An Algo i hm o Compu ing Cocyclic Ma ices
De eloped o e Some Semidi ec P oduc s
V. ´
Al a ez, J.A. A ma io, M.D. F au, and P. Real
Dp o. Ma em´a ica Aplicada I, Uni e sidad de Se illa, A da. Reina Me cedes s/n
41012 Se illa, Spain,
{ al a ez,a ma io,md au, eal}@us.es
Abs ac . An algo i hm o calcula ing a se o gene a o s o ep e-
sen a i e 2-cocycles on semidi ec p oduc o fini e abelian g oups is
cons uc ed, in ligh o he heo y o e cocyclic ma ices de eloped by
Ho adam and de Launey in [7,8]. The me hod in ol es some homolog-
ical pe u ba ion echniques [3,1], in he homological co esponden o
he wo k which G abmeie and Lambe desc ibed in [12] om he iew-
poin o cohomology. Examples o explici compu a ions o e all dihed al
g oups D4 a e gi en, wi h aid o Ma hema ica.
1 In oduc ion
Le Gbe a g oup, Ua i ial G-module. Func ions ψ:G×G→Uwhich sa is y
ψ(a, b)ψ(ab, c)=ψ(b, c)ψ(a, bc), a, b, c ∈Ga e called 2-cocycles [19]. A cocycle
is a cobounda y δα i i is de i ed om a se mapping α:G→Uha ing α(1)=1
by δα(a, b)=α(a)−1α(b)−1α(ab).Fo each Gand U, he se o cocycles o ms
an abelian g oup Z2(G, U) unde poin wise mul iplica ion, and he cobounda ies
o m a subg oup B2(G, U). Two cocycles ψand ψa e cohomologous i he e
exis s a cobounda y δα such ha ψ=ψ·δα. Cohomology is an equi alence
ela ion and he cohomology class o ψis deno ed [ψ]. I ollows ha he quo-
ien g oup Z2(G, U)/B2(G, U) consis ing o he cohomology classes, o ms an
abelian g oup H2(G, U), which is known as he second cohomology g oup o G
wi h coefficien s in U. Fo each n≥0 one may define he cocycle analogous
in dimension n(n-cocycle). In spi e o he impo an ole played by cocycles in
Algeb aic Topology, Rep esen a ion Theo y and Quan um Sys ems, he p oblem
o explici ly de e mining a ull ep esen a i e se o n-cocycles o gi en Gand
Udoes no appea o ha e been adi ionally s udied by cohomologis s, a leas ,
ill he las decade.
A 2-cocycle ψis na u ally displayed as a cocyclic ma ix (associa ed o ψ,
de eloped o e G); ha is, a |G|×|G|squa e ma ix whose ows and columns
a e indexed by he elemen s o G(unde some fixed o de ing) and whose en y
All au ho s a e pa ially suppo ed by he PAICYT esea ch p ojec FQM–296 om
Jun a de Andaluc´ıa and he DGESIC esea ch p ojec PB98–1621–C02–02 om
Educa ion and Science Minis y (Spain).
S. Boz a¸s and I.E. Shpa linski (Eds.): AAECC-14, LNCS 2227, pp. 287–296, 2001.
c
Sp inge -Ve lag Be lin Heidelbe g 2001
288 V. ´
Al a ez e al.
in posi ion (g,h)isψ(g,h). This no ion was ui ully used by Ho adam and
de Launey [6,7,15] p o ing some in e es ing connec ions be ween combina o ial
design heo y and 2-cocycles, as well as connec ions be ween coding heo y and
2-cocycles. I is also appa en ha cocyclic ma ices, associa ed wi h cocycles
wi h coefficien s in K2={−1,1}, accoun o la ge classes o so-called Hadama d
ma ices [8], and may consequen ly p o ide an uni o m app oach o he amous
Hadama d conjec u e.
These ac s ha e yield ha o e he pas decade conside able effo has
been de o ed o compu a ions o cocycles and cocyclic ma ices. Using clas-
sical me hods in ol ing he Uni e sal Coefficien Theo em, Schu mul iplie s,
infla ion and ansg ession, wo algo i hms o finding 2-cocycles ep esen ing
2-dimensional cohomology classes can be wo ked ou . The fi s one [7,8] applies
o an abelian g oup Gand he second [10] o e g oups G o which he wo d
p oblem is sol able.
Ho adam and de Launey’s me hod is based on an explici e sion o he well-
known Uni e sal Coefficien Theo em, which p o ides a decomposi ion o he
second cohomology g oup in o he di ec sum o wo summands,
H2(G, U)∼
=Ex (G/[G, G],U)⊕Hom(H2(G),U).
These connec ions make possible he ansla ion o cocyclic de elopmen on o a
(co)homological amewo k.
This link becomes s onge no ing he “Ba cons uc ion” [19] ela ed o G.
I is a DG-module, which consis s o he Z–modules
M0(G)=Z,M
m(G)=<[g1,...,g
m]: gi∈G, 1≤i≤m>,
and diffe en ial ∂,
∂1([g1])=0,∂
m+1([g1,...,g
m+1])=(−1)m+1([g1,...g
m])+
+([g2,...,g
m+1]) +
m

