A subset A of an abelian group G is a Bh[g] set on G if every element of G
can be written at most g ways as sum of h elements in A. In this work we
present three constructions of Bh[g] sets on product of groups.Comment: 10 page

Ruiz, Diego

Trujillo, Carlos

English

arXiv

A subset A of an abelian group G is a Bh[g] set on G if the elements of G can be written in at most g ways as sum of h elements of A. Given any field F, this work presents constructions of Bh[g] sets on the abelian groups (Fh, +) (Zd, +), and (Zm1 × … × Zmd, +), for d ≥ 2, h ≥ 2, and g ≥ 1

Ruíz, Diego

Universidad Nacional De Colombia - Repositorio Institucional UN

Revista Colombiana de MatemáticasVolumen 50(2016)2, páginas 165-174Construction of Bh[g] sets in product ofgroupsConstrucción de conjuntos Bh[g] en producto de gruposDiego Ruiz1,B, Carlos Trujillo11Universidad del Cauca, Popayán, ColombiaAbstract. A subset A of an abelian group G is a Bh[g] set on G if the elementsof G can be written in at most g ways as sum of h elements of A. Given anyfield F, this work presents constructions of Bh[g] sets on the abelian groups(Fh,+),(Zd,+), and (Zm1 × · · · × Zmd ,+), for d ≥ 2, h ≥ 2, and g ≥ 1.Key words and phrases. Sidon sets, Bh[g] sets.2010 Mathematics Subject Classification. 11B50, 11B75.Resumen. Un subconjunto A de un grupo abeliano G es un conjunto Bh[g]sobre G si todo elemento de G puede escribirse en a lo sumo de g for-mas como la suma de h elementos de A. En este trabajo se presentan con-strucciones de conjuntos Bh[g] sobre los grupos abelianos(Fh,+),(Zd,+), y(Zm1 × · · · × Zmd ,+), para d ≥ 2, h ≥ 2, y g ≥ 1, con F cualquier campo.Palabras y frases clave. Conjuntos de Sidon, Conjuntos Bh.1. IntroductionLet g and h denote positive integers with h ≥ 2. Let G be an abelian additivegroup denoted by (G,+). The set A = {a1, . . . , ak} ⊆ G is a Bh[g] set on G ifevery element of G can be written in at most g ways as sum of h elements inA, that is, if given x ∈ G, the solutions of the equation x = a1 + · · ·+ ah, witha1, . . . , ah ∈ A, are at most g (up to rearrangement of summands). If g = 1, Ais a Bh set, while if g = 1 and h = 2, A is a Sidon set.Let Fh(G, g) denote the largest cardinality of a Bh[g] on G. If g = 1 wewrite Fh(G). Furthermore, if G is the direct product of d ≥ 2 abelian groupsand A is a Bh[g] set on G, sometimes we say that A is a d−dimensional Bh[g]set on G. For N ∈ N, let [0, N −1] := {0, 1, . . . , N −1}. If Zd denotes the set of165166 DIEGO RUIZ & CARLOS TRUJILLOall d−tuples of integer numbers and [0, N − 1]d denotes the cartesian productof [0, N − 1] with itself d times, we defineF dh (N, g) := max{|A| : A ⊆ [0, N − 1]d, A ∈ Bh[g]}.The main problem on Bh[g] sets consists on establishing the largest cardinalityof a Bh[g] set on a finite group G. With analytical constructions it is possible tocharacterize lower bounds for Fh(G, g), while using counting and combinatorialtechniques, it is possible to characterize upper bounds. In this work we focus onconstructions to obtain known lower bounds for Fh(G, g) on particular groupsG ((Fh,+),(Zd,+), and (Zm1 × · · · × Zmd ,+) for any field F and d ≥ 2, h ≥ 2,g ≥ 1), while other works are focused on upper bounds [1], [4], [11].Different works have introduced constructions of Bh[g] sets