
t h a t / = 1 = e . B u t if 3 11 a t h e n a - 3 α i a n d aχ + 6 ^ 0 ( 3 ) by ( i i i ) of L e m m a 1.
Were / = 0 , t h e n B + 2a ι = 0 ( 3 ) b y ( 1 5 ) , w h i c h i m p l i e s B = ax ( 3 ) . B u t t h e n
B 3 + a B 2 + b = a 3 + b έ θ ( 3 ) ,
a contradiction to (8). Hence again f = e.
In both Cases 6b and 6c, 2 | β by (7). Now
2(3B + α )ω 3 ) I + |α jy(-ω 2 + ω 3 ) | - 2 | α ι ; ( ω 3 ) | =
D
Since 2 11 2 (3β + o), we have / > e - 1.
We now consider in particular Case 6b when ff + o = 0 (4) . Then 3β =
- 2a + 2e ιQ, where 0 is odd. Thus
2 7 ( β 3 +aB2 + 6 ) = 4 α 3 + 2 7 6 - 3 ρ 2 α 2 2 e - 2
+ ( ? 3 2 3 e - 3 .
Hence if f ~ e — 1, then
# - 3 α + 2 e - 1 = 0 ( 2 3 )
by (14), and if this is satisfied then / = e - 1. For, the first term in (13) has
numerator divisible by 2 2 e + 1 , and 2 e | | ( 3 β 2 + 2aB) and 2° | | ( 3 β + o ) so that
2 e + 1 | [ c 2 c 3
2 ( 3 β 2 + 2 α β ) + O C 2
2
C 3 ( 3 β + α ) ] .
Hence in lowest terms \a{j \ has a denominator divisible by no power of p greater
than e — 1.
We finally discuss Case 6c when α + c = 0 (4) . Then β = 2 6 " 1 + C2 e, where
we may assume that 2 e + 2 | C, and b = 2 2 ^ e " ι ^c. Hence
If / = e - 1, then by (14) this expression must be = 0 ( 2 2 e + 1 ) , so that
If this is satisfied then / = e - 1 because the first term of (13) has numerator
divisible by 2 2 e + 1 , and 2 e 11 ( 3β 2 + 2αβ) and 2° 11 (3β + a) so that
CUBIC FIELD 631
REFERENCES
1. A.A.Albert, A determination of the integers of all cubic fields, Ann. of Math.,31 (1930), 550-566.
2. J. Sommer, Vorlesungen ilber Zahlentheorie, Berlin, 1907.
3. E. v. Zylinski, Zur Theorie der aus serwe sentliche D is krminantenteiler alge-braischer Korper, Math. Ann. 73 (1913), 273-274.
UNIVERSITY OF MICHIGAN
ON EIGENVALUES OF SUMS OF NORMAL MATRICES
H E L M U T W I E L A N D T
1. Problem, notations, results. A well-known theorem due essential ly to
Bendixson [ l , Theorem I I ] s ta tes that if X and Y are hermitian nxn matrices
with eigenvalues
ζγ <_ ζ2 <_ < ζn and η ι <C 77 2 < <_ η^ ,
then every eigenvalue λ of X + iY is contained in the rectangle
What is the exact range of λ, for given ζv and ηv? We shall solve the following
slightly more general problem, referring to normal instead of hermitian matrices.
Let Cί i, , Cί j, βι9 9 β be given complex numbers. Describe geometrically
the set A of all numbers λ which may occur as eigenvalues of A + B, where A
and B run over all normal nxn matrices with eigenvalues OC i, , CXn and
βί9 *, βn respectively.
To state the results concisely let us denote by the terms circular region and
hyperbolic region every set of complex numbers ζ + iη which may be described,
using some real constants α, bf c9 d9 by
(1) aξ+bη + c{ξ2± η2)+d > 0 .
where + refers to the circular, - to the hyperbolic case. We denote by \ Qiv\
and \βv\ύie sets whose elements are Cίi, ,(Xπ and β ^ , / ^ respectively.
For every two sets Γ, Δ of complex numbers we denote by Γ 4- Δ the set whose
elements are all γ 4- δ, where y G Γ , 8 € Δ. Our main result is
THEOREM 1. // Oli, •••, 0Cπ, βι9 ,βn are arbitrary complex numbers,
then the set A defined above can be represented as an intersection:
Received April 16, 1954. This paper was prepared under a National Bureau of Stand-ards contract with the American University.
