ERRORS IN LINEkR NONHOMOGENEOUS
AILEBRAIC SYSTEI
by
ROBERT DONP.LD QEDER
A TiE3IS
submitted to
OREGON STA TI COLLEGE
in partial fulfillment of
the requirements for the
degree of
EASTER OF SCIENCE
June
l9O
APMtC
Professor of Mathematics
In Charge of Ihajor
Head of Department of Mathematics
Chairman of School Graduate Committee
Deai
of Graduate School
Date thesis is presented
Typed by helen C. Pfeifla
May t, 19S0
ACKNWLEDGMENT
To Doctor Arvid T. Lonseth, the author
owes a debt of gratitude for his guidance,
suggestions and constructive criticism in the
preparation of this thesis.
Also, for the encouragenent given to me,
the writer wishes to expres s his gratitude to
his w)íe,
TABLE OF CONTENTS
CHAPTERI.
INTRODUCTION................ page
i
Statenientoftheproblem
.....
......
Method of Investigation
An Inequality of K.O. Friedrichs
Relatod Investigations . . . .
CHAPTER
Ii
UPPER AND
ILWR
I
,
.
.
.
.
.
.
.
.
.
.
.
.
.
.
III.
TriE ALGEBRAIC CASE
.......
.......
I(k) -D(k)I
ErrorinaDeterminant
(k,y)
Limitationof(xj-xjj
6
6
8
9
9
.
Fori of solution of Algebraic Systems.
Limitation of
Limitation of
S
5
..............
............
........... li
DETENINANT
BOUNDS FOR A
Inequality of Hadamard
Sufficient Condition that a Deterriinant be not zero.
Positive Determinants ..............
Inequality of Friedrichs . .
CHA.PTiR
1
.
.
.
.... .....
-D(k,y)I
U
.
12
. . 13
. . . . . . . 15
.
.
.16
CHAPTERIV.}UMRICALEX.pLß............,18
Example. .
.
. .
Table of values
LITuiATJt CITED.
.
.
*
. . . .
*
. .
.
.......
.
. .
.
. . .
.
. . . .
. .
.
.
.
.
.
*
.
.
.....
18
20
21
ERRORS IN LINEAR NONHOMOGENEOUS ALGEBRAIC SYSTEMS
CHAPTER I
INTRODUCTION
STATENT
OF THE PROBLßM.
The purpose of this thesis is to
consider a system of linear nonhomogeneous algebraic equations in n
unknowns, x1, X2, - - - -, Xn,
31
where the n2 coefficients
much the solutions X1,
are subject to error, and to find by how
- - - -,
,
of the approxinating equations
j=1
x's.
may differ frau the original
rors in the coefficients
frequently occur in the application of mathematics to natural
plie-
niiena, where they may appear as errors cf observation; they may also
arise when decimal coefficients aro "rounded 0fft
THOD OF INVESTIGATION
.
This problem has been solved before and
partial or complete solutions may be found in papers by Etherington(3),
Lonseth() and
Moulton(6).
The results of Lonseth and
Moulton are
compared with the results of this paper in a numerical example in
Chapter four.
The method of this paper is quite different
above.
fri those mentioned
It follows Tricomi's solution(8) of the ar.1ogous problem with
respect to a linear integral equation of Frociholm type and second kind.
2
Tricorni.
-
x(s)
1)
considered the integral equation for x(s)
K(s,t)x(t)cit = y(s),
wilere K(s,t) and y(s) are known
oils
on the square O
parameter A
O
1,
s
functions.
Also, K(s,t) is continu-
i and y(s) is continuous on O
s,t
s
1
The
is restricted to values such that the Fredholm f orrnula
(9, p.211) for x(s) holds.
If the kernel K(s,t) is replaced by
(s,t),
where
(s,t)
-
,
K(s,t)k
E
o, and "small,"
I
on the square O
1, and if
s,t
is correctly restricted, the
equation
-
i(s)
2)
(s,t)(t)dt
y(s),
O
s
1,
Jo
is solvable for 5(s) by Fredho].m's
formula.
Triconii
derives an expros-
sion which limits the maximum value of
Ji(s) -x(s)I
Suppose
D(
(s,t)
and
\)
is
,
the Fred.holnì denontinator
n/2
If
and
K
Y
s
O
xn
for K(s,t),
(
)
that
*
are upper bounds for K(s,t)J and y(s)
,
respectively,
J
and L
J
?..
