Prova Elementar da Forma Canônica de Jordan

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REVISTA DE LA 
UNION 
MATEMATICA ARGENTINA 
Dkector: Dario 1. Picco 
Rec:tactores: A. Diego, E. Gentile, M. Herrera, 
D •. Herrero, C. Trejo, O. Eo Vlllamayor 
Secretarlos de Redaccl6n: M. L. GastAmtnza, A. O. de POUIa 
VOLUMEN 27, NUMERO 1 
1974 
BAHIA BLANCA 
1976 
Revista de la 
Union Matematica Argentina 
Volumen 21, 1974. 
AN ELEMENTARY PROOF OF THE JORDAN CANONICAL FORM 
Enzo R. Gent; i e 
Let V be a finite dimensional vector space over a field K. Let t 
be an endomorphism of V. Then, as it is well known, V can be 
written as a direct sum of cyclic subspaces. If mt(X) E Klxl de-
notes the minimal - polynomial of t , in studying the structure 
of t one is reduced to consider the case where mt(X) = p(X)a, 
with p(X) an irreducible polynomial in Klxl and where a is a nat-
ural number. 
A cyclic subspace of V admits the following matrix representation: 
P N 
P N 
(C) 
P N 
P 
where P is a block consisting of the companion matrix of p(X) and 
where N is the block 
of the same siz.e as P. 
o 
o 
o 
o 
o 0 
o 
o 
o 
o 
o 
o 
o 
The rational canonical form of t consists of the matrix obtained 
by assembling blocks of the type (C). If p(X) = X - k, k E K (for 
instance if K is algebraically closed) then (C) becomes a Jordan 
block ,and the canonical form is called the Jordan canonical form. 
The proof of the Jord.an canonical form depends essentially on the 
canonical form of a nilpotent endomorphism, fact tediously proven 
in most books in linear algebra. 
, 
2 
In this Note we intend to give a direct proof of the Jordan canon-
ical form. Nevertheless the ideas in the proof permit to prove the 
structure theorem of finitely generated torsion modules over a 
principal domain, which shall be done elsewhere. 
To start with, we set some terminology. Throughout this Note, sub-
space means subspaae invariant (or stab~e) under t. Furthermore 
morphism means morphism aommuting with t. Recall that a subspace 
W of V is a direct summand of V if and only if there is amorphism 
f: V --+ W whose restriction fl W to W coincides with Idw' 
Let v E V. With (v > we denote the cyclic subspace of V generated 
by v. Any element of (v > can be written as a(t)v for some 
a (X) E K Ix I. 
From now on assume that t is a nilpotent morphism. For any v E V, 
v # 0 we define the order of v as the highest positive integer 
o(v) satisfying 
to(v)v = 0 
Notice the following property of o(v): for any a(X) E Klxl, 
a(t)v = 0 implies that Xo(v) divides a(X). 
In fact, this follows from the property of being Klxl a principal 
domain and standard arguments on polynomials. 
Let v E V be an element of order s. We consider in (v the fo1-
lowing sequences of subspaces: 
o c ( ts-1v > c ... c ( tiv > c ... c ( tv > c {v > 
We claim that if x E (v then 
In fact, part "if" is trivial. On the other hand, let tix = O. 
Write x = a(t)v. Then 0 = tix = tia(t)v and therefore XS divides 
Xia(X) which implies that a(X) is divisible by xS - i , a(X) 
= Xs-ir(X). Finally x = ts-ir(t)v E (ts-iv > as we wanted to prove. 
Next we start to prove that V is a direct sum of cyclic subspaces. 
Let v E V be an element of highest order in V. This implies that 
tSz = 0 for any z in V. 
Let (v > be the cyclic subspace of V generated by v. We shall de-
fine a projection of V onto {v > • 
For this, let W be a subspace of V satisfying the following prope£ 
ties 
3 
i) {v} c W 
ii) There is a morphism f: W --+ (V) such that fl(v} 
iii) W is maximal with properties i) and ii). 
Obviously if W = V nothing has to be proved. Assu~e then V , W. 
Choose u E V - Wand consider the subspace 
W' = W + (u) • 
W' contains (v) and we shall extend f to W'. 
Let J be the ideal of all polynomials a(X) in Klxl satisfying 
a(t)u E W. 
J is generated by a monic polynomial g(XJ. Since tSu = 0 E W it 
follows that XS E J, therefore g(X) divides XS, so g(X) = Xd 
for some d ~ s. Notice that tdu E W therefore f(tdu) E (v ) . 
But since t s - d f(tds) = f(tSu) = 0, by an earlier remark we get 
that f (tdu) E ( tdv ) , that is 
for some x E (v ) . 
We set 
f': W+(u} ----..(v} 
f': w + a (t)u l---- f (w) + a (t)x 
and we claim that f' is a well d'efined morphism of W' onto (v ) • 
Let w,w' E W, a(X),a' (X) E Klxl satisfy 
w + a(t)u = w' + a' (t)u 
Hence 
(a' (t) - a(t))u w - w, E W 
implies that 
a' (X) - a(X) E J, that is a' (X) - a(X) = b(X)Xd 
Therefore 
few) - few') f(b(t)tc1U ) = b(t)tdx .. (a' (t) - a(t))x 
4 
f(w) + a(t)x = few') + a' (t)x 
which says that f' is well defined. Clearly f' is a morphism of W' 
onto (v > that extends f. Since W is properly contained in W', we 
have a contradiction. Therefore f is a projection of V onto (v 
and we can write 
V = (v > e V' 
But by an inductive argument, V' is a direct sum of cyclic sub-
spaces, so V is a direct sum of cyclic subspaces and this was our 
claim. 
REMARK. Notice that the present proof holds for any endomorph i sm 
t whose minimal polynomial is mt(X) = p(X)a, with p(X) irreducible 
in Klxl. As we remarked at the beginning the general situation 
a . 
mt(X) = IT Pi(X) 1 of different irreducible factors reduces trivi-
ally to the case above. 
Recibido en marzo de 1974 
Universidad de Buenos Aires 
Argentina 
Revista de la 
Union Matematica Argentina 
Volumen 27, 1974. 
ACTIONS ON A GRAPH 
Antonio Diego 
ABSTRACT. Part of the theory of flows and tensions on a graph is 
extended to any kind of actions, i.e. to any subspace of the 
space of real functions defined on the arcs; in particular, the 
theorems on the existence of flows or tensions under bilateral 
restraints. 
1. INTRODUCTION. 
Our basic setting is a graph G = (X,S); here it means that 
sex x X verifies: (i,i) ~ Sand (i,j) E S implies (j,i) E s, 
i,j E X. We assumeG connected. 
We consider functions f,g, ... defined on S and we denote by f.g 
L f ij gij' the usual scalar product. 
By Ewe denote the linear space of anti-symmetric functions: 
f.. + f.. = 0, (i,j) E S. 
~J J ~ 
The subspaces of flows and tensions,<I>,e C E, are defined by: 
'II E <1>, if L 'II .. = 0, i E X, and 0 E e, if o .. = t. - t i , t defined j ~J ~J J 
on X (0 = 6t) . <I> and e are orthogonal complements in E. 
If FeE is defined as the set of solutions f of the linear sys-
tem: Aa.f = 0,~ EM, its orthogonal spa'ce G = pi in E, is gene-
rated by the p'a, ~ E M, where J.I~j • Aj i - A~j When M '= X and the 
A~k vanish except for i, writing A~k = Aik , we have p.~j 
= -A ij , J.I~i = Aij and J.I~k = 0 for the remaining (h,k) E S: Then, 
the elements g E G are of the form gij = tjAji - tiAij (g = AA t ), 
where ~ is a function on X. The case A.: 5 1 corresponds to F = <1>, 
~J 
G = e. Given an orientation to G - i.e. a subset U of S contain-
ing for each (i,j) E S one and only one of the pairs (i,j), (j,i)-
and a positive mi , i E X, defining Aik= mi , if (i,k) E U, and 
6 
x .. = 1, if (i,j) ~ U, we obtain the spaces of multiplicative 
~J 
flows and tensions ([ 11 pp.22S). 
Actions of a certain kind can be thought as the elements f of a 
subspace F of E - J ij representing the · intensity of the action f . 
trasmited from i to j through the link (i,j) E S, 
Certain notions and results of the theory of flows and tensions 
on a graph, can be extended to any subspace F of E. Doing that, a 
unified linear treatment of the outstanding cases F = ~, F = e 
- that may be useful - is obtained. 
In 2 we give the notiQn of elementary action, corresponding to 
the notions of elementary cycles and cocycles, and a proposition 
on the decomposition of any action in elementary ones. It gives 
the known decomposition of a positive flow (tension) - on an ori-
ented graph - as a positive linear combination of elementary cy-
cles (cocycles) ((1] pp.143). 
In 3 we prove the analogue of Hoffman and Roy's theorems ([2], 
[3]) for actions of any kind, using the appropiate geometric ver-
sion of the consistence theorem of a system of linear inequalities 
(Farkas-Minkowsky). 
2. ELEMENTARY ACTIONS. 
I 
For fEE we denote s(f) = :{(i,j)/f .. > OJ, the (effective) sup-
~J 
port of f. 
It is seen that, for f,g E E: 
(A) ~~s(g) C s(f) implies s(f-Xg) C s(f), properly, for the posi-
f · i 
X = max{-i.i / g > o} A function f E F, f # 0, is 
gij ij • 
tive number 
said to be an eLementary funation of F if for any g E F, s.(g) c 
c s (f) implies g = H. I 
This means that s(f) is) a minimal set of {s(g)/g E F}. In fact, 
s (f) minimal implies, flor each g E F with s (g) C s (f), that 
s(f-Xg) C s(f), proper~y, (A); since f - Xg E F it follows 
s(f -.Xg) • 0, f ·· Xg. , The converse is ~lear. 
I 
Of course, if f is an l~lementary function of F, so is xl, X ~ O. 
PROJ.>OSITION, Any . fEE. f ; , ot i.~ a 8Urn , of eLementary funation8 
I 
7 
fn of F suah that s(fn) C s(f). 
Proof. Let gl be an elementary function of F such that s(gl) C 
C s(f). From (A), for some Xl > 0, it is s(f-Xlg l ) c s(f), proper-
ly. If f - Xlg l "I 0, we apply to f - XlglE F the same argument 
and we get an elementary g2 E F, X2 > 0, such that s(g2) c 
c s(f-Xlg l ) and s(f-X l g l -X 2g2) C s(f-Xlg l ), properly. After a fi-
nite number of steps we have elementary gl, ... ,gk E F, Xl ,X 2, ... , 
Xk > 0, such that f - Xlg l -
lows with fn = Xng a , 1 ";a"; k. 
- Xkgk = O. The proposition fol~ 
For a set Z C S, such that (i,j) E Z implies (j,i) ~ Z, we define 
~ = ~(Z) by ~ij = 1, -1, 0 according to (i,j) E z, (j,i) E Z or 
(i,j),(j,i) ~ Z, respectively. ~ E E and s(~) Z. 
If Z = {(i,j),(j,k), ... ,(h,l),(l,i)} is a cycle, ~ is a flow, if 
Z = {(i,j)/i E A, j ~ A} (A, X - A "I 0) is a cocycle, ~ is the 
tension ll( -1 A) • 
If the sequence i,j,k, ... ,h,l of the cycle Z has not repeated ele-
ments Z is an elementary cycle. If A and X - A are connected, in 
the graph Gz obtained eliminating the (i,j),(j,i), with (i,j) E Z, 
the cocycle Z is said to be an elementary one. 
It is clear that ~ = ~ (Zl is an elementary flow, when Z is an ele-
mentary cycle. For an elementary cocycle Z, if 0 llt is such 
that s(O) c Z = s(r), the connectedness of A and X - A in Gz im-
plies tl A = a, t
IX
_
A 
= p, then llt = 0 = (p-a)r, with P ~ a. Hence 
~ is an elementary tension. 