i=1
(−1)i([g1,...,g
igi+1,...,g
m+1]).
The quo ien Ke (∂m)/Im(∂m+1) is known o be he m h in eg al homology
g oup o G,Hm(G). Le R2(G) deno e he quo ien M2(G)/Im(∂3)⊇H2(G).
Taking in o accoun wha ∂3means, i is eadily checked ha he map
φ:Z2(G, K2)→Hom(R2(G),K2)
h→ φ(h)
such ha
φ(h)

(a,b)∈G×G
λ(a,b)(a, b)+Im(∂2)
=
(a,b)∈G×G
λ(a,b)h[a, b]
defines an isomo phism be ween he se o 2-cocycles and Hom(R2(G),K2) [7].
Cocyclic Ma ices o e Semidi ec P oduc s 289
The p oblem o compu ing a se o gene a o s o 2-cocycles hence ansla es
o he p oblem o de e mining a se o cobounda y,symme ic and commu a o
gene a o s, such ha
Z2(G, K2)∼
=B2(G, K2)⊕Ex Z(G/[G, G],K2)⊕Hom(H2(G),K2).
A minimal se o symme ic gene a o s may be calcula ed om a p ima y
in a ian decomposi ion o G/[G, G]∼
=H1(G), as a K onecke p oduc o back
negacyclic ma ices [7]. A minimal se o cobounda y gene a o s is de i ed om
he mul iplica ion able o Gby means o linea algeb a manipula ions. Bu
i is a om clea how o ge a minimal se o commu a o gene a o s, in
gene al. One should y o compu e he second homology g oup o Gby means
o (M2,∂
2,∂
3). Indeed, ∂2is no needed o fini e g oups G, since H2(G)isa
di ec sum o fini e cyclic g oups as i is he case. This p ocedu e is no sui able
in p ac ice, since ma ices in ol ed a e la ge in mos cases.
On he o he hand, Flanne y calcula es hese summands as he images o ce -
ain embeddings which a e complemen a y, called infla ion and ansg ession.
Calcula ion o ep esen a i e 2-cocycles associa ed o Ex (G/[G, G],U) (infla-
ion) is canonical. Howe e , calcula ion o a complemen o he image by he
embeddings o infla ion in H2(G, U) as he image o ansg ession is no canon-
ical, anyway. As a ma e o ac , i depends on he choice o a Schu comple-
men . This is a po en ial sou ce o difficul ies in compu a ion o ep esen a i e
2-cocycles associa ed wi h elemen s o Hom(H2(G),U). This me hod has al eady
been implemen ed in [11], using he symbolic compu a ional sys em MAGMA.
Using a a diffe en app oach, G abmeie and Lambe p esen in [12] al e na e
me hods o calcula ing ep esen a i e 2-cocycles o all fini e p–g oups om
he poin o iew o Homological Pe u ba ion Theo y [13,14,20]. The compu e
algeb a sys em Axiom has been used in o de o make calcula ions in p ac ice.
He e we p esen a me hod o explici ly de e mining a ull se o ep esen a-
i e 2-cocycles o he elemen s o he second cohomology g oup H2(G, Z) whe e
Gis Z ×χZs. All gene al s a emen s gi en in his pape a e applicable o any
semidi ec p oduc o fini e abelian g oups, bu o simplici y in he exposi ion,
o his class, only he case Z ×χZswill be p esen ed.
Ou me hod could be seen as a mix u e o bo h he algo i hms gi en by
Flanne y in [10] and G abmeie –Lambe in [12]. Indeed, we compu e ep esen a-
i e 2-cocycles p oceeding om Hom(H2(Z ×χZs),K2). This al e na e me hod
is based on some Homological Pe u ba ion echniques de eloped in he wo k
o au ho s [3,1] on he de e mina ion o “homological models”( hose diffe en ial
g aded modules hG wi h Hn(G)=Hn(hG), see [4] o ins ance), o semidi ec
p oduc s o fini e abelian g oups wi h g oup ac ion. The algo i hm is s aigh -
o wa d enough o be p og ammed in any compu e algeb a sys em, as we ha e
done in Ma hema ica[2].
The main s eps a e o define unc ions and F:M2(Z ×χZs)→V2and
di:Vi→Vi−1, whe e Via e ce ain “pe u bed” simple algeb as. These will
be defined in such a way ha o any ep esen a i e 2-cycle zin he quo ien
ke d2/Im d3, he ele a ion o z h ough Fwill define a ep esen a i e 2-cocycle.
This is he homology analogous o he wo k o G abmeie –Lambe in [12].
290 V. ´
Al a ez e al.
I should be no ed ha explici o mulae o ep esen a i e 2-cocycles o
H2(Z ×χZs,Z2) a e gi en in [21]. The app oach explained in his pape co e s
he mo e gene al case o any semidi ec p oduc o fini e abelian g oups.
Simila algo i hms may be conside ed o each many o he se ings, p og ess-
ing om any fini e g oup wi h known homological model.
2 The Algo i hm
Le Z ×χZsbe a semidi ec p oduc , χa g oup ac ion such ha
(a1,b
1)·(a2,b
2)=(a1+χ(b1,a
2),b
1+b2),a
1,a
2∈Z ,b
1,b
2∈Zs.
Le conside he ollowing auxilia y se s
V2=Z[x2,xy,y2],V
3=Z[x3,x
2y,xy2,y3],
B2={[n, m]⊗[]: 1≤n,m< }∪{[n]⊗[m]: 1≤n< ,1≤m<s}∪
∪{[]⊗[n, m]: 1≤n,m<s},
B3={[n, m, k]⊗[]: 1≤n, m, k < }∪{[n, m]⊗[k]: 1≤n,m< ,1≤k<s}∪
∪{[n]⊗[m, k]: 1≤n< ,1≤m, k < s}∪{[]⊗[n, m, k]: 1≤n, m, k < s}.
We will define Z–linea unc ions g3:V3→B3, i:Bi→Vi o i=2,3,
φ2:B2→B3,ρ3:B3→B2,d3:V3→V2and ∞:B2→V2. Le
g3(x3) = ([1,1,1] + ···+[1, −1,1]) ⊗[],
g3(x2y) = ([1,1] + ···+[1, −1]) ⊗[1],
g3(xy2) = [1] ⊗([1,1] + ···+[1,s−1]),
g3(y3)=[]⊗([1,1,1] + ···+[1, −1,1])
2([n, m]⊗[])=x2,i n+m≥ ,
2([n]⊗[m])=(nm)xy,
2([ ] ⊗[n, m]) = y2,i n+m≥s,
3([n, m, k]⊗[])=kx
3,i n+m≥ ,