for particularvalues of h, and g. On (Z,+), the most obvious construction of Sidon sets isgiven by Mian–Chowla using the greedy algorithm [2]. This result is generalizedby O’Bryant for any h ≥ 2 and any g ≥ 1 in [10].Other constructions of Bh sets are due to Rusza, Bose, Singer, and Erdös& Turán. Rusza constructs a Sidon set on the group(Z(p2−p),+)for p prime.Bose’s construction initially consider h = 2 but could be generalized for anyh ≥ 2 and any prime power q on the group(Zqh−1,+). Similarly to Bose, Singerconstructs a Bh set with q + 1 elements on(Z(qh+1−1)/(q−1),+). Actually thisconstruction can be established using Bose’s construction [8]. Finally, based onquadratic residues modulo a fixed prime p, Erdös & Turán construct Sidon setson (Z,+) [10].In dimension d = 2 some constructions are due to Welch, Lempel, Golomb[6], Trujillo [12], and C. Gómez & Trujillo [8]. Welch constructs Sidon setswith p− 1 elements on the groups (Zp−1 × Zp,+), (Zp × Zp−1,+), generalizedin [7] to the groups (Zq−1 × Fq,+) and (Fq × Zq−1,+), respectively, where Fqis the finite field with q elements. Golomb constructs Sidon sets with q − 2elements on the group (Zq−1 × Zq−1,+) (Lempel’s construction is a particularcase of Golomb). Trujillo in [12] presents an algorithm to construct Sidon setson (Z× Z,+) from a given Sidon set on (Z,+). Finally, C. Gómez & Trujilloconstruct Bh sets on(Zp × Zph−1−1,+)[8].In higher dimensions, Cilleruelo in [4] presents a way of mapping Sidon setsin N to Sidon sets in Nd for d ≥ 2, from which is possible to obtain a relationbetween the functions Fh(Nd) and F dh (N).In this work we present constructions of d−dimensional Bh[g] sets (d ≥ 2)on special abelian groups. The first construction uses the elementary symmet-ric polynomials and the Newton’s identities to generalize a construction doneinitially for d = 2 [3]. In the second construction we generalize Trujillos’s algo-rithm given in [12] to any dimension d and all h ≥ 2, g ≥ 1, obtaining lowerbounds for F dh (N, g) from a known lower bounds for Fh(Nd, g). Finally, usingVolumen 50, Número 2, Año 2016CONSTRUCTION OF BH [G] SETS IN PRODUCT OF GROUPS 167a homomorphism between abelian groups, we construct d−dimensional Bh[g′]sets from d−dimensional Bh[g] sets, with g a divisor of g′.The remainder of this work is organized as follows: For any finite field F, Sec-tion 2 describes a construction of Bh sets on(Fh,+), where Fh denotes the setof all h−tuples of elements of F. Section 3 presents a construction of Bh[g] setson(Zd,+), and in Section 4 we construct Bh[g] sets on (Zm1 × · · · × Zmd ,+).Furthermore, we present a generalization of a Golomb Costas array construc-tion. Finally, Section 5 describes the concluding remarks of this work.2. Construction of Bh sets on(Fh,+)Let p be a prime number. Note that A := {(x, x2) : x ∈ Zp} is a B2 set on(Zp × Zp,+) [3]. In this section we generalize this construction using h−tuples(h > 2). First we introduce the following notations and definitions.Let n be a positive integer. The elementary symmetric polynomials in thevariables x1, . . . , xn, written by σk(x1, . . . , xn) for k = 1, . . . , n, is defined asσk(x1, . . . , xn) :=∑1≤j1<···<jk≤nxj1 · · ·xjn .If k = 0 we consider σ0(x1, . . . , xn) = 1. For n = 3 we haveσ0(x1, x2, x3) = 