Conversely, if op ., 0 is a flow, sLOP) verifies that (i,j) E s(op) 
implies Cj,k) E sLOP) for some k., i (op .. + L op'k = 0, opJ"1' = 
J 1 k,li J 
= -op ij < 0). 
It follows that sLOP) contains a cycle, and then also an elementary 
cycle Z. Hence if op is an elementary flow, t = X.op, X > O. On the 
other ~and, if 0 = llt , 0 is a tension, taking a: min ti < a < 
< max t., the cocycle Z defined by means of A = {i/t. <: a} is con 
1 1 -
,\:ained in s(8). Hence, if 8 is an elementary tension we have t = 
. • X8, X > O. Z has to be an elementary cocycle, otherwise we could 
8 
take a connected component A' of A (alternatively of X - A) in Gz 
and define Z' in terms of A'. But this wQuld imply set') c s(8), 
properly; which is a contradiction. 
Resuming, the multiples A.t, A > 0, t ~ t (Z), for Z elementary cy-
cle (cocycle), are the elementary functions of 41(9). 
3. EXISTENCE THEOREM. 
We consider a finite dimensional linear ~pace with the scalar pr~ 
duct x.y. For a cone Q - Q+Q c Q, AQ ~ Q for every A ~ 0 - the du 
al cone is defined by QO ~ {xix. y ;;. 0, for any y E Q}. 
We need the theorem of consistence of a system of linear inequali-
ties under the following form: 
"Given the polyhedral set C and the polyhedral cone Q, (C, Q -1 0) 
it holds: Q n C -1 0 if and only if, for every x E QO there is a 
c E C such that x.c ~ 0". 
In fact, if Q n C 0, we can separate the closed convex (poly~ 
hedral) set Q - C from 0; i.e. there is x such that x.(c~q) < 0, 
for any c E C, q E Q. Taking Aq, A > 0, instead of q, we conclude 
that x.q ;;. 0, i.e. x E QO. For q ~ a we have x.c <0 for every 
c E C. The converse is clear. 
We will apPlY the theorem to a subspace Q. In this case QO ~ Ql. 
THEOREM. Let c
ij 
+c
j 
i ;;> 0, for any (i, j)E Stand F be a linear 
subspaae of E. In order that there e~ists f < c, f E F, it is nea 
essary and 8uffiaient that. for eaah e~ementary g E G ~ Fl, 
g+.c ;> o. 
+ REMARK. As it is usual: g ~ max (g,O). The condition C .. +C .. ;;. 0 
1J J 1 
is obviously necessary for the existence of an anti-symmetric 
f "' c. 
Proof., Omitting the word "elementary", the equivalence follows 
from the theorem of consistence applie~ to Q = F, C = {xix", c} 
in the linear space E. 
In fact, Q n c ~ 0, i.e. there is f ~ F, f "' c, is equival~nt to 
9 
assert that for any g E G = QO there is x E E, x ~ c such that 
g.x ~ O. This implies g+.c ~ 0, since from (g+-g-)x ~ 0, C ~ x, it 
follows g+.c ~ g+.x ~ g- .x, and then 2g+.c ~ (g++g-)x = Iglx = 0 
(I g I is symmetric). 
+ And conversely,if g .c ~ 0, g E G, then defining, for a given 
g E G, x by x .. = c .. , - c .. , -1/2(c .. -c .. ), according to g . . > 0, 
l.J l.J l.J l.J J l. l.J 
< 0 or 0, we have g.x = 2g+.c ~ O. 
Finally, if g+.c ~ 0 for elementary functions g E G, the same 
holds for any g E G, since from the proposition we can write 
g+ L g:, for elementary functions ga of G. 
a 
REFERENCES 
[11 BERGE,C.and GHOUILA-HOURI,A., P~og~amm~ng, game¢ and ~~an¢­
po~~a~~on ne~wo~~¢, John Wiley and Sons Inc. New York (1965). 
[2J HOFFMAN,A.J., Same ~ecen~ appl~ca~~on¢ 06 ~he ~heo~y 06 l~n­
ea~ ~nequal~~~e~ ~o ex~~emal comb~na~o~~al analy¢~~, Procee-
dings of the Tenth Symposium in Applied Mathematics. A.M.S., 
New York, (1960). 
[31 ROY,B., Chem~nemen~ e~ connex~~~ dan¢ le¢ g~aphe¢: Appl~ca­
~~on aux p~obleme~ d'o~donnancemen~,Metra, Serie speciale, 
N°1, (1962). 
Recibido en marzo de 1974 
Universidad Nacional del Sur 
Bah1a Blanca, Argentina 
En el cu~~o de la ~p~e¢~6n del p~e~en~e ejempla~ 
de la Rev~~~a de la UMA hemo~ ~u6~~do la ~n60~~unada 
p~~d~da de nue~tM~ c.ompai!e~o¢ de t~abaj a y que~~do~ 
am~go¢ Evel~o T. Oklande~ y R~c.a~do Alemany, a cuya 
memo~~aded~camo~ e~~e ~~abaj o. 
Revista de 1a 
Union Matematica Argentina 
Vo1umen 27, 19~4. 
AN ELEMENTARY PROOF OF THE STRUCTURE THEOREM OF FINITELY 
GENERATED MODULES OVER A PRINCIPAL DOMAIN 
Enzo R. Gentile 
Let A be a principal domain. It is a well known result that if M 
is a finitely generated A-module then M can be written as a di-
rect sum of a free submodule and thetorsion submodule. The free 
part is (up to isomorphism) completely determined by its rank. 
So, the main point is to characterize the torsion submodule. The-
refore let M be a finitely generated torsion module over A. We 
intend to give an elementary proof of the classical result sta-
ting that M can be written as a direct s~ of cyclic submodules. 
Let J be the ideal of A, annihilator ~f M. Since A is principal 
we have J = (a ). Let 
n 
a = IT 
i=1 
r. E N 
1 
be a factorization of a in A. The elements Pi are irreducibles in 
A and furthermore Pi is not associated to P
j 
if i ; j . . If, for any 
irreducible element p in A we denote with M the p-primary compo-
p 
nent of M, that is 
M = {x I x E M and pix = 0 for some i E N} 
p 
we have that M splits in a direct sum of its Pi-primary components. 
We Can therefore ,assume that M = Mp for some irreducible element 
p in A. For any m E M, m , 0, we consider the integer oCm) as the 
maximal positive integer i satisfying: 
We call oCm) the order of m. 
It is ~lear that if m has order s then p. generates the ideal i~ A, 
.nnihilator of m, so 
(m) ... A/< p.) 
11 
We have the following filtration of {m } : 
c {pm} c {m 
Each {pim } is characterized by the property: 
xE{m} , iff s-i x E {p m 
In fact, "if" is trivial. On the other hand if pix 0, as x 
we have 0 = pix = pirm. This implies pS I pir, that is, 
and we get finally x E {ps-im }. 
s-i 
P 
rm 
r 
Let m E M be an element of maximal order in M. We shall prove the 
key result that {m} is a direct summand of M. To do this we shall 
define a projection of M onto {m } . The next Lemma is the main de-
vice of our proof. 
LEMMA. Let N eM be a submodule of M and let f: N --+ {m,> be a 
morphism. Then f extends to a morphism of Minto {m } . 
Proof. If N = M we have nothing to do. Let N I M and take a E M-N. 
Let I be the ideal of A defined by 
I {k IkE A and ka E N} 
Then I = (d) for some d in A. Since pSa = 0 (by the maximality of 
5) we have that pS I d , or pS = d.y. By the unique fa~torization 
of A is d pr.u with r EO; s . Without loss of generality we can. as-
sume that u = 1 , that is d = r p 
We have 
p s-r f(da) = f (ps-rda) . f(psa) 0 
but by an earlier remark we can write 
We intend to 
Let us. prove 
have 
f (da) = prx for some x E {m .} 
extend f to N + {a as follows: 
f' : N + {a ~ ( m. ) 
n + ta t----+ fen) + tx 
that f' is well defined. WIth n,n' E N, t,t' 
n + ta = n' + t'a * n - n' = (t' - t)a 
* t' - t e I 
.. t' - t • zd z e A 
E A we 
12 
Therefore 
f (n - n') f(zda) zf(ad) zdx (t' - t)x 
that is 
fen) + tx = fen') + t'x 
and this proves that ~' is well defined. We can repeat this pro-
cess, but after a finite number of steps we have to arrive to M 
(M is a noetherian module !). This concludes the proof of the 
Lemma. 
Applying the Lemma to the situation: N = (m ) , f = 1d< m )' ,,!e get 
a morphism of Minto < m ) which is the ident:j. ty over < m ) . This 
is clearly a projection over (m ). So ~m) is a dire.ct summand of M: 
M = <m) ED M' .We can repeat the process with M'. But by the noe-
therian property of M, the process must stop after a finite num-
ber of steps. Then we ~et that M is a direct sum of cyclic sub-
modules. 
REMARKS. Notice the revealing property of elements of maximal or-
der. The present proof improves the one given in our Note appea-
red in the American Mathematical Monthly, Vol.76, N°1. pp.60-61 
(1 !;I69) • 
Recibido en julio de 1974 
Universidad de Buenos Aires 
Argentina 
Revista de la 
Union Matematica Argentina 
Volumen 27, 1974. 
SEQUENTIAL ESTIMATION OF A TRUNCATION PARAMETER 
Eduardo Warner deWeerth 
1. INTRODUCTION. 
Let Xl be a random variable defined on a probability space 
(ll,B,P) and whose distribution belongs to the family {Pa: a E 8} 
where e is a finite or infinite interval of the real line R. We 
want to estimate the true value of the parameter a on the basis 
of a series of independent observations XI ,X 2 , ... of the varia-
ble Xl' After taking each observation we decide, on the basis 
of the observations that we have already taken, whether to take 
still another observation or to stop and estimate. We suppose 
that it costs us c units to take an observation and that if we 
estimate a by d when its true value is a we lose L(a,d). Our to-
tal loss then, upon taking n observations and deciding d is 
L(a,d) + cn. 
A sequential decision procedure is a pair (8,t) where t is the 
stopping rule which tells us, for each possible sequence of obse~ 
vations x = (x I ,x2 , ... ), when to stop, and 8 = {8 n : n = 1,2, ..• } 
is a sequence of terminal decision functions. The meaning of 8 
is as follows: for each n, if we stopped after taking n observa-
tions (t(x) = n) and have observed x f ,x 2 , ••• ,xn ' we must decide 
a = 8 n (xl' ... ,x
n
). If we use the procedure (8, t), our loss as "a 
function of x E ROO is given by: 
L[a,8
t
(x)(x)] + ct(x) 
if a is the true value of the parameter. The average loss is 
called the risk and is given by: 
R(8,t,e) 
The most desirable sequential procedure would be one that mini-
14 
mizes the risk uniformly in e. Such a procedure. however. is unat-
tainable in all but trivial cases. 
In the Bayesian setup we assume that there is not one true value 
of the parameter but rather that e is a random variable with a 
distribution 'It. The variables X1 .X2 •..• are then assumed to be i!!, 
dependent and identically distributed given e. This determines the 
jo~nt distribution of e,x1,x2"~' . A procedure will be optimal in 
the Bayes sense (Bayes procedure) if it minimizes the Bayes risk: , 
fe R(6.t,e)'It(de). 