3([n, m]⊗[k]) = kx
2y,i n+m≥ ,
3([n]⊗[m, k]) = nxy2,i m+k≥s,
3([ ] ⊗[n, m, k]) = ky
3,i n+m≥s,
φ2([n, m]⊗[])=−([1,1,m]+···[1,n−1,m]) ⊗[]
φ2([n]⊗[m]) = −([1,1] + ···[1,n−1]) ⊗[m]+[n]⊗([1,1] + ···[1,m−1]),
φ2([ ] ⊗[n, m]) = −[]⊗([1,1,m]+···[1,n−1,m])
ρ3([n, m]⊗[k])=[χ(k,n),χ(k,m)] ⊗[]−[n, m]⊗[],
ρ3([n]⊗[m, k])=[n]⊗[k]−[χ(m, n)] ⊗[k],
D3(x2y)= xy,
D3(xy2)=−sxy.
Cocyclic Ma ices o e Semidi ec P oduc s 291
These mo phisms a e unde s ood o be ze o o he wise. Le define
d3=D3+ 2ρ3g3− 2ρ3φ2ρ3g3+ 2(ρ3φ2)2ρ3g3−···,
and ∞:B2→V2,
∞= 2− 2ρ3φ2+ 2(ρ3φ2)2−···
Geome ic se ies o hese ypes con e ge o define a map, as i is p o ed in he
mo e gene al se ing o gene alized semidi ec p oduc s o fini e abelian g oups
in [1]. The ac is ha ρ∗dec eases he dimension on he second componen , and
φ∗ei he inc emen s he dimension only on he fi s componen o dec eases he
alue o he elemen in he second componen . Hence he composi ion φi−1ρi
becomes nilpo en .
No ice ha he se s Bidefined abo e consis o he p oduc s
Bi=
0≤j≤i
(Mj(Z )⊗Mi−j(Zs)).
The e is a connec ing map F2:M2(Z ×χZs)→B2, so ha
F2[(a1,b
1),(a2,b
2)]=[]⊗[b2,b
1]+2[χ(b2,a
2)]⊗[b1]+2[χ(b2,a
2),χ(b2b1,a
1)]⊗[]−
−[χ(b2b1b2,a
2),χ(b2b1b2b1,a
1)] ⊗[]−[χ(b2b2,a
2)] ⊗[b1].
Theo em 1. Assume he no a ion abo e.
1. H2(Z ×χZs)=H2(V2),which is compu ed om d3.
2. The map F= ∞◦F2:M2(Z ×χZs)→V2induces an isomo phism in
homology, such ha o any z∈H2(V2) he ele a ion o z h ough Fdefines
a cocyclic ma ix o e Z ×χZs.
In [1] he au ho s find a homological model o semidi ec p oduc s o fini e
abelian g oups. In pa icula , a ending o he g oups Z ×χZs, i is p o ed ha
H2(M2(Z ×χZs)=H2(V2). Mo eo e Fis shown o induce an isomo phism in
homology.
Ne e heless he o mula o Fis no explici ly gi en he e, since i is compli-
ca ed o gi e an explici o mula o ∞in he gene al case o semidi ec p oduc s
o g oups.
I is a ema kable ac ha o e e y fini e g oup G,H2(G) is a fini e abelian
g oup [5]. This way, i is only needed d3in o de o compu e H2(V2) by means
o Veblen’s algo i hm [22].
This p ocess consis s in calcula ing he in ege Smi h no mal o m Do
he ma ix M ep esen ing d3wi h ega ds o basis B={x3,x
2y,xy,y3}and
B={x2,xy,y2}.
Le U={u1,u
2,u
3,u
4}and V={ 1,
2,
3}define hese change basis, such
ha DU,V=PMB,BQ, o app op ia ed change basis ma ices Pand Q.
Now we explain wha we mean wi h “ele a e z h ough F”.

292 V. ´
Al a ez e al.
We wan o de e mine all cocyclic ma ices o e Z ×χZs. Tha is, all ep-
esen a i e 2-cocycles o Z ×χZs. Thus i suffices o calcula e which xin
M2(Z ×χZs) a e shown o gi e non i ial homological in o ma ion in H2(V2).
Fo each gene a o zin H2(V2), he ele a ion o z h ough F ela es o he
se o elemen s in M2(Z ×χZs) which p ojec s on o zwi h 2-homological in-
o ma ion. This can be achie ed in wo single ele a ions: one om V2 o B2, he
o he om B2 o M2(Z ×χZs).
¿F om he heo em abo e, an algo i hm o calcula ing ep esen a i e 2-
cocycles may be de i ed in a s aigh o wa d manne .
No ice ha map Fshould be called he uni e sal 2–cochain, ollowing G ab-
meie –Lambe’s no a ion in [12].
Algo i hm 1 Inpu Da a: a semidi ec p oduc Z ×χZs.
S ep 1. Compu e d3:V3→V2, he diffe en ial o he homological model o
Z ×χZsin dimension 3.
S ep 2. Compu e H2(Z ×χZs)and ep esen a i e cycles om d3.
S ep 3. Ele a e he ep esen a i e cycles om H2(Z ×χZs) o M2(Z ×χZs)
ia F.
Ou pu Da a: Se o commu a o gene a o s o a basis o cocyclic ma ices
o e Z ×Zs.
I should be aken in o accoun ha S ep 2 o en equi es o compu e he
Smi h no mal o m o he ma ix co esponding o d3, which is always o size
4×3, independen ly o indexes and so he ac o s. This is he undamen al
imp o emen in he calculus o he commu a o gene a o s, since he size o
ma ices which a ises om he complex (M∗,∂
∗) depends on he o de o he
g oup ( he ma ix co esponding o ope a o ∂3is o size ( s)3×( s)2 o he
semidi ec p oduc Z ×χZs).
I may be possible o ex end he Theo em 1 and i s associa ed algo i hm
o o he ce ain amilies o g oups, wi h homological models al eady known,
such as cen al ex ensions [18], fini ely gene a ed o sion ee nilpo en g oups
[16], me acyclic g oups [17] and many o he s. I is only needed o find explici
o mulae o he analogous o maps F2and F.
3 An Example: Dihed al G oups D2 ·2
In his sec ion we apply Algo i hm 1 in he pa icula case o dihed al g oups.
AMa hema ica p og am is used, which au ho s p o ide in [2].
I should be no ed ha dihed al g oups D ·2 o odd alues o do no p o ide
2-homological in o ma ion, since H2(D ·2) is known o be ze o in his case.
Le D2 ·2={(0,0),(1,0),...,(2 −1,0),(1,1),...,(2 −1,1)},
χ(0,n)=n, χ(1,n)=2 −n, ∀n∈Z2 .
Cocyclic Ma ices o e Semidi ec P oduc s 293
An explici o mula o Fcan be wo ked ou o hese g oups, so ha i we
define λ:Zk×Zk→Z2,k≥2, as λ[x, y]=1i x+y≥kand 0 o he wise, i is
eadily checked ha
F[(a1,b
1),(a2,b
2)] = b1b2y2+2b1χ(b2,a
2)xy +2b1(χ(b2,a
2)−1)x2+
+2λ[χ(b2,a
2),χ(b2b1,a
1)]x2−λ[χ(b1,a
2),a
1]x2−a2b1xy −b1(a2−1)x2.
Le conside he cases =1,D2·2={(0,0),(1,0),(0,1),(1,1)},
=2,D4·2={(0,0),(1,0),(2,0),(3,0),(0,1),(1,1),(2,1),(3,1)},
and =6,D12·2={(0,0),(1,0),...,(11,0),(0,1),(1,1),...,(11,1)}.
S ep 1. Compu e d3.
d(V3) =1 =2 =6
x30 0 0
x2y2xy 2x2+4xy 10x2+12xy
xy2−2xy −2x2−4xy −10x2−12xy
y30 0 0
S ep 2. Compu e H2(D2 ·2) and ep esen a i e cycles om d3.
In o de o compu e H2(D2 ·2) in he cases =1,2,6, i is use ul
o calcula e he Smi h no mal o m D =P M Q o he ma ix M
associa ed o d3, wi h basis change ma ices P and Q , espec i ely.
In hese cases,
=1 =2 =6
D 



200
000
000
000








200
000
000
000








200
000
000
000




Q 

010
100
001



1−20
010
001



−1−60
150
001


Hence, H2(D2 ·2)=Z2 o =1,2,6 and he ep esen a i e cycle is he
fi s elemen in he new basis Uo Z[V2].
In o de o ansla e o he basis Bo Z[V2] he homological in o ma-
ion which H2(D2 ·2) p o ides, i suffices o selec he odd en ies o
each o he columns o Q co esponding o each ep esen a i e cycle in
he basis U( ha is, o selec which elemen s o Z[V2] wi h ega ds o
basis Bha e an odd en y in he posi ion co esponding o a ep esen-
a i e cycle wi h coo dina es in basis U). The homological in o ma ion
is concen a ed in elemen s wi h coo dina es (−,n,−)B o odd alues
o nin he case = 1, in elemen s (n, −,−)B o odd alues o nin he
case = 2, and in elemen s (n, m, −)B o n, m o dis inc pa i y in he
case =6.
294 V. ´
Al a ez e al.
S ep 3. Ele a e he ep esen a i e cycles om H2(D2 ·2) oM2(D2 ·2) ia F.
I suffices o de ec which elemen s o M2(D2 ·2) a e ca ied ou ia F o
elemen s (−,n,−)B o odd n( = 1), (n, −,−)B o odd n( = 2), and
(n, m, −)B o n, m o dis inc pa i y ( = 6). These elemen s indica e
he posi ions in he |D2 ·2|×|D2 ·2|commu a o cocyclic gene a o
ma ix which a e no i ial.
In he case = 1, we ob ain he ollowing elemen s:
[(0,1),(1,0)],[(1,1),(1,0)],[(0,1),(1,1)],[(1,1),(1,1)].
Fo =2,
[(1,0),(3,0)],[(1,0),(3,1)],[(2,0),(2,0)],[(2,0),(3,0)],[(2,0),(2,1)],
[(2,0),(3,1)],[(3,0),(1,0)],[(3,0),(2,0)],[(3,0),(3,0)],[(3,0),(0,1)],
[(3,0),(2,1)],[(3,0),(3,1)],[(0,1),(2,0)],[(0,1),(2,1)],[(1,1),(1,0)],
[(1,1),(2,0)],[(1,1),(1,1)],[(1,1),(3,1)],[(2,1),(1,0)],[(2,1),(1,1)],
[(3,1),(1,0)],[(3,1),(3,0)],[(3,1),(1,1)],[(3,1),(3,1)].
In he case = 6, he elemen s which a e ca ied ou ia F o elemen s
(n, m, −)B o n, m o dis inc pa i y a e hose [(a1,b
1),(a2,b
2)] such
ha 


b1=0,a1+a2>11;
o
b1=1,a1<a
2.
Ou pu da a: se o commu a o gene a o s o a basis o cocyclic ma ices
o e D2 ·2, =1,2,6. Assuming K2= 1, we ob ain
=1 =2 =6
A1A1
B1B1A2A2
B2B2A6A6
B6B6
whe e
A1=11
11,B
1=1K
1K,A
2=



1111
11 1K
11KK
1KKK




,B
2=



11K1
1KK 1
1K11
1K1K




,
A6=







11··· 11
11··· 1K
.
.
..
.
..
.
..
.
.
11···KK
1K···KK







,B
6=







1K···KK
11···KK
.
.
..
.
..
.
..
.
.
11··· 1K
11··· 11







.
No e ha A6is usually called back negacyclic.
In gene al, i may be p o ed ha o >2 he compu a ion o H2(D2 ·2)
educes o he ma ices
D =



200
000
000
000




and Q =

−1− 0
1 −10
001


Cocyclic Ma ices o e Semidi ec P oduc s 295
so ha H2(D2 ·2)=Z2and he homological in o ma ion is concen a ed in
elemen s wi h coo dina es (n, m, −)B o n, m o dis inc pa i y.
Hence, he se o commu a o gene a o s o a basis o cocyclic ma ices o e
D2 ·2 educes o A A
B B ,whe e A is he co espondan back negacyclic
ma ix and B consis s in he ma ix whose ows a e he ones o A displayed in
e e se o de .
I should be no ed ha he cocyclic ma ices o e dihed al g oups ha e al-
eady been ound om Flanne y’s echniques in [10].
Rema k 1. The case = 2 is also s udied in [7], whe e he commu a o gene a o
is said o be 











11 1 1 111 1
11 1BB11 1
11BBB11B
1BBBB1BB
11 1 1 111 1
1BBBB1BB
11BBB11B
11 1BB11 1












wi h B2=1.
Bo h ma ices diffe in he (Hadama d) p oduc o a cobounda y gene a o
Cand a symme ic gene a o S, which a e
C=












11111111
1A1AAA11
1111A1A1
1A1AA11A
11A111A1
1A11A111
1AAAAA1A
111AA111












,S=












11111111
1D1D1D1D
11111111
1D1D1D1D
11111111
1D1D1D1D
11111111
1D1D1D1D












,
wi h A=D=−1.
The ma ix Ca ises om any o he se map αk:D4·2→K2,k∈{1,−1},
α(0,0)=1,α(1,0) = −1,α(2,0) = −1,α(3,0)=1,
α(0,1) = k, α(1,1) = k, α(2,1) = k, α(3,1) = −k.
Re e ences
1. V. ´
Al a ez, J.A. A ma io, M.D. F au and P. Real. Homology o semidi ec p oduc s
o fini e abelian g oups wi h g oup ac ion. P ep in o Dp o. Ma em´a ica Aplicada
I, Uni e si y o Se ille (Spain, 2001).