1,σ1(x1, x2, x3) = x1 + x2 + x3,σ2(x1, x2, x3) = x1x2 + x1x3 + x2x3,σ3(x1, x2, x3) = x1x2x3.Note that the elementary symmetric polynomials appear in the expansion of alinear factorization of a monic polynomialn∏j=1(λ− xj) =n∑k=0(−1)kσk(x1, . . . , xn)λn−k.Note also that if pk(x1, . . . , xn) = xk1 + · · · + xkn, the Newton’s identities aregiven bykσk(x1, . . . , xn) =k∑i=1(−1)i−1σk−i(x1, . . . , xn)pi(x1, . . . , xn), (1)for each 1 ≤ k ≤ n and for an arbitrary number n of variables.Theorem 2.1. Let F be a field with characteristic zero or p > h. The setA := {(x, x2, . . . , xh) : x ∈ F},is a Bh set on(Fh,+).Revista Colombiana de Matemáticas168 DIEGO RUIZ & CARLOS TRUJILLOProof. Let s ∈ Fh. Suppose there exist two different representations of s assum of h elements of A as followss = (a1, . . . , ah1 ) + · · ·+ (ah, . . . , ahh) = (b1, . . . , bh1 ) + · · ·+ (bh, . . . , bhh),ai, bi ∈ F for i = 1, . . . , h. Note that for all k = 1, . . . , h,∑hi=1 aki =∑hi=1 bki .Because pk(a1, . . . , ah) =∑hi=1 aki and pk(b1, . . . , bh) =∑hi=1 bki , using (1)recursively we have σi(a1, . . . , ah) = σi(b1, . . . , bn), for all i = 1, . . . , h, that isa1 + · · ·+ ah = b1 + · · ·+ bh,a1a2 + · · ·+ ah−1ah = b1b2 + · · ·+ bh−1bh,· · ·a1 . . . ah = b1 . . . bh,which implies that the elements of the sets {a1, . . . , ah} and {b1, . . . , bh} areroots of the same polynomial q(x) on F[x], i.e.,q(x) = (x− a1) · · · (x− ah) = (x− b1) · · · (x− bh).That is, {a1, . . . , ah} = {b1, . . . , bh} (F[x] is a unique factorization domain).Thus, cannot be possible to have two different representations of s ∈ F as sumof h elements of Fh and A is a Bh set on(Fh,+). XConsider the case when F is the finite field Fq, with q = pn for some n ∈ Nand p prime. Note that the groups (Fpn ,+) and(Fnp ,+)are isomorphic, becauseif θ is a root of an irreducible polynomial of degree n over Fp in an extensionfield, the functionφ : Fpn → Fnpa0 + · · ·+ an−1θn−1 7→ (a0, . . . , an−1)(2)defines an isomorphism between them.Corollary 2.2. For all p > h prime and for all n ∈ N there exists a Bh setwith pn elements on(Zhnp ,+).Proof. It follows immediately from Theorem 2.1 and the isomorphism φ givenin (2). XWe illustrate these results in the following example.Example 2.3. Consider h = n = 2 and p = 3. Let p(x) = x2 + 1 be anirreducible polynomial on Z3. Suppose that θ is a root of p(x) in an extensionfield of Z3. The field with 9 elements is given byF9 = {a+ bθ : a, b ∈ Z3}= {0, 1, 2, θ, θ + 1, θ + 2, 2θ, 2θ + 1, 2θ + 2}.Volumen 50, Número 2, Año 2016CONSTRUCTION OF BH [G] SETS IN PRODUCT OF GROUPS 169Using Theorem 2.1 we know thatA ={(0, 0), (1, 1), (2, 1), (θ, 2), (θ + 1, 2θ), (θ + 2, θ),(2θ, 2), (2θ + 1, θ), (2θ + 2, 2θ)}is a Sidon set on (F9 × F9,+) =(F29,+). Furthermore, using Corollary 2.2 wehaveB ={(0, 0, 0, 0), (0, 1, 0, 1), (0, 2, 0, 1), (1, 0, 0, 2), (1, 1, 2, 0),(1, 2, 1, 0), (2, 0, 0, 2), (2, 1, 1, 0), (2, 2, 2, 0)}is a Sidon set on (Z3 × Z3 × Z3 × Z3,+) =(Z43,+).3. Construction of Bh[g] sets on(Zd,+)In this section we present a construction of Bh[g] sets for all h, g ≥ 2 on(Zd,+). This construction generalizes a construction introduced by Trujilloin [12] which allows to obtain Sidon sets on (Z× Z,+) from a Sidon set on(Z,+). Our generalization also allows to