Assuming that Pe(dx1) = f(x1.e)p(dx1). where p is a a-additive 
measure. and 'It(de) = tjI(e)de, .the Bayes risk for a sample of fixed 
size n is: 
(1.1) J {J L[e,6 (x1.···.x )]f(x1.e) ... f(x .e)p(dx1)···Il(dx )}. e Rn n n n n 
• 1jJ(e)de + cn = 
J {f 1[e,6 (x1' .. ·.x )]tjI(slx1.···,x )de}Q (dx1.···.d:x: )+cn Rn 9 n n n n n 
is the condition 
al distribution of e given 11 •... rXn and Qn is the marginal distri 
bution of Xl'" .• Xn' 
Suppose that for each n there exists a measurable function of the 
observations, en = en(X'l •...• Xn).such that 
(1 .2) J L(e.e )tjI(elx:1.···.X )de 9 n n min f L(e .d)1jJ(e'\X 1 ..... X )de d . '9 · n 
From (1. 1) and (1. 2) it follows easily (cf.[ 1]) that the seq1,lence 
of terminal decision functions of a Bayes procedure must be: 
{e }. 
n 
The corresponding stopping rule is the one that minimizes: 
(1.3) 
15 
where Q is the marginal distribution of (X1'X
Z
",,) and 
f 1(1),6 (X1'''''x )].(e\x1, .. ·,x )de 8 n n n 
is called the posterior (Bayes) risk after n observations. 
Bickel and Yahav introduced in [2] the notion of asymptotically 
pointwise optimal (A.P.O.) stopping rules. A stopping rule t 
Y +Ct 
will be called pointwise optimal if Q[ t .;;; 1] = 1 for any 
Y~+ct' 
other stopping rule t'. 
Pointwise optimal rules exist only in essentially deterministic 
situations. One such case obtains when there exists a random va-
riable V, 0 < V < = such that 
(1. 5) Y n 
V 
n ' n=1,2, .... 
In this case it is easy to see that the stopping rule: 
"stop as soon as _V-,--_.;;; c" 
n (n+ 1) 
is pointwise optimal. 
In what follows we will call a function from the interval (0,=) 
to the set T of all possible stopping rules for the sequence 
{Yn } also a stopping rule. Then, a stopping rule t(c) is call~d 
A.P.O. if 
Yt(c)+ ct(c) 
lim ----~-~-~------
c+O inf{~ +cn: n=1,2, ..• } n . 
In their paper [3] Bickel and Yahav proved. 
THEOREM 1.1. If 
(1.6) Yn> 0 a.s. Q fo~ all. n 
a.s. Q 
(1.7). nBYn -- V a.B. Q. whe~e B > 0 and 0 < V <. a.s. Q 
then the stopping ~uZ.e t(c):stop fo~ the fi~stn ' suoh that 
(1.8) Y [1 - (~)B] ~ c 
n n+1 
16 
is A.P.O. 
REMARK. The theorem as stated in [3J is somewhat more general. 
It is not hard to see that any rule s(c) such that 
tltl_ 
iCc) 
a.s. Q 
is also A.P.O. In particular, the rule t'(c) defined by: "stop 
for thefirst n such that Y __ 13_" c" is A.P.O. The rule t' (c) 
n n+l 
is obtained when we replace in (l.a) the expression 1 - (-1!.....)t3 
. by its firs t order approximation _13_ • 
n+l 
n+l 
The relation between the deterministic case (1.5) and (1.7) is 
clear. 
Bickel and Yahav proved furthermore (cf. also [4J) that the rela-
tion (1.7) was fulfilled (in the case of quadratic loss" for 13=1) 
under very general conditions that may be described roughly as 
those insuring the existence and asymptotic normality of maximun 
likelihood estimators. 
It is the purpose of this work to extend the results of Bickel and 
Yahav to a case that clearly does not satisfy the above conditions, 
the case of the estimation of a truncation parameter. The model we 
consider is as follows. 
Let h be a strictly positive and continuous function on the open 
interval a = (a o '=)' Here ao can be finite or -=. For each a E a 
we assume that 
c(a) = f: h(x)dx < = 
o 
Then, for each a E e" the function ~, 0 " x " a, is the den· 
C (a) 
sity function of a probability distribution Pa concentrated on 
the interval (eo,a), and we seek to estimate the truncation point 
a sequentially. 
In section 2 we prove that (1.7) (with 13 = 2) holds in this model 
for loss functions of the type L(a,d) = B(a)(a-d)2 where B(e) is 
a positive, continuous function of e such that 
Je(1+e2)B(e)~(e)de < =. It then follows from the results uf Bickel 
17 
and Yahav that the stopping rules: 
are A.P.O. 
t(c): stop for the first n such that 
t' (c): stop for the first n such that 
Y _2_,.;; c 
n n+1 
In section 3 we show, using a theorem of Bickel and Yahav [3], 
that the rules t(c) and t'(c) are also asymptotically optimal in 
the sense of Kiefer and Sacks [5] . 
It is interesting to remark that in our development the function 
~ plays a role analogous to that played by the Fisher informa-
C (e) 
tion (or information matrix) in the work of Bickel and Yahav. 
It is clear, furthermore, that if eo is finite we can always as-
sume that eo = O. 
Finally, we wish to remark that all our results, obtained for a 
distribution truncated above, can easily be translated into the 
corresponding result for a distribution truncated below. 
2. A LIMIT THEOREM. 
We turn now to the truncation parameter model described in sec-
tion 1 and use the same notation as there". We assume that the 
loss function is 
(2.1) L(e ,d) = B(e) (e-d)2 
where B(e) is a positive and continuous function of e. The prior 
density ~(e) is a positive continuous and bounded function of 9 
and we assume that B(e) and ~(e) are such that 
(2.2) I~B(e)(1+e2)~(e)de < ~ and 
o 
[ "e2~(e)de < .. . e 
o 
18 
We 4enote by an the Bayes estimator based on n observations an~ 
we will now show that it exists and compute its value. The c;ondi-
tional distribution of a given X1""'Xn has the density 
(2.3) (w(e)c-n(e)I[a~en]);f; ,(A)C-n(A)dA 
n 
where Ia denotes the indicator of the set A and 
~n = max(x1,···,xn ) is as before the M.L.E. based on n observa-
tions. It follows from C1.1) that the Bayes .estimator is the one 
that attains the minimun in the expression 
Y 
F,rom 
n 
inf 
d 
Yn i:f f; Bce)ca-d)2.ce)c-ncalde/l; .CA)C-n(A)dA -
n n 
= i:f f; ca-d)2(B(a).ca)Cn ce) I r; BCA)wCA)C-nCA)dA)Ii~. 
n n 
f; BCA),CA)C-nCA)dA 
n 
J; .cA)CnCA)dA 
n 
it is clear that 
(2.4) a 
n 
and therefore 
(2. S) 
We now prove 
f;a CB(9).(e)C-nca)/J; BCA)~CA)C-nCA)dA)da 
n n 
Y 
n J
'" Bce)(e-e )2.ca lx1 ... ·,x )de e n n 
o 
LEMMA 2.1. en -+ e a.B. Pe for every a e e. 
19 
But 
Since the term on the right goes to Oas N -+- co the lemma is pro-
ved. 
In what follows we assume throughout that 9 is fixed and that we 
are dealing with a fixed sequence of observations for which 
9 -+-9. 
n 
,.. -1,.. n 
THEOREM 2.2. Let "n(s) = (C(9 n )/C(sn +9 n)) . T hen 
h (9) 
(2.6) J"'(1+s 2)IB(Sn- 1+e )~(sn-l+a )" (s)_B(9)~(9)e-C(9)slds + 0 o n n n 
a s n -+ 00. 
Proof. We can write the integral in (2.6) as the sum of the fol-
lowing two integrals, where 8 is a positive number to be determi 
ned later. 
no _h(9)s 
J 
(1+s2) IB(sn-1+e )Hsn-1+e )" (s)-B(9)H9)e C(9) Ids o n n n (2.7) 
(2.8) 
We first consider (2.7) 
(2.9) 
where e < ~ < sn- 1+ 6 
n n 
If 0 ~ s ~ n8 and n > N(8), we get 
9 - 8 < ~n < ~ < ~n + 8 < 9 + 8 
Therefore 
(2.10) " (s) .;;; e-W8 where w = inf {h....(& : Ix-al < 6} 
n C (x) 
It is clear, furthermore. that we can assume 6 sufficiently small 
and N sufficiently large so that, in addition 
20 
(2.11) 
-1 A B(sn +e
n
) < 1 + B(e) for 0 ~ 5 < n6 $nd p > N . 
It then follows that if K is the upper bound of 1/1, 
~(e) 
(1+5 2 ) I B(sn- 1+8
n
H (sn- 1 +8
n
)"n (5) -B(e)l/l(e)e- c(e)"1 I( O(:u~6) 
(2.12) _hills 
.;;; (1+5 2 )[ (l+B(e))Ke-ws + B(e)1/I(e)e c(e) ]1 [ ou] 
Since it is clear from (2.9) that 
"n(s) 
it follows from (2.12) and the dominated convergence theorem that 
the integral (2.7) goes to 0 as n --+ m . 
We now consider integr~l (2.8). From (2.9) it is clear that 
~here w is as before. 
Therefore, putting s = nee-a ) 
n 
h(e) 
(2.1~) Jm (1 +5 2) I B(sn- 1+a n) 1jJ (sn-
1+e n)"n (5) -B (e)1/I (e)e -c(e) S Ids .;;; 
nil 
+ B(e)1jJ(e) Jm 
n5 
ne-nowfm A 
5+9 
n 
+ 
Due to (2.2) this last expression goes to 0 as n --+ m and ~he 
proof of the theorem is thus concluded. 
We state for later use 
21 
COROI.LARY 2.3. 
(2.14) 
(2.15) 
Proof. Immediate. 
THEOREM .2. 4. Pe as n - a>. 
Proof. It follows from (2.4) that 
n(e -~ ) = Ja> nee-a )B(e)f(e)C-n(e)de/Ja> B(l)f(l)C-n(l)dl 
n n ~ n ~ 
n n 
and making the substitution s = nee-an) We get 
n(e -e ) n n 
The theorem now fOllows from a direct application of 
Corollary 2.3. 
Proof. Putting s n(e-a ) 
n 
f
a> - 2 
. A B(e)(e-e ) Helxl, ... ,x )de e n n 
n 
J
a> A - A 2 
.. B (e)[ n(e-e ) -nee -e )] He Ix l , •.• ,x )de e n n n n 
n 
fa>B(Sn-l+e )[s-n(e -e )]2f (sn- l +e )v (s)ds o n n n n n 
11: . j 
J:f(Vn-
l +9 n)Vn (V)dV 
By Theorem 2.4. for n > N sufficiently large 
and 
Therefore 
22 
nee -0 ) < £1iL + 1 
n n h(8) 
[ (ill.l+ 1) 2 + 2]B( -1 A ) ( -1 A) () < h(8) s sn +8 n ~ sn +en vn S. 
and it follows from Theorem 2.2, the dominated convergence theo-
rem (see Loeve [6] p.162) and (2.15) (with B = 1), that 
as was to be proved. 
It follows from Theorem 2.5 and Theorem 1.1 that the stopping ru-
les t(c) and t'(c) defined in section 1 are A.P.O. in our trunca-
tion parameter model with loss functions of the type L(e,d) = 
= B(e)(e-d)2 and subject to the stated conditions on • an~ B. 
3. ASYMPTOTIC OPTIMALITY. 
Following Kiefer and Sacks [5] , we say that a stopping rule tec) 
is asymptotically optimal if 
(3.1) lim sup[E(Yt(c)+ct(c))/in£{E(Ys(c)+cs(c)):s(c) ET}] < 1 
c .... O 
where T is the set of all stopping rules. 
Then we have 
THEOREM 3.1. (Bickel and Yahav (31) •. Under the oondiHons of 
Theorem 1.1 and if 
(3.2) sup nBE(Y ) < • 
n n 
23 
then the stopping rules t(c) and t'(c) are asymptotically optimal. 
The following corollary is an immediate consequence of Theorem 
3.1, upon applying (1.1) and the definition of Bayes risk (1.4). 
COROLLARY 3.2. (Bickel anq Yahav [3]). If the conditions of Theo 
rem 1.1 hold and if there exists a sequence of estimates 8 such n 
that 
then the ru les t (c) and t I (c) are asymptotically optimal. 
We now apply this corollary to our truncation parameter model 
and prove 
THEOREM 3.3. In the truncation parameter model hlith loss func-
tion L(e,d) = B(e)(e-d)2, if the conditions of section 2 are sa-
tisfied and furthermore 
(3.3) hiil > a > 0 for every e c(e) 
then the rules t(c) and t'(c) are asymptotically optimal. 
Proof. Integrating by parts, 
(3.4) lim 
A+e 
o 
-1 Making the substitution: x = ~+sn , we get 
q.S) 
zJe (e-x)(~)ndx 
e Cf8T 
o 
because by (3.3) 
-1 
(C (Han») n 
C(e) 
24 
-1 en[log c(e+sn )-log c(e)] 
for every n(eo-e) < s < Q (e+sn- 1 < ~ < e). 
From (3.4) and (3.5) it fo1~ows that 
where K is a constant. 
Then 
n2J: B(e)Ee[ (e-s n )2]1/J(6)d6..;; KJ: B(e)1jJ(e)de < CD 
o 0 
by (2.2). 
The theorem now follows from Co~ollary 3.2. 
4. REMARK. 
An important corollary can be obtained from Theorem 2.2. Puttin~ 
B_1 in (2.Q) and disregarding the s2 we obtain: 
(4.1) 
_hills 
J
" IHsn-
1+& )v (s) - Ha)e C(e) Ids -+ 0 as n -+ .. 
o n n 
If we now divide by 
(4.2) 
we get 
(4.3) ds '" 
-+ 0 a,s n -+ • 
Furthermore 
(4.4) 
2S 
_.hills 
- ~ e c(e) Ids -+ 0 
c(e) 
because of (4.3) and (2.15). 
as n -+ co 
The limit theorem for the posterior distribution embodied in 
(4.4) is an analogue of the Bernstein-Von Mises theorem (cf. 
Bickel and Yahav [4] , Theorem 2.2). 
ACKNOWLEDGEMENT. This research is part of my Ph. D. Thesis com-
pleted at the University of California, Berkeley, under the gui-
dance of Prof. Peter J. Bickel. The problem was suggested as well 
by Prof. Bickel to whom I wish to express my deepest gratitude. 
26 
REFERENCES 
[1] ARROW,K., BLACKWELL,D. and GIRSHICK,M.A., Ba.ye.~ a.nd m.in..una.x 
~otuz.ion~ 06 ~e.que.nz.ia.t de.e.i~.ion p40bte.m~, Econometrica 17, 
pp. 213-244 (1949). 
[2] BICKEL,P.J. and YAHAV, J.A., A~ympzoz.iea.tty po.inzw.i~e. opz..una.l 
p40ee.du4e.~ .in ~e.que.nz.ia.t a.na.ty~.i~, Proc. Fifth Berk. Symp. 
Math. Stat. Prob. (1965). 
[3] BICKEL,P.J. and YAHAV,J.A., A~ympzoz.iea.tty opz..una.t Ba.yu a.n,., 
m.in..una.x p40ee.du4e.~ .in ~e.que.nz.ia.t e.~z.£ma.z.ion, Ann. Math. 
Statist. 39, pp. 442-456(1968). 
[4] BICKEL,P. J. and YAHAV ,J .A., Some. eonz4.ibuz.ion~ ZO zhe. a.~ymp ... 
zoz.ie Zhe.04Y 06 Ba.ye.~ ~otuz.ion~, Zeit. Wahr. verw. Geb. 11, 
pp. 257-276 (1969). 
[5] KIEFER,J. and SACKS,J., A~ympzoz.iea.tty opt.£ma.t ~e.que.nz.ia.t 
.in6e.4e.nee. a.nd de.~.ign, Ann. Math. Statist. 34, pp. 705-750 
(1963). 
[6] LO~VE,M., P40ba.b.it.izy Zhe.04Y, 3rd. edi, n. Van Nostrand Co. 
Inc. (1963). 
Recibido en marzo de 1974 
Universidad Nacional del Sur 
Bahra Blanca, Argentin~ 
En e.t eu4~0 de. tao .£mp4e.~.i6n de.t p4e.4e.nze. e.je.mpta.4 
de. ta.Re.V.i4Za. de. tao UMA he.m04 4u64.ido tao .in6Mzuna.da. 
pl4d.ida. de. nue.4Z40~ eompa.ne.404 de. z4a.ba.jo y que.4.ido~ 
a.m.ig04 Eve.t.io T. O~ta.nde4 y R.iea.4do Ate.ma.ny, a. euya. 
me.mo~.ia. de.d.iea.m04 e.4Ze. z4a.ba.jo. 
Revista de la 
Union Matematica Argentina 
Volumen 27, 1974. 
EL GRUPO DE WITT DE CIERTAS CLASES DE ANILLOS DE ENTEROS 
Ignacio Kaplan y Horacio H. O'Brien 
En este articulo calculamos el grupo de Witt del anillo de ente-
ros de una extensi6n finita de Q, con la condici6n de que 2 no 
sea una unidad, y que tal anillo sea de Bezout. 
En este trabajo tratamos de generalizar ciertos resultados de 
( 2) • 
Sea entonces K una extensi6n finita de Q, A el anillo de enteros 
de K, tal que 2 no sea inversible en A, y A es un anillo de idea 
les principales. 
1. Denotamos con Q(A) el grupo de extensiones cuadraticas de A; 
dado que 2 no es inversible en A, se tiene por [3) un epimorfis-
mo 
dis:Witt(A) ---+ Q(A) ---+ 0 
dis(P,q) = Z(Co(P,q)) 
siendo Z(C (P,q)) el centro de la parte homogenea de grado 0 del 
o 
4lgebra de Clifford asociada. 
Sea M(A) el nucleo de tal homomorfismo;rsigue que el diagrama: 
0 - M(A) ---+ Witt (A) ---+ Q(A) - o · 
1 1 JI 
0- N(A) ---+ H(A) ---+ Q(A) ---+ 0 
1 1 
0 0 
es exacto y conmutativo, siendo las aplicaciones verticales las 
can6nicas y H(A) el grupo de isomorfismos de algebras de Clifford. 
2.1. PROPOSICION. Sea (P,q) un A m6du~0 de rango 4 y disorimi-
nan .te 1. Entonoes q admite un aero no trivia~ en toda oomp~eta­
taoi6n no arquimediana de K. 
28 
DemoBtraaion. Sea L una tal extensi6n de K yAel anillode 
v v 
valuaci6n correspondiente. 
Como L es una extensi6n finita de Q , para cierto primo p, se 
v p 
tiene que Br (Av) = O. [1] 
En particular N(Av) = O. Entonces (P ~ Av ' q 8 Av) es un Av m6-
dulo cuadratico de range 4 y discriminante trivial, cuya algebra 
de Clifford es nula. 
Por 10 tanto (P ® Av,q ® Av) 
y obviamente tenemos la tesis. 
2.2. COROLARIO. Si A no eBta aontenido en R, y (P,q) es un A 
moduZo auadratiao de rango 4 y disariminante triviaZ entonaes 
(P,q) es un eBpaaio hiperb6Ziao. 
Demostraaion. Dado que q admite un cero no trivial en toda com-
pletaci6n, sigue que q admite un cero no trivial en A [4]. 
Por ser A un anillo de Bezout, (P,q) admite un espacio hiperb6-
lico como sumando directo. [Z] 
Entonces (P,q) = H(A) 1 (P' ,q'), donde (P' ,q') es un A m6dulo 
cuadratico de rango 2 y discriminante trivial. 
Dado que la aplicaci6n can6nica Witt(A) ---+ Witt(K) es un mQ-
nomorfismo, se tiene que (P' ,q') = H(A), de donde obtenemos la 
tesis. 
2.3. COROLARIO. Si A eBta aontenido en R, y (P,q) eB un A modu-
Zo auadratiao de rango 4 y diBariminante 1, entonaes (P,q) es 
un espaaio hiperboZiao si y soZo si (P 8 R,q ® R) es una forma 
reaZ semidefinida. 
3.1. PROPOSICION. Si A no esta aqntenido en R entonaes Witt(A)-
= Q (A) • 
Demostraaion. Es claro que basta demostrar que M(A) = O. 
Dado que todo A m6dulo cuadratico de range mayor 0 igual que 4 
y discriminante 1 admite un cero no trivial en A, basta consi-
derar el caso de range 2 y discriminante trivial. 
Pero en tal caso sigue trivialmente que (P,q) = H(A), Y conse-
cuentemente tenemos 1a tesis. 
4.1. PROPOSICION. Si A seta aontsnido sn R sntonaee ta aptiaa~ 
29 
aion aanoniaa M(A) ---+ M(R) es inyeativa. 
Demostraaion. Sea (P,q) un A modulo cuadratico de range 2n y di! 
criminante trivial, tal que (P ® R,q ® R) es nulo en WitteR). 
Si la dimension real de P & R es 2, el problema es trivial. 
Si tal dimension es 4, sigue que q ® R es una forma cuadratica 
semidefinida y por (2.3) tenemos el resultado. 
Finalmente si la dimension es mayor 0 igual a 6, se tiene que q 
admite un cero no trivial en toda completaci6n de K, y por 10 
tanto admite un espacio hiperb6lico sobre A como sumando direc· 
to. 
Iterande el procedimiento tenemos el resultado, pues resulta 
ser (P,q) un espacio trivial sobre A. 
4.2. COROLARIO. Si A esta aontenido en R entonaes existe una 
suaesion exaata de La forma: 
o -+ nZ ---+ Witt(A) -+ Q(A) -+ 0 
donde n es muLtipLo de 4. 
Demostraaion. Es inmediataa partir del heche que M(R) 4Z. 
4.3. COROLARIO. Si A esta aontenido en R entonaes N(A) 0 6 
N(A) = Z2' 
Demostraaion. En efecte, N(A) es un subgrupo de Br2 (A) por 
[3] y en este case es imagen de un grupo ciclico. 
4.4. PROPOSICION. Si N(A) 
sion e::r:aata 
o entonaesM(A) at y z'a 8uae-
o -+ az ---+ Witt(A) ---+ Q(A) ---+ 0 
8e parte. 
Dem08t:roaoi6n. Sea (P ,q) un A m6dulo cuadrhico pe'rteneciente a 
MeA). 
Si el rango de P es 2, sea iv1 ,v2} una base de P y sea 
II C
za zbOl1 la matrizde dq en tal base, donde dq es la for-
ma bilineal asociada a q. 
30 
Dado que 4ab-c 2 = -1, sigue que x 
q y por 10 tanto (P,q) = H(A). 
- 2v1 + (c+1)v2 es un cero de 
Si rango de P es 4, dado que N(A) = 0, sigue que C(P,q) = 0 en 
Br2(A);entonces por [21 es un A modulo cuadratico trivial. 
Si rango de P=6, sigue que q admite un cero no trivial en toda 
completaci6n de K, y por 10 tanto .10 admi te en A. 
Sigue que (P,q) = (P',q') 1 H(A) , donde (P',q') es un A modulo 
cuadratico de rango 4 perteneciente a M(A}. 
Por 10 anterior se deduce que (P,q) esun espacio trivial . 
Si n=8 
2 0 - 1 0 0 0 0 0 
0 2 0 - 1 0 0 0 0 
- 1 0 2 - 1 0 0 0 0 
0 - 1 -1 2 -1 0 0 0 
0 0 0 - 1 2 - 1 0 0 
0 0 0 0 - 1 2 - 1 0 
0 0 0 0 0 - 1 2 - 1 
0 0 0 0 0 0 - 1 2 
es la matriz correspondiente ·a un modulo cuadratico de rango 
8 y discriminante 1. [ 21 
Por ser (P ® R,q ® R) un R m6dulo cuadratico definido, se tiene 
que (P ,q) -I 0 en Witt(A). 
La primer par t e del teorema se deduce entonces del diagrama 
exacto y conmutativo: 
0- M(A) - M(R) 2Z 
L L 
0 N(A) - N(R) Z2 
1 L 
0 0 
Veamos que la sucesion se parte. 
Sea B un elemento de Q(A), y sea (P,q) un A m6dulo cuadratico 
de rango 2 tal que Z(Co(P,q)) = B. 
Dado que N(A) = 0, se deduce que es el unico m6dulo de rango 2 
(salvo isometrias) con esa propiedad. 
En efecto, sea (P',q') un modulo de rango 2 con discriminante B. 
Sigue entonces que C(P,q)C(P',q') en H(A). 
Dado entonces que rango P = rango P' = 2, sigue trivialmente que 
31 
C(P,q)(l) = C(P' ,q')(l)' Y obtenemos el resultado parcial. 
En las condiciones anteriores 2(P,q) = 0 en Witt(A). 
Sigue que la aplicaci6n 
Q(A) ---+ Witt(A) 
B ---+ (P,q) definida anteriormente 
es un homomorfismo. 
Para esto es suficiente ver que si Bl Y B2 son dos elementos de 
Q(A) y (P1,ql) ,(P2 ,q2) son las imagenes respectivas, entonces 
(P1,ql).1 (P 2 ,q2) = (P,q) donde este modulo cuadratico es ima-
gen de Bl x B2 . 
Dado que (P1,ql) 1 (P 2 ,q2) 1 (P,q) es un elemento de. range 6 en 
M(A) se deduce que es ' nulo en Witt(A). 
Dado que 2(P,q) = 0 en Witt(A) , tenemos la tesis. 
4.5. COROLARIO. Si A esta aontenido en R, N(A) = 0 y Q(A) = 0, 
entonaes Witt(A) = 8Z. En partiauZar,se tiene eZ resuZtado de 
[21 : Wi tt (Z) = 8Z. 
4.6. PROPOSICION. Si N(A) = Z2 entonaes M(A) = Z2' . 
Demostraai6n. De todas las consideraciones anteriores, se deduce 
que M(A) = 4Z 6 M(A) = 8Z. 
Si M(A) = 8Z, entonces el m6dulo cuadratico de range 8, (P,q) 
asociado a la matriz de la proposici6n 4.4 , es un generador 
de M(A). Veamos que esto no puede suceder. 
En efecto, dado que la aplicaci6n M(A) ---+ N(A) es un epimor-
fismo, sea (P,q) un A modulo cuadratico de dimensi6n minima, tal 
que C(P,q) # 0 en N(Al. 
Sigue que (P,q) = n(P,q), 10 cual es un absurdo pues obtenemos 
que C(P,q) = nC(P,q) = 0 en N(A). 
Se deduce que rango P • 4, Y obtenemos el resultado. 
4.7. PROPOSICION. Sea A aontenido en R y (P,q) un A m6duZo ou£ 
drdtioo de rango 2. taZ que (P e R,q e R) tiene determinante n! 
gativ,O . 
Entonoes 2(P,q) • 0 en Witt(A) • 
. Demostraoidn. Sigue trivialmente que (P,q) ! (P,q) es unespacio 
hiperb6lico en toda completaci6n de K. 
32 
Por 10 tanto 2(P,q) admite un cero no trivial en A. 
Sigue entonces que 2(P,q) H(A) 1 (P',q'), donde este m6du10 es 
de rango 2. 
Es faci1 ver que (P' ,q') H(A), Y tenemos e1 resu1tado. 
4.8. COROLARIO. Si A esta aontenido en R, N(A) 
suaesi6n exaata 
Z2' 'entonaes 1.a 
o ---+ 4Z ---+ Witt(A) ---+ Q(A) ---+ 0 
se parte si y s61.0 si no existen m6du1.os de rango 2 auyo deter 
minante sea positivo. 
B IBLlOGRAF lA' 
[1] AUSLANDER,M., GOLDMAN,O., The B4aue4 g40Up 06 a commu~a~~ve 
4btg, (T.A.M.S.), vol.97, 1960. 
[2] LAROTONDA,A., MlCALl,A., VILLAMAYOR,O., SU4 te g40upe de 
WU~ (1972). 
[3] MlCALl,A., VlLLAMAYOR,O., SU4 te~ atg~b4e~ de Ct~6604d, 
Ann. Scient. Ec. Norm. Sup., (1968). 
[4] O'MEARA, Quad4a~~c F04m~. 
Recibido en julio de 1974 
Universidad Nacional de Rio Cuarto 
Argentina 
Revista de la 
Union Hatematica Argentina 
Volumen 27, 1974. 
ON THE PARTITIONS OF AN INTEGER 
Raul A. Chiappa 
As is well known (see for example [11, [21) the following formu-
la (I) gives by recurrence the number of "partitions" - the "par-
tages" of French authors - of a positive integer n; i.e., the nu!!!. 
ber of non decreasing sequences of p6sitive integers whose sum is 
n. 
Formula (I) is 
7f 
n 
I (_l)i-l(7f 
i~1 3 ·2 . l. -l. n---
2
-
where we take 7f
n 
= 0 for n < 0, and 7fo = 1. 
(I) 
We present here a direct calculation - direct, in the sense that 
no generating function or Euler identities are used - Further-
more, for the proof we introduce a general lemma, which seems to 
merit some attention in itself. 
Let ~j,n denote the number of such sequences whose first element 
is j (j ~ 1). Then, 7f
n
_ 1 ~l for n ~ 2; and 7f = L~. for ,a n j~IJ,n 
n ~ 1. Clearly,~. = 1 whenever j = n or [n/31 < j ~ [n/21 and 
J ,n 
~j ,n = 0 when n # j > [n/21 ([ xl denotes "integral part of x"). 
The following array gives the non zero values of~. and 7f for 
J ,n n 
1 <;n<;14. 
2 3 4 S 6 7 8 9 10 11 1 Z 13 14 
2 
3 
4 
5 
6 
7 
2 3 5 7 11 
2 2 
1 
15 22 
4 4 
1 2 
1 
30 
7 
2 
1 
42 
8 
3 
1 
S6 
12 
4 
2 
77 
14 
5 
2 
1 
101 
21 
6 
3 
2 3 5 7 11 15 22 30 42 56 77 101 135 
34 
For convenience, we extend the function ~ to all nEZ, setting 
~. = 0 whenever j ~ 1 and n ~ O. We then have the equalities: 
J,n 
A) lT n _ 1 = ~l n , 
~. J,n 
for n , 0 
We now proveC. It is clear, when j , p, ~. }: ~h .. J,P 
h~j 
,P-J 
Then, for p , q we have: 
~ = }: ~ ~ + }: ~ ~ + ~ q-l,p-l h~q-l h,p-q q-l,p-q h~q h,p-q q-l,p-q q-p 
and for p = q, ~q,p = ~q-l,p-l = and ~ = O. q-l,p-q 
In passing, we note that Property C and the values~. = 0 for 
J,n 
j ~ 1, n ~ 0, and ~l 1 , ~. 2' = 1 for all j ~ 1, determine uni-J, J 
quely the function ~. 
The following formula (II) is equivalent to (I), and will provi-
de us with our basic approach. 
IT = 2 IT - IT + n n-l n-3 
}: (_1)i-l (A 
i~2 
+ A 
3i2 -i 
n---
2
-
where As ",- . lT s - lT s _1 ' lT n = 0 for n < 0 and lTD = lTi 1. 
(II) 
Clearly, evaluating lT n - lT n_I from (I) provides (II). On the oth-
er hand, evaluating with the aide of (II) the sum of the values 
lT
n
, lT
n
_ I , lT n _ 2 , ... we have (I). 
Let us see now that the evaluation of lT n (n,O) - i.e., the sum 
of the elements of the n-th column of the array ~ - can be redu~ 
ced to the evaluation of a difference: subs tract th~ sum of the 
values of ~ on the set L = {(j,n-j-1) ; j ~ 1} from twice the sum 
of the elements of the (n-1)-th column of the array ~. 
Specifically, for n#O and from A), B) and C) we get: 
'If 
n }: ~J',n 
j~l 
Therefore, using again A), B) and C) we have: 
7r 
n = 2 11' n-l -
35 
7r -
n-3 
for n ~ 2 
The use of Property D, see below, reduces the evaluation of 
(1) 
L ~o 0 1 to the consideration of the elements of the array for 
j~3 J,n-J-
another set T, which will be determined implicitly by the resulting 
equations. 
First, it will be convenient to visualize in the following array 
the sets Land T, whose elements are represented by means of "+" 
and "0,, respectively. 
n-13 n-12 n-11 n-10 n-9 n-8 n-7 n-6 n-5 n-4 n-3 n-2 n-1 n 
0 0 0 0 0 0 0 0 + j = 1 
0 0 0 0 + 2 
·0 0 0 + 3 
0 0 + 4 
0 + 5 
+ 6 
The above mentioned Property D is the following: 
L 
l~h~q-l 
D) ~q,p = ~1,p-(q-l) - ~ h,p-q 
For the proof, it suffices to reiterate (q-1) times Property C. 
From D, we obtain: 
L ~Jo n- Jo-l = L (~l n-2 Jo - ~1,n-2Jo-l) - L L ~h,n-2Jo-l (2) . j~3' j23' h~2 j~h+l 
For convenience, we regroup the terms as follows: 
'PI, m L (~l m-2s - ~l m-2s-1) 
S20' , 
(3) 
for h ;;;. 2 
From C, we obtain 'P 2 ,m = 'P1,m-l ' and more generally, from D 
'P r ,m = 'P 1 ,m-r+l - L 'Ph,m-r for r ~ 2 (4) 
2~hsr-l 
Also, from A), C), and (3) we get: 
'P 1 •m- i + 1 - 'P 2 ,m-i .. .7r m-i - 7r m-i-l D. m-i (5) 
36 
The last equality and the following proposition will permit us to 
express <p , and thus L 1/1. • I ' in terms of llk (k <; n-7). 
r,m j~3 J,n-J-
PROPOSITION. Fo~ r ~ 3, we have: 
<Pr,m II + L (-1) k II : m-r we:W m-r-(wI+w2+w3+"'+w~) 
r 
where W r is the set of sequences of positive integers w. (1 <; i <; k) J. 
such that 
Proof. For r=3, the set Wr is empty, and thus, from (4),{5) 
'P -<P =b. l,m-2 2,m-3 m-3 
For r > 3, the use of (4),{5) and an inductive reasoning gives 
'P1,m-r+1 - 'P 2 ,m-r - L 'Ph,m-r 
3~hsr-1 
b. 
m-r 
L (b. + L (-1) t b. ) 
m-r-h ue:U
h 
m-r-h-(u l +u 2+ .•• +u t ) 3sh~r-l 
where, for each h, we have h-1 ~ u l > u 2 > u3 > ... > u t ~ 3. 
Setting wI = h and wi = u i _1 for 2 <; i <; k = t+1, we can write: 
'P ' = II r,m m-r + L 
we:W 
r 
where the w-sequences satisfy the required conditions. 
Having obtained this, we return to (2). If in this formula we sub-
stitute j by (3+s) in the indexes (1,n-2j), (1,n-2j-1) and j by 
(h+1+s) in the indexes (h,n-2j-1) for h ~ 2, we obtain: 
Finally, using (3),(5) and the proposition above: 
For h=l, the set Wh is empty, then: 
(6) 
where 
37 
with h-1 ~ wl > w2 > w3 > ... > wk ~ 3 for any h ~ 4. 
The only remaining task is now to evaluate S. This task is simpli-
fied by using the following Lemma. Although we need it only for 
a=3. b:;l. we will present it for any a > O. b ~ O. 
LEMMA. For given arbitrary integers a > 0, b ~ 0, and every inte-
ger s ~ a, tet Ws denote the set of subsetsof the set 
{a.a+l •...• s}. Then, if r is any funation for whiah there is some 
~ suah that x > ~ impties rex) = 0, we witt have: 
R = 
s~a we:W 
s 
(-1)1 wi r(a(s+b) + 
r(a 2+ab) + L (-1)k{r(t(3(k+a)2 
k~l 
+ r(lC3(k+a)2 + (k+a) + a(2b-a-1)))} 
L 
W.e:w 
~ 
(k+a) + a(2b-a-1))) + (8) 
Proof. The sums in (8) are finite. as follows easily from a > 0 
and the above conditions on the support of r. 
For fixed s. we shall proceed to associate to each w E Ws - with 
one exception - an element w· E W .• where j satisfies either 
J 
s+l or j = s-l and in such a way that a(s+b) + L" Wi 
a(j+b) + 
W.e:w 
~ 
L w!. with cardinals Iwl and Iw'l differing 
w~ EW' 1 
~ 
in one. 
Therefore. as r(a(s+b)+L Wi) is equal to r(a(j+b) , +L wi). these 
terms c~n be omitted in the evaluation of R. since they appear 
with opposite signs. Hence. for fixed s. the sum 
L (_1)1 wi r(a(s+b) + L Wi) will be reduced to a single term. 
we:W w.e:W 
S ~ 
To accomplish this. we start by identifying each w E Ws' w ; 0, 
with the decreasing sequence of its elements. That is to say, if 
k is the number of elements in w (w E We)' we will have 
w = C"':l.w2 ;"w3 ' ...• wk), with s ~ wl > w2 > w3 > ... > wk ~ a. 
Denoting by: 
Wl = {w / w 
s a} 
38 
w2 {w / w (WI' w2 ' ... , Wk) wk > a wI < s} s 
W3 {w / w (WI' w2 ' ... Wk) wk > a , wl = s} s 
we get: W = WI U W2 U W3 U {0 } , where 0 will denote the empty s s s s s s 
set as element of W s 
Clearly', W2 W2 2 are empty. a' a' and Wa+I 
We now define the rule of associat i on and, for the sake of clari-
ty, we divide the problem in two parts. 
PART I. For fixed s ~ a, let as : W~ 
ping defined by: 
- 2 Ws+I U {0 s+I } be the map-
Obviously, as is onto and one-to-one. 
We will then have : 
R = r (a(a+b) + I 
s<!a+I 
I 3(_1)lwl r(a(s+b) + I wi) 
WEW w. e:w 
s L 
because for 5 ~ a the terms in WI compensate with those in s 
2 
Ws+I u {0 s+l} , and for s ~ a+1 the same 6ccurs with W
2 H\} U s 
I Moreover, that in R appear, only the and Ws _I . we see now, terms 
corresponding to 0a and W
3 (s > a+1) . s 
PART II. First, let f, 1 ~ f ~ k, be the greatest integer such 
3 that for WE Ws ' w = (w I ,w2 ' ... ,wk) we have wi = s - (i-1) whe-
never 1 ~ i ~ f. 
For convenience, we set: wk = a+g and, since it wil1 be needed la 
ter, we single out the case f=k. Thus, the additional assumption 
g=f clearly implies s = 2k + a - 1 and reciprocally;. while g = f+1 
implies s = 2k + a and reciprocally . That is to say, f=k=g if and 
only if (s-a) is odd, while f = k = g-l if and only if (s-a) is 
even. Thus, for both we have k = [s-2+ 11 , where [ 1 denotes the 
function "integral part of". 
Next, we consider in W3 two subsets:W4 , whose elements are the s~ 
s s 
quences such that wk = a + g ~ a + f but excluding that one with 
39 
f = k = g(if any); and w~, where wk = a + g > a + f, but e;x:clu-
ding that one with f = k = g-l (if any). 
455 
Cl.early, Wa+I ' Wa+I ' · and Wa+2 are empty. 
The above considerations enable us to complete the rule of asso-
ciation by defining the followirtg mapping: 
For fixed s ~ a+1, let 13 : W4 --+ WS+1 be given by: 888 
13
8 (wI' w2 ' , wk )= (wi, wi, ... , wit_I) 
{ w. 
+ when 1 ,.; i ,.; g 
w! = 1. 
1. when g+l ,.; i ,.; k-1 w. 
1. 
where 
Clearly, I3
s 
is one to one. Moreover, it is onto, since for 
w' E WS 8+1 
gives f3 8 (w) 
w. 
1. 
w' = · (wi, wi, . ,. , wit), the sequence 
j 
w! - 1 if ,.; i ,.;f 
1. 
= w! if f+1 ,.; i ,.; k 
1. 
a + f if i k + 
w' . 
By means of f3 8' we see that for s ~ a+1 the terms corresponding 
to W: are equal to the terms corresponding to 5 W8+1 ' and for 
s ~ a+2 the same thing happens with the terms of WS and 4 8 W8_1 · 
Therefore, in the evaluation of R they compensate and hence can 
be omitted. Thus, for e:och W3 , only one element will remain for 
8 
evaluating R. Using this, we have the following reduced expres-
sion: 
r(a2 (-1) k 
k 
R + ab) + L r(a(s+b) + L wi) 
8~a+1 i-I 
where w. = s - (i-1) , 1 ,.; i ,.; k , and k = [ s-r1] 
1. 
According to these restrictions, we have, for s = 2k + a -
k 
a(s+b) + L w. - l (3(k+a)2 - (k+a) + a(2b-a-l)) 
i-I 1. 
and for s 2k + a 
(9) 
40 
k 1 2 
a(s+b) + I w. - Z (3(k+a) + (k+a) + a(2b-a-l)) 
i=1 1 
Replacing these expressions iri (9), we obt~in the asserte4 equali-
ty (8). 
Returning to our problem, we substitute in the Lemma the values 
a=3, s=h-l, b=l, r(j) = t::. 3.' Then, the resulting value .o-f R is 
n- -J 
precisely the value of S given by (7). 
If moreover, we set k+3 = i, we obtain: 
S " I:' (_l)i-l {" = Un-IS + [. U 
h4 
} 
3i2+i n----
2 
(10) 
Finally, by replacing in (1) the results from (6) and (10), we 
obtain (II), as asserted. 
REFERENCES 
[1] M.HALL, Comb~na~o4~al Th~04Y, Ed. Blaisdell, 1967. 
[2] P.BERTIER, Pa4~ag~~, pa4~~~~, pa4~~~~on~, d~comp~~~ ~~ 4~p4~­
~~n~a~~on~. METRA, 1967, (VI-I) pg. 103-129. 
A\,;KNOWLEDGMENTS 
I wish to express my sincere thanks to Prof. A.Diego and M.L. 
Gastaminza for many stimulating discussions. 
Recibido en octubre de 1974 
Universidad Nacional del Sur 
Bahia Blanca, Argentina 
En el CU4~O d~ la ~p4~~~6n d~l p4u~n~~ ~j~pla4 
d~ la R~v~~~a d~ l.a UMA h~o~ ~U64~dola ~n604~unada 
pl4d'(;da d~ nuu~4o~ compaii~40~ d~ ~4abaj 0 y qU~4~do~ 
im~go~ Ev~l~o T. Okl.and~4 y R~ca4do Al~any, a cuya 
m~o4~a d~d~camo~ ~~~~ t4abajo. 
Revista de la 
Uni6n Matem~tica Argentina 
Volumen 27. 1974. 
CURVAS Y CUATERNIONES 
Luis A. Santa16 
RESUMEN. Es bien sabido que las rotaciones alrededor de un 
punto, 
en E3 ' se representan biyectivamente por los puntos del es
pacio 
eliptico S3. A una curva 'Y de S3 correspondera una familia 
de ro 
taciones dependientes de un parametro. El objeto de esta no
ta es: 
1° Calcular la curvatura y la torsion de la curva 'Y de S3 e
nfu~ 
cion del vector V (velocidad instantanea) de la rotacion co
rres-
pondiente de E3 (formulas (4.15) y (4.18)). 
2° Aplicar el resultado al caso de las "rotaciones de Fren
et", 0 
sea, a las rotaciones correspbndientes al triedro de Frene
t 
(T.N,B) de una curva de E3 . Resultan 
as! los teoremas del final 
del trabajo. 
1. INTRODUCCION: RESULTADOS CONOCIDOS. 
Sea E3 el espacio euclidiano de 3 dimensiones. 
Un cuaternion es un mlmero hipercomplejo 
(1 • 1 ) 
donde a i son num
eros reales (componentes del cuaternion) y las 
unidades i. j, k cumplen, para el producto. las relaciones 
(1 .2) - 1 
de las cuales ~e deducen 
(1 .3) i = jk = -kj j = ki = - ik k = ij = -ji 
de manera que los cuaterniones forman un algebra asociativa
, no 
conmutativa. La primera componente ao se llama la compo
nente es-
calar y las aI' a 2 • a3 las comp
onentes vectoriales. 
A todo cuaternion a asociamos el vector 
42 
(1.4) 
que tiene pOT componentes a 1 , a2 , a 3 respecto de una terna de vec 
tores ortonormales I, J, K. En todo 10 que sigue las letras mi-
nusculas (sin indices) indicaran cuaterniones y las mismas letras, 
mayusculas, indicaran los vectores correspondientes. Reciproca-
mente, a todo vector (1.4) haremos corresponderel cuaterni6n 
"puro" (con la parte escalar nula) que tiene por componentes veS 
toriales las componentes del vector. Observese que en esta corres 
pondencia, a los cuaterniones unidad i, j, k, corresponden los 
vectores de la base I, J, K. 
Representaremos por a* = aO-a1i-a2j-a3k al cuaterni6n aonjugado 
de a. La norma de un cuaterni6n es entonces 
(1 .5) N(a) * * a2 2 2 2 aa = a a = 0 + a 1 + a 2 + a3 
y el inverso de a, es 
(1 .6) 
Se observa que es 
-1 a a* 
N (a) 
(1 .7) N(ab) = (ab)(ab)* 
Para un cuaterni6n puro es 
abb*a* 
(1 .8) u* -u N (u) = u (-u) 
Y si es de norma unidad, se cumple 
(1 .9) - 1 
N (a)N (b) 
2 -u 
Es bien sabido que una rotaci6n X -+ X' en E3 alrededor del ori-
gen, se puede representar mediante cuaterniones por la expresi6n 
(1.10) -1 x, = q*xq = q xq 
donde x es el cuaterni6n puro correspondiente al vector X y q es 
un cuaterni6n de norma unidad. 
El cua.tern16n -q representa evidentemente la misma rotaci6n. 
Comodar q equivale a dar sus componentes qi ' que por ser q de 
norma unidad satisfacen la relaci6n q~ + q~ + q~ + q~ = 1,resul-
ta que a todo punto de la esfera unidad de E4 corresponde una ro-
43 
tacion de E3 y a puntos diametralmente opuestos corresponde la 
misma rotacion. 5e puede demostrar facilmente que esta correspon-
dencia es biyectiva y como la esfera unidad de E4, con los puntos 
diametralmente opuestos identificados, es el espacio proyectivo 
P3, se puede enunciar: 
_ Las rotaciones de E3 alrededor de un punto, estan en corresponde~ 
cia biyectiva con los puntos del espacio proyectivo P3 . Mas exac-
tamente, este espacio P3 es el espacio del grupo de las rotaciones 
de E3 alrededor de un punto. 
En el espacio proyectivo P3 se puede introducir la distancia ¢ 
entre dos puntos a, b por la formula 
(1.11) cos ¢ = t(a*b + b*a) ( a, b ) 
El espacio proyectivo P3 con esta metrica se llama el espaaio eLip-
tiao 53' 
Tenemos as! una correspondencia biyectiva entre las rotaciones 
de E3 alrededor de un punto y los puntos de 53' A una familia de 
rotaciones dependientes de un parametro correspondera una curva 
de 53' Nuestro objeto es estudiar esta correspondencia. 
Para ello necesitaremos algunas formulas que vinculan los vecto 
res de E3 con los cuaterniones (que son los puntos de 53)' 
5e introduce la notacion 
(1.12) 
y dos puntos se l1aman aonjugados (ortogonales) si su distancia 
es w/2, 0 sea 
(1.13) (a,b) = (b,a o 
5i a,b son cuaterniones puros, es 
(1.14) (a,b) = A.B 
donde A.B indica el producto escalar de vectores de E3 . 
Entre 3 vectores A, B, C de E3 y sus correspondientes cuaternio-
nes puros a~ b, c se puede comprobar la relacion 
(1.15) (A B C) = t (cba - abc) 
donde (A B C) es el producto mixto (A A B).C, 0 sea, el determinan 
45 
que son las f6rmulasde Frenet para las curvas del espacio elip-
tico S3' 
Para futuros usos, conviene calcular ql, q2, q3 en funcion de 
qO q. Se tiene, por ser q de norma unidad 
(2.6) 
(2.7) 
De aqui 
(2.8) 
(q,q) = q*q = qq* = 1 
1 
q 
q' 
I( q , ,q ') 
( q , ,q ') q It _ (q', q It) q , 
( , ') 3/2 q ,q 
( q ,q' ) o 
Para hallar q2 procederemos por coeficientes indeterminados, po-
niendo 
(2.9) 
de donde, dadas las condiciones de ortonormalidad (2.1), resulta 
(2.10) 
( , It) 
( q It 1 ) = .:....:I.q --,-, .::I.q--,--"" 
,q (") 1 / 2 q ,q 
( qlt ,q ) 
y por tanto 
(2.11) 
1 ( q , ,q It) 
/J (qlt - (qlt ,q) q - .:....:I._--'-_:l._--!- q') 
( q' ,q ') 
El coeficiente /J se determina por ser q2 de norma unidad, resul-
tando 
(2.12) ( q" ,q It) _ (q. q It) 2 _ (q'. q It) 2 
(q' ,q ') 
La expresi6n de q3 es mas complicada y no la vamos a necesitar. 
3. LA CURVATURA Y LA TORSION DE UNA CURVA DE 53' 
A partir de las f6rmulas de Frenet (2.5) se introducen las siguie~ 
tes definiciones 
(3.1) 
(3.2) 
(3.3) 
46 
ds = elementode arco 
d 
2 
wO 
1 
2 
w 3 
wO 
1 
k curvatura 
w= torsi6n 
Para calcular k a partir de la curva q 
que a partir de (2.5) se deduce 
(3.4) 1 1 2 1 2 w 2 = (dq ,q) = (q ',q) d t 
(q',q,)1/ 2 dt 
q (t) 
0 · 
q (t), se observa 
[ ( 9 ' ,9 ') ( 9" ,q ") - (9', 9 ") 2 _ (9', 9 ')( g" ,9 ) 2] 
v (q',q,)3/2 
[ ( 9 ' .9 ')( 9" , 9 ") - (9'.9')( 9 , 9 ") 2 - (g', 9 ") 2] 1 I 2 
(q',q,)2 
y por tanto 
(3.5) k 2 = (9',9')(9",9") - (9'.9')(9,9,,)2 - (9',9,,)2 
(q',q,)3 
Teniendo en cuenta que 
(q,q') = 0 ( q , q ") + (q' ,q , ) o ( q,q") -(q',q') 
se puede escribir tambien 
(3,.6) k 2 = (9', 9 ')( g" , 9 ") ( 9 , ! 9 ") 2 _ 1 
(q',q,)3 
Comparando (3. 5) con (2.12) resul ta 
(3.7) 
Para el c§lculo de w procederemos de manera indirecta. Sabemos 
que es 
q' = (q',q') 1/2 ql 
y pongamos q'" = ( .. ) gO + ( •• ) ql + ( •• ) q2 + 7jq3 , donde los va 
47 
10res de los parentesis no interesan. Por ser e1 determinante 
1, sera 
(3.8) [q q' q" q'''1 = (q',q,)1/2" 11 
E1 valor de " esta dado por (3.7). Para ha11ar 11 observemos que 
o y tambien 
(3.9) 
puesto que (ql, ,q3) = 0 por tener ql, solamente componentes se-
1 2 gun q, q , q . 
Por otra parte se tiene 
(3.10) w 
,,( , ') 1/2 q ,q 
y comparando con (3.8) y ap1icando (3.7) resu1ta 
(3.11) [g g' g" g"'1 
(q',q,)3 
que, puesto que ya se conoce k, nos da e1 valor buscado de w en 
funcion de q y sus derivadas. 
4. RELACIONES ENTRE LAS CURVAS OE S3 Y LAS FAMILIAS DE ROTACIONES 
EN E3 • 
Una rotacion en E3 a1rededor del origen, se puede considerar de-
finida por 1a posicion de 1a terna ortonorma1 de vectores 
(El, E2, E3) transformada, por 1a rotacion dada, de 1a base ini-
cia1 I, J, K. Por 1a rotacion definida por e1 cuaternion q = qO, 
los vectores Ei son los correspondientes a los cuaterniones 
(4.1) l" e = q* i q e 2 = q* j . q e 3 = q* k q 
o sea, 
48 
e l ( 2+ 2 2 2) . + 2 ( ) . qo ql-q2- q3 1 qlq2- qOq3 J + 2(qlq3+qOq 2)k 
(4.2) e 2 2(qOq3+q l q2)i + 
2 2 2 2 . 
(qO-ql+q2- q3)J + 2(q2 q3-q Oql)k 
e3 2(Q1Q3- q Oq2)i + 2(QOQ1+Q2 Q3)j + 
2 2 2 2 
(QO-Ql-Q2+Q3)k 
Es decir, la terna ortonormal de vectores (E l , E2, E3) que defi-
ne la rotacion correspondiente al cuaternion de norma unidad Q, 
es la de los vectores Ei que tienen las mismas componentes que 
los cuaterniones e i , dadas por las formulas anteriores. 
Consideremos l a familia de rotaciones correspondientes a los pun-
tos de la curva Q(t) = QO(t). Por ser (E l , E2, E3) una terna or-
tonormal, es Ei.dE i = 0, Ei.dEj + Ej.dE i = 0 (i # j) y por tanto 
los vectores dEi/dt se expresan en la base (E l , E2, E3) en la 
forma 
(4.3) 
Los coeficientes a, ~, ~ tienen claro significado geometrico. 
Ellos son, efectivamente, las componentes respectivas de la base 
movil (E l , E2, E3) del vector potaai6n instant&nea 
(4.4) H 
El nombre proviene de que el movimiento, en cada instante, es una 
rotacion alrededor de H, puesto que dEi/dt H 1\ Ei. 
Teniendo en cuenta que las componentes de Ei son las de e i en 
(4.2) , resulta 
a dE
2 
E3 2(Qbql qiqo + QiQ3 - Qj q2) -
dt 
(4.5) fJ dE
3 
El 2(QOQ2 - Qi Q3 - QiQo + QjQl) 
dt 
~ 
dEl E2 2(qOq3 + Qi Q2 QiQl - Q3 QO) -
dt 
De aQui se deduce 
49 
2 qo a ql + (j q2 + 'Y q3 
2 qi -a qo - (j q3 + 'Y q2 
'(4.6) 
2 qi a q3 -(i qo - 'Y ql 
Estas ecuaciones pueden condensarse en 
(4.7) -2 q' :0 v q v :0 -2 q' q-l 
donde 
(4.8) v :0 ai + (ij + 'Yk 
es el cuaterni6n correspondiente al vector 
(4.9) v :0 aI + (jJ + 'YK 
Nuestro objeto es c~lcular ahora la curvatura y _la torsi6n k, w 
de la curva q(t;) de S3' en funci6n del cuaterni6n v (4.8) cuyas 
componentes a, (i, 'Y tienen significado geometrico en el espacio 
de las rotaciones E3 . 
Para ello, de (4.7) se deduce 
(4.10) 
q':o_1. vq 
2 
q" -(1/2)(v'q + vq') - 1. (v'q - 1. v v q) 
2 2 
q'" 1 (v"q v'vq 1 v'q + 1. v v v q) - - -v 
2 2 4 
q'* 1 q* v* 
2 
q"* (q*v' * 1 - q* v* v*) 
2 2 
q"'* - 1.(q*v"* q*v*v'* 1. q*v'*v* + 1 q*v*v*) 
2 2 4 
Introduciendo el vector V definido por (4.9), tenemos 
(4.11) v2 :0 v*v :0 V v* VV' :0 (v*v' + v'*v) 
2 
50 
que son, ambas, expresiones escalares, a sab~r: y2 = a2 + p2 + ~2. 
Y Y' = aa' + PP' + ~~'. Por otra parte, siendo v un cuaterni6n 
puro, es 
(4.12) v + v* = 0 v v = v* v* 
y por ser q de norma unidad, es 
(4.13) q*q = qq* = q*'q + q"'q' = 0 ,(q',q) 
Con estas relaciones, de (4.10) se deduce f~cilmente 
(q',q,)=ly2 
4 
con 10 cual queda 
(4.14) 
( q" ,q") 
o sea, en notaci6n vectorial 
(4.15) 
(y2)2 + ly,2 ,(q',q") 
16 4 
Para hallar la torsi6n w, aplicamos la relaci6n (1.16), 
(q,q') =0 
1 y Y' 
4 
(4.16)[q q'q"q"'] = l(q"q'*qq"'*+q"'q"'q'A"*-·qq'*q"q",*.q"'q"*q'q*) 
4 
Teniendo en cuenta los valores (4.10) y las relaciones (4.11), 
(4.12) y (4.13), despues de un c~lculo un poco largo, pero sin 
artificios, resulta 
(4.17) [q q'q"q"'] (Y Y'Y") + 1 ((Y y,)2 _ y2 y,2) 
8 16 
donde en el segundo miembro se trata de operaciones con el vec-
tor Y. 
Por tanto, teniendo en cuenta (3.11) y (4.14), resulta 
(4.18) [g 9 '9"9"'] 
(q',q,)3 
de donde, conociendo k por (4.15), se puede expresar w en funci6n 
de Y y sus derivadas. 
51 
5. ROTACIONESDE FRENET. 
L1amaremos rotaaiones de Frenet a las rotaciones a1rededor de -un 
punto de E3 que corresponden a1 movimiento del triedro de Frenet 
(T,N,B) de una curva del espacio E3 . Esto quiere decir, que los 
versores T,N,B que definen e1 movimiento (tangente, normal prin-
cipal y binormal de una curva) satisfacen a las ecuaciones de 
Frenet 
dT KN 
ds 
(5.1) dN - KT + TB 
ds 
dB - TN 
ds 
Es decir, comparando con (4.3) resu1ta que e1 vector V tiene las 
componentes a = T , P = 0 , ~ = K, 0 sea, es e1 vector 
V = TI + KK, siendo ",T 1a curvatura y 1a torsion de 1a curva de 
E3 • 
Segun esto, 1a curva de S3 correspondiente a una rotacion de 
Frenet, tendra 1a curvatura y 1a torsion dada5 por 
(5.Z) k 
Z(KT I - KIT) 
(,,2 + T2)3/2 
y, si k # 0, entonces 
(5.3) w - 1 
be aqui 5e deduce: 
1 0 A Las heLiaes de E3 (KIT 
(k=o) de S3. 
Z d arc tan ~ 
ds K 
constante) aorresponden Las reatas 
ZO A Las aurvas de E3 que no son heLiaes. aorresponden aurOas de to~ 
sion aonstante (w=-1) de S3. Para estas aurvas. La aurvatura to-
taL Ik ds es un muLtipLo de 4w (inaLuido 0). 
Podemos preguntar, ~cua1 es 1a condicion para que 1a rotacion de-
finida por las ecuaciones (4.3) sea un movimiento de Frenet? 
Para e110 sera necesario y suficiente que exista un cambio de ba-
se ortonorma1, sea 
52 
(5.4) 
con la matriz (a i h) ortogonal, tal que el versor rotacion instan-
tanea H (4.4) no tenga componente segun AZ, 0 sea, se verifique 
AZ.H = 0, 10 que equivale a 
(5.5) o 
de donde se deduce (puesto que las a i . son constantes) 
J 
(5.6) 
y para que este sistema (5.5), (5.6) tenga solucion debe ser 
(5.7) (V V'V") ex ' !3' 'Y ' o 
a: II (3 " 'Y" 
o 
Rec1procamente, si esta condici6n se cumple, el sistema (5.5), 
(5.6) tiene soluci6n y se tiene el versor AZ, que completado con 
AI, A3 de manera que formen una terna ortonormal, determinan un 
triedro de Frenet. Por tanto se puede enunciar: 
3° La aondiai6n neaesaria y sufiaiente para que una rotaai6n de-
finida por el sistema (4.3) sea una rotaai6n de Frenet, es que 
sea (V V'V") = 0, siendo V el veator (4.8), lo que equivale a la 
aondiai6n (5.7). Segun (4.18), la aondiai6n es equivalente a que 
sea w=-1, siendo w la torsi6n de la aurva de 83 asoaiada a la r~ 
taai6n, ~iempre que esta aurva sea de aurvatura k distinta de aero. 
BIBL IOGRAF IA 
[1) BLASCHKE,Wi1he1m, K~nema~~k und Qua~e~n~onen, Mathematische 
Monographien, Deutscher Verlag der Wissenschaften, Berlin, 
1960. 
En este trabajo se citan otros de Clifford, Hje1ms1ev, Study re1a 
cionados con 1a representacion de rotaciones por cuaterniones y -
por puntos de 53' Para problemas referentes a 1a integracion del 
sistema (4.3) ver tambien e1 cliisico tratado de DARBOUX, Th€.o~~e 
de~ S~~6ace~, Parte I, pggs.I-64, Gauthier-Villars, Paiis, 1914. 
Recibido en octubre de 1974 Universidad de Buenos Aires 
Argentina 
Revista de la 
Union Matematica Argentina 
Volumen 27, 1974. 
ACTIONS OF COMPACT, CONNECTED 
LIE GROUPS ON SPIN MANIFOLDS 
Cr;st;an U. Sanchez 
ABSTRACT. We study actions of compact connected Lie groups on 
Spin manifolds with only two fixed points obtaining results 
about the equivalence of the representations of the group on 
the tangent spaces at the fixed points. 
For semifree actions we get: Let X be a connected 2t-dimensional 
Spin manifold. Let G be a compact connected Lie group acting di,f 
fereptiably on X, with only two fixed points and semifreely. 
Then the two representations of G at the fixed points are equiv~ 
lent. 
1. I NTRODUCT ION. 
As the title indicates our objective is to study actions of co~ 
pact, connected Lie groups on Spin Manifolds. We consider the 
specific case of an action with only two fixed points attemp-
ting to obtain information on the relationship between the re-
presentations of the group on the tangent spaces at the fixed 
points. 
The basic tools for this work are the results of Atiyah and 
Hirzebruch [3] (see (2.1) and (2.2)). 
In the case of a semi-free actions our results generalize a well 
known theorem · of Milnor ([ 1], p.478) for the particular case of 
connected groups. 
Our notation will be essentially from [1], [2] and [3]. 
We also indicate bY ,F(H,X) the fixed point set of H in X and by 
F(H,x,X) the connected component of F(H,X) containing the point 
x EX'. 
I wish to thank ·Professor Ted Petrie for his encouragement. 
54 
2. SPIN MANIFOLDS. 
Let X be a compact, oriented n-manifold with a Riemannian metric. 
Let Q be the oriented orthonormal frame bundle of the cotangent 
bundle to X. 
A Spin-structure of Xis a principal Spin(n) bundle P over X 
with a covering map w: P --4 Q of degree 2 such that the diagram 
P x Spin (n) -----+1 P 
1 w x A w 1 
Qx SO(n) ---_I Q 
co~utes, where A: Spin(n) ----+ SO(n) is the standard double 
covering and the horizontal arrows are the actions on the bun-
dles. A compact, orientable manifold with a Spin-structure is 
called a Spin Manifold. 
If we have an action of a compact Lie group G on X (n=2t) by 
orientation preserving isometries and an action of G on P com-
muting with the action of Spin(2t) from the right and compa-
tible with w we can consider its Spinor-index Spin(G,X) which 
is by definition an element in R(G) ([1), p.479, [3], p.1S). 
The evaluation of Spin(G,X) in an eleme~t g of G is denoted by 
Spin(g,X). 
We shall indicate now two results from [3] and one from [11 
which are necessary for our work. 
(2.1) THEOREM. (Atiyah-Hirzebruch) LetX be a aonneated Spin 
Manifold of dimension 2t and G a aompaat aonneated Li"e group 
aating non triviaHyorz X and aating on its Spin-struature P. 
Then Spin(G,X) = O. 
(2.2) THEOREM. (Atiyah-Hirzebruch) Let X be a aompaat, aonnea-
ted, orientable differentiable manifold with Riemannian metria , 
on whiah a aompaat aonneated Lie group G operates effeatively 
by isometries. Suppose P is a Spin-struature for X. Then there 
exist aanoniaa Hy a Lie group Gland a homomorphism h: G 1 - G, 
whia~ is either identity 01' ~ double a6vering, suah that G1 
aats on P induaing via h the given aation of G on X. 
(2.3) THEOREM. (Atiyah-Bott) Suppose that f:X - X is anisom£ 
55 
try of the 2t-dimensional compact oriented manifold X with only 
isolated fixed points x. Suppose further that X admits a Spin-
structure P and that f has a lifting f 'to this Spin-structure. 
The Spin-number Spin(f,x) is then given by the expression 
Spin(f,X) = L vex) 
where x ranges over the fixed points of f and 
vex) ± (i/2)t 1r~=lt cosec(8 .12) 
J= J 
where 8 1 ".8 t is a coherent system of angles for dfx ([ 1] ,p.473). 
We can prove now 
(2.4) THEOREM. Let X be a connected 2t-dimensional Spin manifold 
and G be a compact, connected Lie group acting differentiably on 
X with only two fixed points x and y. Assume that each subgroup 
of prime order H of G satisfies the following condit i on: 
i) Either F(H,X) = {x,y} or F(H~x,X) = F(H,y,X) 
Then the representations of the group on the tangent spaces at 
the fixed points are equivalent. 
Proof. We can think that G acts effectively and by isometries. 
From (2.2) we obtain a Lie group G1 (which is connected by cons-
truction [3], p.22) and a homomorphism h:G 1 --+ G, which is the 
identity or a double covering, such that G1 acts on P and indu-
ces via h the action of G on X. 
The action of G1 on X is by orientation preserving isometries 
and its action on P commutes with the action of Spin(2t) from 
the right and is compatible with 1r. We can consid,er then the 
Spinor-index of the action of G1 on X and as a consequence of 
(2.1) we have Spin(G 1,X) = O. 
Since we want to prove the equivalence of the representations 
and the elements of prime order form a dense set in G, it is 
enough to prove the equivalence of the restrictions of the re-
presentations to each cyclic subgroup of ,prime order p. Let H 
be a cyclic subgroup of prime order p;2 of G with generator f 
and assume that F(H,X) = {x,y}. 
Put Hl • h",l(H), then h-l(f) .. {f1,f2} (we could have one or two 
elements inthis set). Take f l , the~ Sign{fl,X) • 0 and we can 
apply (2.3) to obtain a condition on the coherent systems of 
angles of df and df . This condition will suffice to prove the x y 
equivalence of the representations. 
56 
Let us identify H with Z and f with the preferred generator. The 
p 
expression of vex) in (2.3) can be written as 
(2.5) v (x) = ± w J . J J
. = t [ exp (i8 . /2) - exp ( - i8 . /2) ] 
j=l (1 - exp i8
j
)(1 - expC -i8 j ))' 
Note that exp(i8 . ) ~ 1 since x is isoiated fixed point of f. 
J 
Let us put now hk exp(-i8 k/2). We have 
(2.6) vex) =± w~:~[h~h~ 1] 
Now the condition Sign(f1.X) = 0 gives vex) + v(y) 
turn implies 1 v (y) 12. 1 v (x) 1- 2 = 1. 
o which in 
In Iv(x)1 2 we have all the eigenvalues of dfx ' therefore 
(2.7) 
where aX i .s the number of eigenvalues of df which are equa;t to 
6 X 
exp(iw/p) ) and Z* is the set of congruence classes mo-
p 
dulo p which are prime to p. 
We have then 
(2.8) 
Put now a 
6 
(2.9) 
aX and wri te 
6 
sa -a 
w h S (h 26 _ 1) 6 
6e:Z* 
where q ~ 
6e:Z* 
p 
p 
sa 
S 
Clearly q E 0 (mod p). then (2.9) can be written as 
(2.10) 
(2.11) 
w 
6e:Z* 
p 
= -a 
6 
-a 
(h 26 _ 1) 6 = 1 
and wr.i te 
Since Ib 
s 
(2.12) 
o we have 
1r 
se;Z* 
p 
57 
Now by a theorem of Kummer ([ 1], p.477) we have b s 
and then a = o. 
0, s E Z* 
P 
s 
We have proved then the equivalence of the restrictions of the 
representations to H in the case F(H,X) = {x,y}. If this is 
not the case then F(H,x,X) = F(H,y,X) and in this situation the 
equivalence is obvious. The theorem is then proved. 
(2.13) REMARK. If the group G is a torus condition (i) of (2.4) 
can be replaced by 
i) Either dim F(H,X) = 0 or some of the subspaces F(H,x,X), 
F(H,y,X) has . positive dimension. (dim F(H,X) means the maximum 
of the dimensions of the components). 
If the action of G is semifree, that is free outside the fixed 
point set and F(G,X) = {x,y}, the theorem can be written as 
(2.14) THEOREM. Let X be a aonneated 2t~dimensional Spin mani-
fold. Let G be a aompaat aonneated Lie group aating differen-
tiabZy on X, with only two fixed points and semifreely. Then 
the two representations of G at the fixed points are equiva-
lent. 
REFERENCES 
[1] M.F.ATIYAH and R.BOTT, The Le6~ehetz 6~xed-po~nt theo~em 
6o~ eLL~pt~e eompLexe~ II, Ann. of Math. 88, 451-491. 
[2] M.F.ATIYAH and I.M.SINGER, The ~ndex 06 elL~pt~e ope~ato~4 
111, Ann. of Math. 87, 546-604. 
[3] M.F.ATIYAH and F.HIRZEBRUCH, Sp~n-man~6oLd4 and g~oup ae-
t~on4. Essays on Topology and Related Topics. Memoires de-
dies a G. de Rham. Springer Verlag 1970, 18-28. 
Recibido en agosto de 1974 
I.M.A.F. Universidad de Cordoba 
Cordoba, Argentina 
INDICE 
Volumen 27, Numero 1, 1974 
An elementary proof of the Jordan canonical form 
Enzo R. Gentile ..... 0'0.......................... 1 
Actions on a graph 
Antonio Diego 
An elementary proof of the structure theorem of finitely 
generated modules over a principal domain 
5 
Enzo R. Gentile ..................... '. . . . . . . . . . . 10 
Sequential estimation of a truncation parameter 
Eduardo W. de Weerth ........................... 13 
EI grupo de Witt de ciertas clases de anillos de enteros 
1. Kaplan y H. H. O'Brien .................. . .. . .. 27 
On the partitions of an integer 
Raul A. Chiappa ....................... '.......... 33 
Curvas y cuaterniones 
Luis A. Santal6 ...... j........................... 41 
Actions of compact, connected Lie groups on Spin manifolds 
Cristian U. Sanchez ............................ .. 53 
01 
Reg. Nac. de la Prop. 
Int. NQ 1.044.738 
o.S 
f~ 
~~ 
0" «: 
C; 
~o 
I'I~ Q)" 0;:1 
olrll 
[(it-> 
0 
TARIFA REDUCIDA 
CONCESION N9 9120 
FRANQUEO PAGADO 
CONCESION N9 3626 
AUSTRAL IMPRESOS 
VILLARINO 739 - B. B. 
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