Pacific J. Math. 5 (1955), 633-638
633
634 HELMUT WIELANDT
where Γ runs over all circular regions which contain \ βv\
In the special case considered by Bendixson the result may be simplified
as follows.
THEOREM 2. If &ι, , (Xn are real and /31, , βn purely imaginary, then
Λ = Π ΔΔ
where Δ runs over all hyperbolic regions which contain \<XV\ + ί βp\.
2. Proofs. We recall the following theorem [ 3 , Theorem 2 ] : Let λ 1 ? , λΛ
be complex numbers, y and z complex n xl matrices, y*y = 1. Denote by
M{y,z) the point in real 3-space with rectangular coordinates [\iy*z, & y*z9
z*z ] and by P{ζ), for every complex number ζ, the point [ R £ , &ζ, \ζ\2]
If, and only if, the convex closure of the n points P(λv) contains M(y9z)9 then
there exists a normal nxn matrix L with eigenvalues λ-u 9 λn such that
Using the notations introduced in § 1 we prove for arbitrary CCV, βv:
LEMMA 1. Let ζ be a complex number. Then ( E Λ if, and only if, the con-
vex closure C of P ( d i ) , , P ( CLn) has a point in common with the convex
closure C ' of P ( ζ - βχ ), ,P{ζ- β n ) .
Proof, ( a ) Let ( G Λ , Then there are normal matrices A, B with spectra
&1> * * •> an a n d βt9 * '> βn
s u c n that
{A +B-ζI)y=0
for some normalized vector y. Putting
we conclude from the necessity part of the theorem quoted above that M{y,z)
is a common point of C and C".
(b) On the other hand, let there be some point
Then iΛ2+^2 < ,» hence we have N = M(y,z) for some vectors y, z such
ON EIGENVALUES OF SUMS OF NORMAL MATRICES 6 3 5
that y*y = 1. By the sufficiency part of the theorem quoted above there are
normal matrices A9 B with eigenvalues (Xv, βv^ ζ- βv such that Ay = z = By.
Define B - ζl - B. Then B has the eigenvalues βv and satisfies Ay = ( ζl — B) y,
hence ζ is an eigenvalue of A + B (with γ as a corresponding eigenvector).
We transform the "three-dimensional" Lemma 1 into a "two-dimensional"
form.
LEMMA 2. Let ζ be a complex number. Then ζ<£A if, and only if, there
exists a circle or a straight line separating CX 1 ? , dn from ζ— βι, , ζ— β .
Proof. From Lemma 1 we know that ζ£ A if, and only if, there exists a
plane separating C from C', that is, separating
P ( α i ) , . . , P ( α Λ )
from
Piζ-βJ,..., P(ζ-βn).
T h i s m e a n s t h a t ζ£A if, a n d o n l y if, t h e r e a r e r e a l c o n s t a n t s a9b9c9d s u c h
t h a t for v = 1, , n
a R α v + b&av + c\av\2 + d > 0 ,
( 2 )
& β v ) + c \ ζ - β v \ 2 +d < o .
This proves Lemma 2. We turn to the proof of Theorem 1.
(a) Let ζ G Λ, and let Γ be any circular region containing /3 1 ? 9 βn
We have to show that £ E { α v ϊ + Γ. Now ζ— Γ is a circular region containing
ζ — βχ9 , ζ — βn By Lemma 2 there is at least one Cίp such that
(b) Let ζ£ A, Then Theorem 1 claims that there is a circular region Γ
containing \βv\ such that ζfl:\CLv} +T. Indeed, interchanging A and β in
Lemma 2 we see that there is a circular region Γ containing β 9 , β , but
none of the ζ— CCV
Theorem 1 being proved, we turn to Bendixson's case where QLV = ξv is
real, and βv-ir]v with ηy real. We have to show, for any complex number ζ,
636 HELMUT WIELANDT
that ζ jέ Λ if, and only if, there is a hyperbolic region which contains all points
ζn + iΉvi but does not contain ζ. Since this statement obviously is not affected
by a translation applied simultaneously to the ζ + iη and to ζ, we may assume
without loss of generality that ζ- 0.
From (2) we know that 0 ^ Λ if, and only if, there are real numbers α, b, c9 d
such that
aξv + cξl+d > 0
(3) ( v = l , . . . , n ) .
-bηv+cηl + d > 0
This condition is equivalent to the existence of real numbers, α, b9 c such that
(4) aξμ+bηv + c(ξ^-r^)>0 (μ,v=l,...,n).
This inequality is equivalent to the existence of a hyperbolic region containing
all points ζ + i ηv, but not 0.
3. Remarks, (a) It is seen from Theorem 1 that in the determination of
Λ only the distinct (Xv and the distinct βv matter. Multiplicities are of no im-
portance.
(b) If ί α ^ i C ίcίvίand ί/3 piC { βv\, then ΛC Λ.
Proof. For every λ G Λ there are normal r x r matrices A and B with eigen-
values dp and βp such that λ is an eigenvalue of A + B. Define
φ,), B.[\)\ <*„/ \ A,/
Then λ is an eigenvalue of A + B. The eigenvalues of A and B coincide with
CCi, , 0Cn and βγ, , βn except for the multiplicities; hence λ £ Λ by ( a ) .
( c ) Λ is a closed bounded set, since the se t s ίCί^S + Γ are closed and
there is a bounded circular region Γ containing \βv\.
( d ) In Bendixson's case every connected component of Λ is simply con-
nected.
Proof. By Theorem 2 every point of the complement Λ* of Λ is the end
ON EIGENVALUES OF SUMS OF NORMAL MATRICES 637
point of some half line which is entirely contained in A*. Moreover by (c) , A*
contains the exterior of some circle. Hence A* is connected.
(e) If n — 2 in Bendixson's case then Theorem 2 implies that A is the
intersection of Bendixson's rectangle with the rectangular hyperbola passing
through the vertices of that rectangle. This result has been previously ob-
tained by W. V. Parker [2, Theorem 1 ].
(f) The foregoing remarks lead to a simple procedure for constructing
A in Bendixson's case. Let (with a slight change of notation) ξγ < ζ2 <•••
< ζ be the distinct eigenvalues of the hermitian n x n matrix X, and let
?7t < η2 < < η^ be the distinct eigenvalues of the hermitian nxn matrix
y. We define p.μ ( μ - 1,
which passes through.
, m - 1) to be that part of the rectangular hyperbola
which lies in the rectangle with these vertices. Similarly we define H^ί K =
1, •••,&--1) interchanging the role of the <f's and the 77's. Then (b) and (e)
show that ΞμC^ A and II/< C A. It is easily seen that the union of all Ξ μ ' s and
H/<'s consists of one or more closed Jordan curves each of which is contained
in the closed exterior of every other curve, and that no point exterior to all
curves belongs to A. On the other hand, the interior of every curve is con-
tained in A, by (d) . Hence λ is the largest bounded region whose boundary
is the union of3i , , *Bm-1 , H l 5 , H^. t .
As an example we construct the range A of the eigenvalues of X 4- iY where
X and y are hermitian matrices whose eigenvalues are 0, 1,4,8 and 0,2,3
(with arbitrary positive multiplicities).
638 HELMUT WIELANDT
REFERENCES
1. I. Bendixson, Sur les racines d'une equation fondamentale, Acta Math. 25 (1902),359-365.
2. W.V.Parker, Characteristic roots of matrices, Amer. Math. Monthly 60 (1953),247-250.
3. H. Wielandt, Die E ins chlies sung der Eigenwerte normaler Matrίzen, Math. Ann.121 (1949), 234-241.
AMERICAN UNIVERSITY
UNIVERSITY OF TUBINGEN
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Pacific Journal of MathematicsVol. 5, No. 4 December, 1955
Richard Horace Battin, Note on the “Evaluation of an integral occurring inservomechanism theory” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
Frank Herbert Brownell, III, An extension of Weyl’s asymptotic law foreigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
Wilbur Eugene Deskins, On the homomorphisms of an algebra ontoFrobenius algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
James Michael Gardner Fell, The measure ring for a cube of arbitrarydimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
Harley M. Flanders, The norm function of an algebraic field extension.II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
Dieter Gaier, On the change of index for summable series . . . . . . . . . . . . . . . . . 529Marshall Hall and Lowell J. Paige, Complete mappings of finite groups . . . . . 541Moses Richardson, Relativization and extension of solutions of irreflexive
relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551Peter Scherk, An inequality for sets of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . 585W. R. Scott, On infinite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589A. Seidenberg, On homogeneous linear differential equations with arbitrary
constant coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599Victor Lenard Shapiro, Cantor-type uniqueness of multiple trigonometric
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607Leonard Tornheim, Minimal basis and inessential discriminant divisors for
a cubic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623Helmut Wielandt, On eigenvalues of sums of normal matrices . . . . . . . . . . . . . 633
PacificJournalofM
athematics
1955Vol.5,N
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