construct d−dimensional Bh[g] setsfor all h, g ≥ 2 and any dimension d, from which it is possible to determine away to map Bh[g] sets on (Z,+) into Bh[g] sets on(Zd,+).Let d,N be positive integers greater than 1. Let A denote a subset of Z+. Ifa ∈ A, [a]N = (nk, . . . , n1, n0)N represents the integer a = nkNk+· · ·+n1N+n0in base N notation, where k is a nonnegative integer and 0 ≤ nj ≤ N − 1, forj = 0, 1, . . . , k. We denote the set obtained from the representation of eachelement of A in base N as [A]N . Because every positive integer can be writtenuniquely in base N , then|A| = |[A]N |.Note that if A ⊆[0, Nd − 1], then [A]N ⊆ [0, N − 1]d.Theorem 3.1. If A is a Bh[g] set contained in [0, Nd − 1], then [A]N is aBh[g] set contained in [0, N − 1]d.Proof. Let s be a d−tuple in Zd obtained as sum of h elements in [A]N .Suppose there exist g + 1 representations of s as followss = [a1,1]N + · · ·+ [a1,h]N = · · · = [ag+1,1]N + · · ·+ [ag+1,h]N , (3)where ai,j ∈ A for all 1 ≤ i ≤ g + 1, 1 ≤ j ≤ h. Consider the representation ofeach ai,j ∈ A in base N as [ai,j ]N =(n(d−1,i,j), . . . , n(0,i,j)). Note that for any1 ≤ i ≤ g + 1[ai,1]N + · · ·+ [ai,h]N =(n(d−1,i,1), . . . , n(0,i,1))+ · · ·+(n(d−1,i,h), . . . , n(0,i,h))=(n(d−1,i,1) + · · ·+ n(d−1,i,h), . . . , n(0,i,1) + · · ·+ n(0,i,h)).Revista Colombiana de Matemáticas170 DIEGO RUIZ & CARLOS TRUJILLOFurthermore(n(d−1,i,1) + · · ·+n(d−1,i,h))Nd−1 + · · ·+(n(0,i,1) + · · ·+n(0,i,h)) = ai,1 + · · ·+ai,hwhich implies from (3) thata1,1 + · · ·+ a1,h = · · · = ag+1,1 + · · ·+ ag+1,h. (4)Because A is a Bh[g] set, using (4) we know there exist `,m with ` 6= m and1 ≤ `,m ≤ g + 1, such that{a`,1, . . . , a`,h} = {am,1, . . . , am,h}.Since representation in base N notation is unique we have{[a`,1]N , . . . , [a`,h]N} = {[am,1]N , . . . , [am,h]N},That is, it is not possible to have g+1 representations of s as sum of h elementsof A. Therefore [A]N is a Bh[g] set contained in [0, N − 1]d ⊂(Zd,+). XExample 3.2. Note that A = {1, 2, 7} is a Sidon set on (Z8,+). In [12] Trujilloconstructs a B2[2] set on (Z,+) as followsB := A ∪ (A+m) ∪ (A+ 3m) = {1, 2, 7, 9, 10, 15, 25, 26, 31},with m = 8. Because B ⊆ [0, 25 − 1], using Theorem 3.1 we have that[B]2 ={(0, 0, 0, 0, 1), (0, 0, 0, 1, 0), (0, 0, 1, 1, 1), (0, 1, 0, 0, 1), (0, 1, 0, 1, 0),(0, 1, 1, 1, 1), (1, 1, 0, 0, 1), (1, 1, 0, 1, 0), (1, 1, 1, 1, 1)}is a B2[2] set contained in [0, 1]5. Note also that B ⊆ [0, 62 − 1], so[B]6 = {(0, 1), (0, 2), (1, 1), (1, 3), (1, 4), (2, 3), (4, 1), (4, 2), (5, 1)}is a B2[2] set contained in [0, 5]2.4. Construction of Bh[g] sets on (Zm1 × · · · × Zmd ,+)This section extend a construction of Bh[g] sets given in [5] for h = 2 and d = 1,to all h ≥ 2 and any dimension d > 1. First, we introduce the following result.Lemma 4.1. Let G and G′ be two abelian groups and let φ : G → G′ definea homomorphism. If A is a Bh[g] set on G and |Ker(φ)| = g′, then φ(A) is aBh[gg′] set on φ(G), where gg′ denotes the product between g and g′.The proof is given in [9].Now, let m1, . . . ,md and g1, . . . , gd be positive integers. Using Lemma 4.1we have the following result.Volumen 50, Número 2, Año 2016CONSTRUCTION OF BH [G] SETS IN PRODUCT OF GROUPS 171Theorem 4.2. Let A be a Bh[g] set on (Zm1 × · · · × Zmd ,+). If g1, . . . , gd aredivisors