J(K
+
),
E
being
sorne
"small" positive number, then
Tricorni proves the following theorem:
Tricorni' s Theorem.,
E
and
If
for
E
is
a positive nuirber such
!(s,t) -K(s,t)< E
that
3
then
(f y
Ji(s) -x(s)J
ïhe coerricieflt3
XL(L)i1'(L) +
La'2(L) +í1(L)c1n(L)]
-II'(L)
E]
in the algebraic systern can always be written
aj.z
in the £oxa
-
ajj =
Ii ir
gij
=
s
i
{
?hen tze
i.
s are real mribcrs, tim the
ïstems
Xj-EkjXjYj
3)
1)
where ijj =
E j
"ii ±
two integral eçation
ana1oes
"sìil,' are algebraic
E
,
i) arxï 2
) ,
with
¿1
of the
1.
it is supposed that the detei,ninants D,(k) and
(k) of the two
systems are not zero se that both systcs can be solved uniquely.
stead of the transcendental integral functiri
(j)
(x) and
fl (z)
,
In-
two polyniials
are used, where
Q
n
s)
(
)
=
(Ïmn
t
)
and
x((t)2()
n
6)
¿p
n
(x)
When !kjjJ<s
paper
rw
i
IEijI<
be stated as
e.nd
I
I
IYil< Y, the rrincipal reu1t of this
Theorem 1.
(k)I
6<
and
is a positive number such that
If
il'
[<E
-
,
then
(Xj -
J(k)I
<
Coroflary 1.
(a)
-
[IUnOc)I
+ E)]
e
If
the coefficients in equation L) satisfy
I
(b)
)(K)cpt(K+e)]
+
xj
for i
,
-
1, 2,
-, n;
r
(1 - ku)ÌT11(k)
6
where
TT(k)
i-1
rrki -
)
-
)
J
then
i-iYE
-
Xj
I
xii
(1-i11)T7(k)
Corollary 2.
(a)
[K
"(K +) + 9)(JÇ) q'(K
[1_iiirk - E
+
+
)]
12
the coefficients in equation
Li.)
satisfy
1-1jl1JIjjI,fori1,2,
e
(b)
E
(K)
'(K+
e(K) = (1-K)(1-2K)...(1-(n-1)K),
where
()
'
(n-1)K
then
-
xi - xj<
'
1;
Jj(K)çt(K +) +q(K) q'(K
e(K)
%(K)
[
-c'(K
A table of values for 9)3(x),
be found at the end of Chapter IV.
i
+e)J
+ L)]
q"3(x), p3(x) and
'3(X) villi
5
LN INEÇULLITY O? K.O. FR1Ltt1CHS.1
of an inoquality of K.O.
bound for
t
Fridrchs
certain typo or
ie
given in this paper
It is the belief of the
uality has never been published and
ry
L&T1) ThUSIOATIOiS.
inc1uìes a proof
hieh specifies a positIve lower
detriant.
writer that a proof of this in
the
This tbesi
be new.
Adazs(1) considers the hcxogeneou8
tra.
caso, :ì() = O, of equa tions 1)
nd 2) and derives bounds for the
errors in the characteristic values aìd characteristic functions
associated With the prcblen.
Lonseth derives the inequality
-x<
7)
where
th
+
fAjjI
l
-
is the nonvanishing deterrriinant of the eyste
¿
ajjxj =
i = 1, 2,
,
,
n,
.
i satisíiee the systen
n
: (ajj
+
jj)xj = yj + li'
Lij is the co.factor
f &jj in
L
i
,
= I, 2,
,
n,
and
4t
ijl;IiiI<
I
il
1.
5uested
:pAii
j].
by Dr. A. T. Lonseth, ?rofeasor or
Oregon State College.
athentics,
CHAPTER II
UPPER AND
LVER BOUNDS
FOR A DETErth1INANT
In order to prove Theorem i essential use is made of an
inequality of Hadarnard which specifies an upper bound for a
determinant.
A proof is presented below.
Also shown in this chapter
is a sufficient condition for a determinant to be not zero and a
sufficient condition for a determinant to be positive or zero.
This
enables the writer to prove Friedrichs inequality which cives a
positive lower bound for a certain type of determinant.
INEQUALITY OF
HADARD.
whose elements are a.1, i,j
D2
TT
i=l
Proof
;t
Theorem 2.
1, 2,
If D is the determinant
,
n, then
a
k1
(,
cxjx
Suppose that
p.3l):
where Cii
,
caj,
i , 3=1
is a positive definite quadratic forni1 arrì let
whose elements are c1
cC21
.
C1
¿i
.
c
.
C22-
...
. .
C2
=0
...
C,_
is a polynomial of degree n in
1.
be the determinant
Then the determinant
.
c
¿
,
,P(
It can be shown (2, p.171)
Positive for all real values of the variables x
xl = X2 = . . = Xrl
O.
.
except for
7
P( ?)
that
E
Ecu
=
-,
has n positive roots,
and
.
,
and that
Laking use of the fact that the
geometric mean of n positive nunibers is less than or equal to the
arithmetic mean, it foliovfs that
n
8)
)
If cil
O for all i, then the form
)
n
n
cii
ii
')
L__
-
i,j=l
is also positive definite.
L
L_
i,j=l
Applying 3) to this form it follows that
B
where
B
2.
the determinant consisting of elements
i,j = 1, 2,
¿1
--,
n.
Now
L1l,
B = c11c22 ...
therefore
¿X
.
1122
Cirn
Now suppose that the form
n
n
::
=
i,j=l
where
a11 ... a
...
J(ax1
+
a2x
+
... +
ax)2,
[.1
L!]
a1
so that Cj1
+
a2
+ ... +
a.
Also, if D
O, D
=
is the
determinant of a positive definite form and
n
D2
c11c22 .., c
=
n
Ea
k1
n
,IIa2k
k1
k1
or
ft
D2
k1
i=1
SUFFICIENT CONDITION THAT A DEThRJOENANT B
NOT
ZFJtO.
Lengia 1.
If D is the determinant whose elements are ajj, i,j = 1,
laul
then D
II)a
7
n, and
A
O.
Proof (7, p.672):
assume
. + ax
a11x1 +
=
+ ... +
,
i
ax1
+
=
that
D
O, then the
system of equations
O
O
x,
has a nontrivial solution
lxii
,
1, 2,
,
x2,
---,
x1..
Let
be the
n, and consider the rth equation in the
system:
. . .
+ arrxr +
. . .
+
ax
= O
or
arrxr
axk
-
whence
n
I3rrIIcr(
But since
XrI>
jaIx.1
:
,
)Ar\\X.
this contradicts the hypothesis.
Therefore D
O.
POSITIVE DETIHttNTS.
=
eleluent$ are ajj, i,j
ajj
a
Lemma 2
-
1, 2,
If D is a determinant whose
n, and
,
1
i
then D - O.
Proof (7, p.6Th):
i
j.
The lemma is obviously true if ajj = O for
Since D is a continuous function of n2 variables, D
.
O by
lemma 1.
INEQUALITY OF FRIEDRICHS.
whose elements are ajj, i,j
Ja
Theorem 3.
1,
If D
is a deterrilnant
n, and
,
,
then
n-1
a11(a
Proof:
- 1a211) .....
a11
a12
O
.
D
.
By lemma
Therefore
.
2
.
Ia,l)
k1
Consider the determinant
alI...aTh
a
-E
(a
a13...a
a22-I2i) a23...a2
&
a21 la21!
O ... O
a32
a33...a3fl + a31
a32
.
.
a1
a2
&3
a
a
.aun
a
the second determinant in the sum i
Ian,
a1
--
_i_J..
O
a22-a21)
a,
... a1,j
-
_&._i
a23
...
a2n
.
aun
T
.
a
a13...aTh
= a31
.aun
-Jn
a12
a1
.
a3
.
.
3
n2
a33...a3
n3
...a
nfl
positive or zero1
lo
By repeating the step outlined above to the remaining n-2 rows, the
point is reached, after a finite number of steps, where
a11
O
>1
a12
a2-aj
a13
... a3
a23
...
O
O
o
o
O
O
O
O
a33-1a3Jj--Ia32L
a2
...
n
n-1
ej.A ..U.
.
..
a-E
k1
Ea
II
CHAPTER III
TE
ALGEBRAIC CASE
FORE OF SOLUTION OF ALGEBRAIC SYSTE&S.
D(k)
O
and
By Cramer' s Rule, if
O, then
x1-Ekjx=y
3)
j=1
and
= Yj,
i -
)
i
= 1, 2,
-
J=l
have unique solutions of the form
yD
:l
and
Xj
=
D(k)
where
l-k11
l2
l-k
.. -k
...
i-1
-k
-
andD(k)
D(k) =
n2
and Uj
D
:in
-
''nn
are the cofactors of
- kj
in
l2
2l
l'422
2n
Cnl
4n2
"nn
D(k) and
1k
(k) respectively.
To repeat, the purpose of this oaper is to find a bound for
\
9)
X.
-Xj.-
(k,y)
-
D(k,y)
_____
D(k)
(k)
where
D1( k ,y)
=
EyDj
and
(
k ,y)
kik
12
Inequality
may be written as
9)
I(k,1)
lo)
n(k)-1)n(k)I
k
kOc) JD(k,y)-U(k,y)
(k)
(k)Il
To find a bound for
the
J
maximum
-
value of
to determine bounds for the expressions
and 1D(k,y) -
(k,y).
bounds and cOEnpiete
(k) -
(k)II
lL) it will be necessary
ID(k,y)J,
JU(k) -D(k)),JD)I
The following nìteria1 will develop these
the proof of Theorem
1, Corollary 1, and
Corollary 2.
ERROR Th A DTERThANT.
whose elements are
ajj + E
;
IaiiV
if
a.
and
.
Theorem 1.
n
(a)
If Da(a) is a determinant
is a determinant whose elements are
further
K,
then
flW2[(K +
-D(a)J
)fl
Proof:
U(a) =
E
a12+
E21
a22+ E22
l2
aTh+ Ein
a2+E2
a11
a12+ E12
.
. .
aTh+
a21
a22+22
...
a2+
E2
+
.
.
E nl
an2+ E n2
a+ Eun
.
a1 an2n2
ayml Eun
By continuing the decomposition started above, the point is reached
where
13
E
a12
621
a22 ...
a11 ...
a1,1C
.
+ ... +
Da(a) +
+
.
s
..
a
6ii
+
12
e
s
.
t
Ep
That is,
a13
ari1
.. a
t
t..
n2
a113
t.
a
.
a,_i E1
..
E
C
...
.
E
nl
(a) is equal to Da(a) plus n determinants in each of which
just one column of a's has been replaced by the corresponding
£-coluxnn, plus n(n-1)/2 determinants in which two co1unns of a's
have been replaced by
Consequently, with the aid of
's, etc.
Theorem 2,
¡(a)-D(a)J
<.
(a)-D(a)J
n(n-l)..2
nfh2[n:fl
n2[(k: +
LI1ITATION OF 1D(k)
D(k) may be expanded
2
E+...+
n(n-l)
.
(ri-m+1)
mt
].
It can be shown (9, p.211i)
-D(k)l.
in the following marner.
-k11
1 -
.
)fl
..
k1k1
n
D(k) =
.
+
kj
_1
Let
I
i
.
.
Dn(k)1 <
K , then by Theorem 2,
i +
Since
n(n-1)2K2
nl(
+
(21)2
{
mm/2
+
e
n(n-l) ...
(n+1)m/2K
(,)2
i we may define o0/2 to be 1, and write
that
iL
n
'
\
or, by
U)
fl
(mt)2(n_m))
<
)
V)n(c)
<
(-j)Oc).
(k) to
In the same manner, expand the determinant
LE41
+
+
.
i1 k+
E
and
-JJ(k)I
<
E
<
n
k1
in
-k1 ...
n(n-l)
+
2
-k
.23
[(KE)2
+
(21)2
)m
(n_m+1)mm/2[(K
-
km]
(mt)2
With the aid of the mean value theorem of
E
k+
, by Theorem
+
(K +
)m
=
E
m( K +
E.
)XTh.1
o
differential calculus, write
<
<
i
(K+E.)m_Kl< Em(K+6)m
to get
(k)
I
12)
jD(k)
kj kj
jj
-
-
n(n-l)
jj k+
-k11 '
-k-
nfn1
K
E
..
-k11-E11
Let )k
k1+
n
-D(k)
et
-D(k)JK
n
i ntmm/2 \
I
E((mI)2 (n-ni) 1/)ii(K
m=O\
+)m_1
is
LflITATION OF
(9,
expanded
-D(k,y).
I(k,y)
The determinants Dik may be
p.2114) as
n
kjkkjj
3=1
to a finite number
ji
jk
3---
terris.
ol'
33/2
.
k41
kk
k1
h-
K, by Theorem 2,
+
(n-m+1) (m+1)
(,)2
+
kik k
K3
(202
+
k1
IkI<
Since
1)
K + 2nJ(2
n( n-1)
1
kik
(m+1 )/2m+1
n
jDikt<KE((nit)2(n_m)t
J
or, by 6)
13)
(DikI
£(k,y)
Since
iL')
<4(K).
yD
if lyj<
,
Y, then
YP(x).
ID(k,y))<
In a nanner analogous to that of deriving inequality 12),
t
ik
-
Dik <
i
Elk
/
n
k+ Eik
kj+
j=1
kjk+Ejk
k3+E
L
to a finite number of terms.
Theorem
J
;.k
Now ¡k
)
kik
k1
kik
k4
*
+
': K, ¿e
E
, so by
)
!,
D1K
+
+ 2n
n(n-1)
((K)2
y2}+
. . i (nm+1)(m+1)(1)/2
(t)2
)m+1
.3a±1]
s
.
)l
E mt_(i)(11»'2
(rnt)(n) )+
16
n
-
Again, with the aid of the mean value theorem of
write
(
£)"
(m+].)(K+6E )m,
- Km
e
o
K
j
di'ferential calculus,
<
i
(m+l)(Ke)m
Therefore
n
(r+1)/2\
E
mo(
TSik -
I)
(m')2(n-m)1
(m+1)(K+6)m
jDik_Dikl< £j'(KE).
If
jjÎ<
then
Y,
1) J(k,y) -Lj(k,y)J< n!EP'(K+).
LThITATION OF
inequalities which will establish
n'
11)
ID(k)I
12)
I(k) -D(k)I<
lL)
tD(k,y)I4
i)
IU(k,y) -D(k,y)
l)
The
i are,
the proof of Theorem
<
xj.
-
t(K+E),
nYp(K)
<.
and
ny
q'(K
i-
E).
If
<
,-
substitution of
16)
Ii -
x
e1
U),
12),
iLi)
and
i)
6[(K)ys(i+)
1Un(
[I(k)I
_
in 10) results in
+
(K)
'(K
'(+
17
which is the inequality of Theorem 1.
U(k)
To prove Corollary 1, assume that
is a determinant that
satisfies the inequality of Friedrichs, i.e.
17)
(1 - k11)
lrì(k)
TT(k) =
77(k)
,
where
i - k11)
and that
(1 - k11)
<
TT(k)
9'(k+e)
(k) in 16) by the right hand side of 17)
then replace
(i)
+
-
18)
J
(1-k11)
<
(
k)[.)( k)
to get
-
G
'
(
'
which is the inequality of Corollary 1.
(k) is a determinant that
To prove Corollary 2, assume that
satisfies the inequality of Friedrichs,
kj<
K, (n-l)K < 1, and
ê( K)
E
<t(y + e)
where
8(K)
K)(l - 2K) ... (1 - (n-1)K),
(1
'
then
19)
(k)
Now replace
-
ri(k)
2O)Ix_xjI<
nY
in 16) by the right hand side of 19) to get
+
%(K)[6(K)- Eq)'(K+
which is the inequality of Corollary 2.
N1J1ERIG&L EXMAPL1
EXU1PI.
Consider the system of equations,
3.99x1 - 2.01x2 +
2. 00x1 + 3. O1x
.99x3 = i
- 3. Oli3 = i
+ 2.00x2 +
3.99x3i
Rewrite this systeni as
Ii..
oc1
- 2. OO2 + 1. OQx3 = i
2.OQ1 + 3.00x2 - 3.oaE3
i
1.O1+2,OQx+1.OÇ.bc3-1
Then,n3,K3.99,Yz1,and=102.
The solutions, by Craners rule, to four places are;
= .3272
= .1882
X3
- .3263
-
X3 - .0736
.189t
and
I
- xii = .0009
j.
Ix2 " x2J
.0012
k3
.0002
Lonsetht
- X31
s
inequality gives,
Iii-xii
i;
<
.0052
- x2J < .OlU
k33I
<
.0075
For n = 3, and first approximation,
Ii-xiH
[l+
(ouiton writes,
IAiiIJ
.O731
where
E
is the largest error in the system,
vanishing determinant of the system and
A
is the non-
is the cofactor of a1
inn.
For this example,
Ik-xiIi'
.0052
2H
.0067
Ix2 -
Ix3
.00LS.
X31
By Theorem 1, i e. inequality 20), of this paper, and the table
on the following page,
I±_xiI<
.6)366,i=l,
2, 3
The example indicates that inequality 20) of this paper does not
give as close a bound for the maximum value of
- xii as the one
derived by Lonseth, nor the approximation of 1oulton.
table of values for the polynomial functions i1(x),
However, if a
P(x)
and their
derivatives were made available, the calculation of 16) would be quite
easily done.
The methods of Lonseth and Moulton require the
calculation of the determinant of the coefficients in the approximating
system in addition to n2 cofactors, in
of this determinant.
the maximum value of
computation.
Longet,
and n
in
Moulton's,
Also, the method of this paper gives a bound for
j
- xjj, for all i
1, 2,
- -,
n, in a single
TABLE OF VALUES
)(x)
(x)
X
(ni (m+1)(m+1'2)
=o(mt)
(n-ni)t
X
4)3(X)
(f)13(X)
/23(X)
00
1 0000
3 . 0000
o;
O 0000
3.3065
3.6260
0657
.16%
2.298
3.985
.3039
Li.3039
I8S8
3.3621
!.I2O6
.3!;
1.1576
1.3309
1.2Olj
1.7269
I.9S31
2.2L01
2.15h6
.140
2.7351k
.
.10
.15
.20
2
.30
.145
3.O36i
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
3.2603
3.7016
1.00
2.00
3.00
14.00
5.00
Li.0671
L..)3%3
L.867O
5.3029
5.76314
6.21493
6.7613
7.3000
7.8660
25.5282
59.0325
113.2253
193.0025
L.6621
.O338
1 0000
.7166
.6281
6.9921k
11.014147
1.0018
1.3h78
1.7608
2.6979
2.8162
3.L735
.228O
5.0880
6.0626
7.1609
8.3927
9.7681
11.2976
12.9921
11.5980
25.3923
114.8628
142.61482
147.01493
110.2359
203.86140
1413.2961
1115.22014
21471.1855
1105.7896
2978.9003
S.La83
5.8157
6.2261
6.6195
7.0860
7.5353
7.9977
8.L73O
8.96]J4
9.14627
9.9771
10.501414
1414.3826
68.5691
97.9518
3.5220
1O.221j3
12.1070
1L.179O
16.1479
18.9218
21.6085
2Lt.162
27.6527
31.0262
314.614147
38.5160
5314.1414140
21
LITERATURE CITED
1.
Adame, Rachel 13. On the approxinate solution of Fredholru's
homogeneous integral equations. American Journal of
Mathematics 51: 139 - ]J.8.
1929.
2.
Bocher, Maxime. Introduction to higher algebra.
Macmillan, 1938.
3lSp.
3.
Etherington, I.M.H. On errors in determinants.
Proceedings
of the Edinburgh Lathentìca1 Society 3: 107 - 117.
New York,
1932.
Li.
Hardy, G.H., J.E. Littlewood and G. Polya.
New York, Macmillan, l931. 299p.
Inequalities.
!.
Lonseth, A.T. Systeìs of linear equations with coefficients
subject to error. Annals of i.theiiatical Statistics 13:
332 - 337.
19h2.
6.
Moulton, F.R. On the solutions of linear equations having
sria11 determinants
American Mathematical Monthly 20:
2L2 - 2L9.
1913.
7.
Taussky, Olga. A recurring theorem on determinants.
Mathematical Monthly S6: 672 - 675. 191.9.
8.
Tricorni, Francesco.
9.
Whittaker, E.T. and G.N. Watson.
Macmillan, l9Li8. 59Op.
American
Sulla risoluzione numerica deflo equazioni
integrali di Fredholm. Atti della Reale Accademia
Nazionale dei Lincei, Rendiconti, Classe di Scienze
Fisiche, Matematiche, e Naturali 33: 10 sexa. L83 - L86,
2° sem. 26 - 30. 1921.j.
Modern analysis.
New York,