of m1, . . . ,md, respectively, thenB :={(a1 modm1g1, . . . , ad modmdgd): (a1, . . . , ad) ∈ A}is a Bh[gg1 · · · gd] set on(Zm1g1× · · · × Zmdgd,+).Proof. Using notation used in Lemma 4.1, let G = (Zm1 × · · · × Zmd ,+) andG′ =(Zm1g1× · · · × Zmdgd,+)and define the homomorphism φ : G → G′ asφ(b1, . . . , bd) =(b1 modm1g1, . . . , bd modmdgd). We establish Ker(φ) as follows.Note that (b1, . . . , bn) ∈ Ker(φ) if and only if φ(b1, . . . , bn) = (0, . . . , 0), thatis, if (b1 modm1g1, . . . , bd modmdgd)= (0, . . . , 0).Note also that bi modmigi= 0 if and only if bi = kimigi, for ki ∈ [1, gi] and forall i = 1, . . . , d, which implies that bi modmigi= 0 in exactly gi values. Thus,|Ker(φ)| =∏di=1 gi. Finally, using Lemma 4.1 we have that B = φ(A) is aBh[gg1 · · · gd] set on(Zm1g1× · · · × Zmdgd,+). XGiven q a prime power and F a field, to illustrate Theorem 4.2 we present aconstruction of Sidon sets on (Zq−1 × Zq−1,+), which is based on the discretelogarithm1 on Fq.Proposition 4.3. Let q = pn a prime power. If α, β are primitive elements ofF∗q and a ∈ F∗q , thenG(α, β, a) := {(i, logβ(a− αi)) : i = 1, . . . , q − 1, αi 6= a} (5)is a Sidon set on (Zq−1 × Zq−1,+).Proof. Suppose there exist u, v, w, y ∈ G(α, β, a) such that u + v = w + y.Using (5) we know that there exist i, j, k, ` ∈ [1, q − 1] such that(i, logβ(a− αi)) + (j, logβ(a− αj)) = (k, logβ(a− αk)) + (`, logβ(a− α`)) (6)where αi, αj , αk, α` are not equal to a. From (6) we have(i+ j) ≡ (k + `) mod (q − 1),logβ(a− αi) + logβ(a− αj) ≡ (logβ(a− αk) + logβ(a− α`)) mod (q − 1),1If θ is a primitive of Fq , logθ(x) denotes the unique integer k ∈ [1, q−1] such that θk = xon Fq .Revista Colombiana de Matemáticas172 DIEGO RUIZ & CARLOS TRUJILLOwhat implies that (a− αi)(a− αj) = (a− αk)(a− α`). We have in F∗qαiαj = αkα`,αi + αj = αk + α`,that is, αi, αj , and αk, α` are roots of a polynomial q(x) ∈ F[x] of degree 2 (i.e.,q(x) = (x+αi)(x+αj) = (x+αk)(x+α`)). Therefore, {αi, αj} = {αk, α`} and{i, j} = {k, `}, which implies that is not possible to have two representationsof an element in Zq−1 × Zq−1 as sum of two elements of G(α, β, a). That is,G(α, β, a) is a Sidon set on (Zq−1 × Zq−1,+). XExample 4.4. First we apply Proposition 4.3 to construct a Sidon set on〈Z16 × Z16,+〉. Let q = p = 17, and let α = 3, β = 5 be primitive elements ofZ∗17. With a = 1G(3, 5, 1) ={(1, 14), (2, 10), (3, 2), (4, 1), (5, 4), (6, 13), (7, 15), (8, 6),(9, 12), (10, 7), (11, 11), (12, 5), (13, 3), (14, 8), (15, 9)}is a Sidon set on (Z16 × Z16,+). Now, if g1 = g2 = 2, using Theorem 4.2,A ={(1, 6), (2, 2), (3, 2), (4, 1), (5, 4), (6, 5), (7, 7), (0, 6),(1, 4), (2, 7), (3, 3), (4, 5), (5, 3), (6, 0), (7, 1)}is a B2[4] set on (Z8 × Z8,+).5. Concluding remarksUsing the constructions given in this work we can obtain lower bounds andclosed formulas for F dh (G, g), for some abelian group G and some values of d, hand g.Note from Theorem 2.1 and Corollary 2.2 that Fh2 (Fhq ) ≥ q for q a primepower. Particularly if h = 2 and q = p prime we have F 22 (Zp × Zp) ≥ p, butit is easy to establish that F 22 (Zp × Zp) = p [7]. A natural question to stateis the following: Can we obtain a similar result, as the last one, on the group(Zp × Zp × Zp,+)? That is,F 32 (Zp × Zp × Zp) ∼ p3/2?Now, using Theorem 3.1, for integers d, g,N ≥ 1 and h ≥ 2 we know thatFh(Nd, g) ≤ F dh (N, g)Particularly, if d = 2, h = 2, and g = 1 we have that F 12 (N2) ≤ F 22 (N), whichimplies that good constructions of Sidon sets on Z give good lower bounds forSidon sets on Z × Z. Furthermore, an interesting work consists in to analyzethe behavior of the difference F 22 (N)− F 12 (N2) when N grows.Volumen 50, Número 2, Año 2016CONSTRUCTION OF BH [G] SETS IN PRODUCT OF GROUPS 173Finally, from Proposition 4.3 we can establish that F 22 (Zq−1×Zq−1) ≥ q−2,which lead us to wonder if is it possible to state that F 22 (Zq−1×Zq−1) = q−1?Acknowledgment. We thank to COLCIENCIAS and Universidad del Caucafor the support to our research group “Algebra, Teoŕıa de Números y Aplicacio-nes–ALTENUA ERM” under the projects 110356935047 and VRI–3744. Wealso thank to the anonymous referee for their helpful comments. This work isdedicated to the memory of Javier Cilleruelo (1961-2016).References[1] J. Bravo, D. Ruiz, and C. Trujillo, Cardinality of sets associated to B3 andB4 sets, Revista colombiana de matemáticas 46 (2012), no. 1, 27–37.[2] S. Chowla and A. Mian, Solution of a problem of Erdös and Turán inadditive–number theory, Proc. Nat. Acad. Sci. India Series A 14 (1944),3–4.[3] J. Cilleruelo, Conjuntos de enteros con todas las diferencias distintas, LaGaceta de la RSME 11 (2008), no. 1, 151–170.[4] , Sidon sets in Nd, Journal of Combinatorial Theory Series A 117(2010), no. 7, 857–871.[5] G. Garcia, C. Trujillo, and J. Velasquez, B±2 [g] finite sets, Journal OfAlgebra, Number Theory And Applications 4 (2004), no. 3, 593–604.[6] S. Golomb and G. Gong, The status of Costas arrays, IEEE Transactionson Information Theory 53 (2007), no. 11, 4260–4265.[7] A. Gómez, D. Ruiz, and C. Trujillo, Construcción de conjuntos de Sidon endimensión dos, XVIII Congreso Colombiano de Matemáticas, UniversidadIndustrial de Santander (Colombia), Julio 2011.[8] C. Gómez and C. Trujillo, Una nueva construcción de conjuntos Bh mod-ulares, Matemáticas: Enseñanza Universitaria 19 (2011), no. 1, 53–62.[9] J. Gómez, Construcción de conjuntos Bh[g], Master thesis, Universidaddel Valle, Colombia, 2011.[10] K. O’Bryant, A complete annotated bibliography of work related to sidonsequences, The electronic Journal of Combinatorics–Dynamic Surveys 11(2004), 39.[11] L. Rackham and P. Šarka, Bh sequences in higher dimensions, The Elec-tronic Journal of Combinatorics 17 (2010), no. 1, R35 (electronic), 15.Revista Colombiana de Matemáticas174 DIEGO RUIZ & CARLOS TRUJILLO[12] C. Trujillo, Sucesiones de sidon, Ph.D. thesis, Universidad Politécnica deMadrid, España, 1998.(Recibido en abril de 2016. Aceptado en julio de 2016)Departamento de MatemáticasUniversidad del CaucaCalle 5 No. 4-70Popayán, Colombiae-mail: dfruiz@unicauca.edu.coe-mail: trujillo@unicauca.edu.coVolumen 50, Número 2, Año 2016

Spanish

Construction of Bh[g] sets in product of groups

Diego Ruiz

Carlos Trujillo

CiteSeerX

CONSTRUCTION OF Bh[g] SETS IN PRODUCT OF GROUPS

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

Construction of Bh[g] sets in product of groups

Abstract

Similar works

Full text

Available Versions

Universidad Nacional De Colombia - Repositorio Institucional UN

CiteSeerX

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas