Benjamin Girard
email: benjamin.girard@math.polytechnique.fr

A NEW UPPER BOUND FOR THE CROSS NUMBER OF FINITE ABELIAN GROUPS by

Keywords: Mathematics Subject Classification (2000): 11R27, 11B75, 11P99, 20D60, 20K01, 05E99, 13F05

In this paper, building among others on earlier works by U. Krause and C. Zahlten (dealing with the case of cyclic groups), we obtain a new upper bound for the little cross number valid in the general case of arbitrary finite Abelian groups. Given a finite Abelian group, this upper bound appears to depend only on the rank and on the number of distinct prime divisors of the exponent. The main theorem of this paper allows us, among other consequences, to prove that a classical conjecture concerning the cross and little cross numbers of finite Abelian groups holds asymptotically in at least two different directions.

Introduction

Let G be a finite Abelian group, written additively. By r(G) and exp(G) we denote respectively the rank and the exponent of G. If G is cyclic of order n, it will be denoted by C n . In the general case, we can decompose G (see for instance [START_REF] Samuel | Théorie algébrique des nombres[END_REF]) as a direct product of cyclic groups C n 1 ⊕ • • • ⊕ C nr where 1 < n 1 | . . . | n r ∈ N, so that every element g of G can be written g = [a 1 , . . . , a r ] (this notation will be used freely along this paper), with a i ∈ C n i for all i ∈ 1, r = {1, . . . , r}.

In this paper, any finite sequence S = (g 1 , . . . , g l ) of l elements from G will be called a sequence of G with length l. Given a sequence S = (g 1 , . . . , g l ) of G, we say that s ∈ G is a subsum of S when it lies in the following set, called the sumset of S:

Σ(S) = i∈I g i | ∅ I ⊆ {1, . . . , l} .
If 0 is not a subsum of S, we say that S is a zero-sumfree sequence. If l i=1 g i = 0, then S is said to be a zero-sum sequence. If moreover one has i∈I g i = 0 for all proper subsets ∅ I {1, . . . , l}, S is called a minimal zero-sum sequence.

In a finite Abelian group G, the order of an element g will be written ord(g) and for every divisor d of the exponent of G, we denote by G d the subgroup of G consisting of all the elements of order dividing d:

G d = {x ∈ G | dx = 0} .
In a sequence S of elements of G, we denote by S d the subsequence of S consisting of all the elements of order d contained in S.

Let P = {p 1 = 2 < p 2 = 3 < . . . } be the set of prime numbers. Given a positive integer n ∈ N * = N\{0}, we denote by D n the set of its positive divisors. If n > 1, we denote by P -(n) the smallest prime element of D n , and we put by convention P -(1) = 1. By τ (n) and ω(n) we denote respectively the number of positive divisors of n and the number of distinct prime divisors of n.

By D(G) we denote the smallest integer t ∈ N * such that every sequence S of G with length |S| ≥ t contains a zero-sum subsequence. The constant D(G) is called the Davenport constant of the group G.

By η(G) we denote the smallest integer t ∈ N * such that every sequence S of G with length |S| ≥ t contains a zero-sum subsequence S ′ ⊆ S with length |S ′ | ≤ exp(G). Such a subsequence is called a short zero-sum subsequence.

The constants D(.) and η(.) have been extensively studied during last decades and even if numerous results were proved (see Chapter 5 of the book [START_REF] Geroldinger | Halter-Koch Non-unique factorizations. Algebraic, combinatorial and analytic theory[END_REF] or [START_REF] Gao | Geroldinger Zero-sum problems in finite abelian groups: a survey[END_REF] for a survey and many references on the subject), their exact values are known for very special types of groups only. In the sequel, we shall need some results on some of the groups for which we know the exact values, so we gather what is known concerning them in the following theorem.

Theorem 1.1. -The two following statements hold:

(i) Let p ∈ P, r ∈ N * and α 1 ≤ • • • ≤ α r , where α i ∈ N * for all i ∈ 1, r . Then, for the p-group G ≃ C p α 1 ⊕ • • • ⊕ C p αr , we have:

D(G) = r i=1
(p α i -1) + 1.

(ii) For every m, n ∈ N * with m|n, we have:

D(C m ⊕ C n ) = m + n -1 and η(C m ⊕ C n ) = 2m + n -2.
In particular, we have

D(C n ) = η(C n ) = n.
Proof. -(i) This result was proved by J. Olson in [START_REF] Olson | A combinatorial problem on finite abelian groups I[END_REF] using the notion of group algebra. The special case of elementary p-groups, which says that D(C r p ) = r(p -1) + 1, can be easily deduced from the Chevalley-Warning theorem (see [START_REF] Van Emde | A combinatorial problem on finite abelian groups II[END_REF] for example).

(ii) The value of D(.) for groups with rank 2 is also due to J. Olson (see [START_REF] Olson | A combinatorial problem on finite abelian groups II[END_REF]), and uses the special case η(C 2 p ) = 3p -2 with p prime. The complete statement for η(.) has been proved by A. Geroldinger and F. Halter-Koch (see [START_REF] Geroldinger | Halter-Koch Non-unique factorizations. Algebraic, combinatorial and analytic theory[END_REF], Theorem 5.8.3).

The value of η(.) for Abelian p-groups with rank r ≥ 3 is not known in general, even in the special case of elementary p-groups. It is only known that for every r ∈ N * , we have η(C r 2 ) = 2 r , and it is conjectured that for every odd p ∈ P, we have η(C 3 p ) = 8p -7 and η(C 4 p ) = 19p -18. The interested reader is for instance referred to [START_REF] Edel | Zero-sum problems in finite abelian groups and affine caps[END_REF] and [START_REF] Gao | Thangadurai On short zero-sum subsequences II[END_REF], for a complete account on this topic. Yet, N. Alon and M. Dubiner showed in [START_REF] Alon | Dubiner A lattice point problem and additive number theory[END_REF] an important theorem related to the constant η(.) of elementary p-groups. We will use the following corollary of this result.

Theorem 1.2. -For every r ∈ N * , there exists a constant c r > 0 such that for every p ∈ P, the following holds:

η C r p ≤ c r (p -1) + 1.
In this paper, we will study the cross number of finite Abelian groups. For this purpose, we recall some definitions and also the results known so far, to the best of our knowledge, concerning this constant. Let G be a finite Abelian group.

If G ≃ C ν 1 ⊕ • • • ⊕ C νs , with ν i > 1 for all i ∈ 1,
s , is the longest possible decomposition of G into a direct product of cyclic groups, then we set:

k * (G) = s i=1 ν i -1 ν i , and
K * (G) = s i=1 ν i -1 ν i + 1 exp(G) = k * (G) + 1 exp(G)
.

The cross number of a sequence S = (g 1 , . . . , g l ), denoted by k(S), is defined by:

k(S) = l i=1 1 ord(g i )
.

Then, we define the little cross number k(G) of G: The cross number was introduced by U. Krause in [START_REF] Krause | A characterization of algebraic number fields with cyclic class group of prime power order[END_REF] in order to clarify the relationship between the arithmetic of a Krull monoid and the properties of its ideal class group. For this reason, the cross number plays a key rôle in the theory of non-unique factorization (see [START_REF] Krause | A characterization of algebraic number fields with cyclic class group of prime power order[END_REF], [START_REF] Gao | Geroldinger Systems of sets of lengths II[END_REF], [START_REF] Geroldinger | Zero-sumfree sequences in cyclic groups and some arithmetical application[END_REF], [START_REF] Schmid | Differences in sets of lengths of Krull monoids with finite class group[END_REF], [START_REF] Plagne | On large half-factorial sets in elementary p-groups: maximal cardinality and structural characterization[END_REF], [START_REF] Plagne | On the maximal cardinality of half-factorial sets in cyclic groups[END_REF] and [START_REF] Radziejewski | Weakly half-factorial sets in finite abelian groups[END_REF] for some applications of the cross number, the surveys [START_REF] Chapman | On the Davenport constant, the cross number, and their application in factorization theory, Zero-dimensional commutative rings[END_REF], [START_REF] Geroldinger | Halter-Koch Non-unique factorizations: a survey, Multiplicative ideal theory in commutative algebra[END_REF] and the book [START_REF] Geroldinger | Halter-Koch Non-unique factorizations. Algebraic, combinatorial and analytic theory[END_REF] which presents exhaustively the different aspects of the theory).

k(G) =
For the sake of completeness, we mention that the cross number has been studied in other directions also (see for example [START_REF] Chapman | Geroldinger On cross numbers of minimal zero sequences[END_REF], [START_REF] Geroldinger | On minimal zero sequences with large cross number[END_REF] and [START_REF] Baginski | On cross numbers of minimal zero sequences in certain cyclic groups[END_REF]), and that this concept arose in a natural way in combinatorial number theory (see for instance [START_REF] Geroldinger | On a conjecture of Kleitman and Lemke[END_REF] and [START_REF] Elledge | An application of graph pebbling to zero-sum sequences in abelian groups[END_REF]).

Given a finite Abelian group G, a natural construction (see [START_REF] Krause | A characterization of algebraic number fields with cyclic class group of prime power order[END_REF] or [START_REF] Geroldinger | Halter-Koch Non-unique factorizations. Algebraic, combinatorial and analytic theory[END_REF], Proposition 5.1.8) gives the following lower bounds:

k * (G) ≤ k(G) and K * (G) ≤ K(G),
yet, except for Abelian p-groups (see [START_REF]Geroldinger The cross number of finite abelian groups[END_REF]) and other special cases (see [START_REF] Geroldinger | The cross number of finite abelian groups II[END_REF]), the exact values of the cross and little cross numbers are also unknown in general, even for cyclic groups. In addition, still no counterexample is known for which equality does not hold in the previous inequalities, which would allow us to disprove the following conjecture.

Conjecture 1.3. -For every finite Abelian group G, one has the following:

k * (G) = k(G) and K * (G) = K(G).
Regarding upper bounds, and since the constants k(.) and K(.) are closely related to each other, it suffices, according to the following proposition (see [START_REF] Geroldinger | Halter-Koch Non-unique factorizations. Algebraic, combinatorial and analytic theory[END_REF], Proposition 5.1.8), to bound from above the little cross number so as to bound from above the cross number, but also the Davenport constant. Since k(.) is easier to handle, one usually prefers to study the cross number via the little cross number, and we will do so in this paper.

Proposition 1.1. -Let G be a finite Abelian group with exp(G) = n. Then, the two following statements hold:

(i) k(G) + 1 n ≤ K(G) ≤ k(G) + 1 P -(n) , (ii)
D(G) ≤ nk(G) + 1.
Two types of upper bounds are currently known for k(.). The first one holds for any finite Abelian group, and was obtained by A. Geroldinger and R. Schneider in [START_REF] Geroldinger | The cross number of finite abelian groups III[END_REF] and in [START_REF] Geroldinger | Halter-Koch Non-unique factorizations. Algebraic, combinatorial and analytic theory[END_REF], Theorem 5.5.5, using character theory and the notion of group algebra.

k(G) ≤ d -1 P -(n) + log |G| d .
In particular k(G) ≤ log |G|.

Eventhough this upper bound is general and easy to compute, it does not really fit what we know about the behaviour of the cross number. For example, let r > 1 be an integer. If we consider an elementary p-group with rank r, it is known that k C r p = k * C r p ≤ r, yet log |C r p |/p = (r -1) log p diverges when p tends to infinity.

From this point of view, and in the special case of cyclic groups, a more precise upper bound was found by U. Krause and C. Zahlten in [START_REF] Krause | Zahlten Arithmetic in Krull monoids and the cross number of divisor class groups[END_REF] which, expressed with our notations, gives the following.

Theorem 1.5. -For every n ∈ N * , one has the following:

k(C n ) ≤ 2ω(n).
It should be underlined that this upper bound has the right order of magnitude, since one has k(C n ) ≥ k * (C n ) ≥ ω(n)/2 by definition.

New results and plan of the paper

In this paper, we generalize the work of [START_REF] Krause | Zahlten Arithmetic in Krull monoids and the cross number of divisor class groups[END_REF] to every finite Abelian group so as to obtain a new upper bound for the little cross number in the general case, which no longer depends on the cardinality of the group considered, and which supports the conjecture that the little cross number of a finite Abelian group G with rank r and exponent n is less than rω(n).

For this purpose, we introduce the two following constants. Let G be a finite Abelian group and

d ′ , d ∈ N * be two integers such that d ∈ D exp(G) and d ′ ∈ D d . By D (d ′ ,d) (G) we denote the smallest integer t ∈ N * such that every sequence S of G d with length |S| ≥ t contains a subsequence of sum in G d/d ′ . By η (d ′ ,d) (G) we denote the smallest integer t ∈ N * such that every sequence S of G d with length |S| ≥ t contains a subsequence S ′ ⊆ S of length |S ′ | ≤ d ′ and of sum in G d/d ′ .
To start with, we will prove in Section 3 (Proposition 3.1), that for any finite Abelian group G and every 1

≤ d ′ | d | exp(G), D (d ′ ,d) (G) and η (d ′ ,d) (G) are linked to the constants D(.) and η(.) of a particular subgroup G υ(d ′ ,d) of G.
In Section 4, we will prove the main theorem (Theorem 2.1). This result will be stated at the end of this section. Before giving this general and technical theorem, we emphasize the many consequences it has.

To obtain these results, we introduce the two following arithmetic functions:

α(n) = d∈Dn P -(d) -1 d and β(n) = d∈Dn∩P P -(d) -1 d ,
which will be investigated in Section 5. In particular, simple upper bounds for these functions lead, by applying the main theorem, to the following qualitative result, proved in Section 6.

Proposition 2.1. -For every r ∈ N * there exists a constant d r > 0 such that, for every finite Abelian group G with r(G) ≤ r and exp(G) = n, the following holds:

k(G) ≤ d r ω(n).
Consequently, when considering the cross number of a finite Abelian group G with fixed or bounded rank, Proposition 2.1 gives a qualitative upper bound which depends only on the number of distinct prime divisors ω(n) of exp(G) = n, and which improves, at least asymptotically, the one stated in Theorem 1.4, since the function ω can have arbitrary small values in N * even for arbitrary large n, but mainly since it is known (see for instance Chapter I.5 of the book [START_REF] Tenenbaum | Introduction à la théorie analytique et probabiliste des nombres, Cours spécialisés[END_REF]) that one has:

ω(n) log n log log n .
In addition, more accurate upper bounds for some sequences built with α(n) and β(n), obtained in Lemma 5.1 (see Section 5), enable us to prove the following quantitative result (see Section 6) which states that when r = 1 or 2, one can choose d r in the following way:

d 1 = 166822111 109486080 ≈ 1.
k(G) ≤ α(n) ≤ d 1 ω(n). (ii) For every finite Abelian group G ≃ C m ⊕ C n , with 1 < m | n ∈ N * , we have: k(G) ≤ 3α(n) -β(n) ≤ d 2 ω(n).
Moreover, the asymptotical behaviours of α(n) and β(n), studied in Lemma 5.2, imply several asymptotical results, some of them being sharp, concerning the cross and little cross numbers as well as the Davenport constant. In particular, these results show that Conjecture 1.3 holds asymptotically in at least two different directions. These results will be proved in Section 7, and in order to state them, we will need the following notation. For every r ∈ N * and l 1 , . . . , l r ∈ N * , we set:

E (l 1 ,...,lr) = r i=1 C n i , 1 < n 1 | . . . |n r ∈ N | ∀i ∈ 1, r , ω(n i ) = l i and gcd n i , n r n i = 1 .
Proposition 2.3. -For every r ∈ N * and l 1 , . . . , l r ∈ N * , the following statements hold:

(i) lim P -(nr) → +∞ Cn 1 ⊕ • • • ⊕ Cn r ∈ E (l 1 ,...,lr ) k(C n 1 ⊕ • • • ⊕ C nr ) = r i=1 l i , (ii) lim P -(nr) → +∞ Cn 1 ⊕ • • • ⊕ Cn r ∈ E (l 1 ,...,lr ) K(C n 1 ⊕ • • • ⊕ C nr ) = r i=1 l i , (iii) lim sup P -(nr) → +∞ Cn 1 ⊕ • • • ⊕ Cn r ∈ E (l 1 ,...,lr ) D(C n 1 ⊕ • • • ⊕ C nr ) n r ≤ r i=1 l i .
Concerning the groups of the form C r n , we obtain the following corollary by specifying

n 1 = • • • = n r in Proposition 2.3. Proposition 2.4.
-For all integers r, l ∈ N * the three following statements hold:

(i) lim

P -(n) → +∞ ω(n) = l k(C r n ) = rl, (ii) lim P -(n) → +∞ ω(n) = l K(C r n ) = rl, (iii)
lim sup

P -(n) → +∞ ω(n) = l D(C r n ) n ≤ rl.
It may be observed that Proposition 2.3 and Proposition 2.4 are somehow reminiscent of [START_REF] Geroldinger | The cross number of finite abelian groups III[END_REF], Theorem 2(b), since this result and our Proposition 2.3 give the value of the cross number of "large" groups. However, a more precise look at both results shows that they are of a different nature. Indeed, while A. Geroldinger and R. Schneider's result is not asymptotical but valid only for special groups satisfying some restrictive conditions, ours, although of asymptotical nature, is valid in a wider framework.

The following proposition will also be proved in Section 7.

Proposition 2.5. -For all r ∈ N * , the two following statements hold:

(i) lim ω(n) → +∞ k(C r n ) ω(n) = r, (ii)
lim

ω(n) → +∞ K(C r n ) ω(n) = r.
All these results are deduced from the following proposition, proved in Section 6 under the stronger form of Proposition 6.1, and which is a somewhat rough corollary of the main theorem (Theorem 2.1). For the sake of clarity, we recall that the constant c r is the one which has been introduced in Theorem 1.2.

Proposition 2.6. -Let G be a finite Abelian group with r(G) = r and exp(G) = n. We set H = C r n and also:

ϕ(G, H) = k * (H/G) if G is a direct summand of H, k * (H/G)/n otherwise.
Then, one has the following upper bound for the little cross number k(G):

k(G) ≤ c r α(n) -β(n) + rβ(n) -ϕ(G, H).
The main theorem of this paper (Theorem 2.1) will be proved in Section 4. In order to state it, we will need the following definitions and notations which will be extensively used in Sections 3 and4.

Let G ≃ C n 1 ⊕ • • • ⊕ C nr , with 1 < n 1 | . . . | n r ∈ N, be a finite Abelian group with exp(G) = n, τ (n) = m and d ′ , d ∈ N * be such that d ∈ D n and d ′ ∈ D d .
For all i ∈ 1, r , we set:

A i = gcd(d ′ , n i ), B i = lcm(d, n i ) lcm(d ′ , n i ) , υ i (d ′ , d) = A i gcd(A i , B i ) , and G υ(d ′ ,d) = C υ 1 (d ′ ,d) ⊕ • • • ⊕ C υr(d ′ ,d) .
Then, for every d ∈ D n = {d 1 , . . . , d m } and x = (x d 1 , . . . , x dm ) ∈ N m , we set:

f d (x) = min d ′ ∈D d \{1} η(G υ(d ′ ,d) ) -1 -x d , g d (x) = D(G υ(d,d) ) -1 - d ′ ∈D d x d ′ , and h(x) = d∈Dn x d d -k * (G).
We can now state the main theorem.

Theorem 2.1. -Let G ≃ C n 1 ⊕ • • • ⊕ C nr , with 1 < n 1 | . . . | n r ∈ N,
P G = {x ∈ N m | f d (x) ≥ 0, g d (x) ≥ 0, d ∈ D n },
and

H G = {x ∈ N m | h(x) ≥ 0}.
Keeping the notations of Theorem 2.1, we obtain the following immediate corollary, which gives a general upper bound for the little cross number of a finite Abelian group, expressed as the solution of an integer linear program.

Corollary 2.2. -For every finite Abelian group G, one has the following upper bound:

k(G) ≤ max x∈P G m i=1 x d i d i .
In principle, the wide generality of Theorem 2.1 and Corollary 2.2 leaves a good hope that it could lead to new -and maybe optimal -upper bounds for k(G) in the general case. However, such improvements will require a precise study of the polytope P G , which is certainly a complicated, but not hopeless, task.

On the quantities D

(d ′ ,d) (G) and η (d ′ ,d) (G)
In this section, we will denote by π i , for all i ∈ 1, r , the canonical epimorphism from

C n i to C υ i (d ′ ,d) .
Although this epimorphism clearly depends on d ′ and d, we do not emphasize this dependence here since there is no risk of ambiguity. Moreover, one can notice that whenever d divides n i , we have

υ i (d ′ , d) = gcd(d ′ , n i ) = d ′ , and in particular υ r (d ′ , d) = d ′ . In the sequel, when d ′ = d, we will write υ i (d) instead of υ i (d, d). Lemma 3.1. -Let G ≃ C n 1 ⊕ • • • ⊕ C nr , with 1 < n 1 | . . . | n r ∈ N,
d d ′ n 1 gcd(d, n 1 )
a 1 , . . . , n r gcd(d, n r ) a r = 0 if and only if π i (a i ) = 0 for all i ∈ 1, r .

Proof.

-First, we have the following equalities:

d d ′ n i gcd(d, n i ) = lcm(d, n i ) d ′ = lcm(d, n i )n i d ′ n i = lcm(d, n i )n i gcd(d ′ , n i )lcm(d ′ , n i ) = B i n i A i ∈ N.
Let [a 1 , . . . , a r ] ∈ G be such that:

d d ′ n 1 gcd(d, n 1 ) a 1 , . . . , n r gcd(d, n r ) a r = 0.
For all i ∈ 1, r , one has:

d d ′ n i gcd(d, n i ) a i = B i n i A i a i = 0,
which is equivalent, considering a i as an integer, to the following relation:

A i |B i a i ,
that is to say, dividing each side by gcd(A i , B i ), that one has:

υ i (d ′ , d) B i gcd(A i , B i ) a i ,
which, since:

gcd A i gcd(A i , B i ) , B i gcd(A i , B i ) = 1,
is equivalent to: υ i (d ′ , d)|a i , and the desired result is proved.

Proposition 3.1. -Let G ≃ C n 1 ⊕ • • • ⊕ C nr , with 1 < n 1 | . . . | n r ∈ N,
D (d ′ ,d) (G) = D C υ 1 (d ′ ,d) ⊕ • • • ⊕ C υr(d ′ ,d) , η (d ′ ,d) (G) = η C υ 1 (d ′ ,d) ⊕ • • • ⊕ C υr(d ′ ,d) .
Proof. -Let [a 1 , . . . , a r ] ∈ G d . We know that ord ([a 1 , . . . , a r ]) = lcm(ord(a 1 ), . . . , ord(a r )), and so ord ([a 1 , . . . , a r ]) |d implies ord(a i )|d for all i ∈ 1, r .

By Lagrange theorem, we also have ord(a i )|n i , which implies that: ord(a i )| gcd(d, n i ) for all i ∈ 1, r , and since any cyclic group C n i contains a unique subgroup of order gcd(d, n i ), we can write:

a i = n i gcd(d, n i ) a ′ i with a ′ i ∈ C n i .
We now consider a sequence S = (g 1 , . . . , g m ) of G d with m ∈ N * . According to the previous argument, the elements of S have the following form:

g 1 = [a 1,1 , . . . , a 1,r ] = n 1 gcd(d,n 1 ) a ′ 1,
d d ′ k∈K [a k,1 , . . . , a k,r ] = d d ′ n 1 gcd(d, n 1 ) k∈K a ′ k,1 , . . . , n r gcd(d, n r ) k∈K a ′ k,r = 0,
and by Lemma 3.1, this relation is equivalent to:

k∈K π 1 (a ′ k,1 ), . . . , π r (a ′ k,r ) = 0 in r i=1 C υ i (d ′ ,d) .
Therefore, from the definition of the constant D(.), one can deduce that the smallest integer m ∈ N * such that for every sequence S = (g 1 , . . . , g m ) of G d with length |S| ≥ m, there exists a nonempty subset K ⊆ {1, . . . , m} such that

k∈K [a k,1 , . . . , a k,r ] is an element of G d/d ′ is exactly D C υ 1 (d ′ ,d) ⊕ • • • ⊕ C υr(d ′ ,d)
. This proves the first equality.

If moreover, one expects the additional condition |K| ≤ υ r (d ′ , d) = d ′ to be verified, then, by the definition of η(.), the corresponding smallest possible integer m ∈ N * is exactly

η C υ 1 (d ′ ,d) ⊕ • • • ⊕ C υr(d ′ ,d)
, which proves the second equality.

Proof of the main theorem

Proof of Theorem 2.1. -Let S be a zero-sumfree sequence of G verifying k(S) = k(G), and being of minimal length regarding this property. For every d ∈ D n , we set x d = |S d |, and we suppose that the m-tuple x = (x d 1 , . . . , x dm ) is not an element of the polytope P G ∩ H G . Thus, one has the three following cases.

Case 1. There exists

d 0 ∈ D n such that f d 0 (x) < 0. Therefore, it exists d ′ 0 ∈ D d 0 \{1} verifying x d 0 ≥ η C υ 1 (d ′ 0 ,d 0 ) ⊕ • • • ⊕ C υr(d ′ 0 ,d 0 ) which means, by Proposition 3.1, that x d 0 ≥ η (d ′ 0 ,d 0 ) (G)
. So, the sequence S contains X elements of order d 0 , with 1 < X ≤ d ′ 0 , the sum of which is an element of order d0 dividing d 0 /d ′ 0 . Let S ′ be the sequence obtained from S by replacing these X elements by their sum. In particular, we have |S ′ | = |S| -X + 1 < |S|. Moreover, S ′ is a zero-sumfree sequence and verifies the following equalities:

     |S ′ d 0 | = |S d 0 | -X, |S ′ d0 | = |S d0 | + 1, |S ′ d | = |S d | ∀d = d 0 , d0 . Since 1 d0 - X d 0 ≥ 0,
one has the following inequalities:

k(S) = d∈D exp(G) x d d ≤ d∈D exp(G) \{d 0 , d0 } x d d + x d0 + 1 d0 + x d 0 -X d 0 = k(S ′ ).
So, we obtain k(S ′ ) = k(G) and |S ′ | < |S|, which is a contradiction.

Case 2. There exists d 0 ∈ D n such that g d 0 (x) < 0. As a consequence, we have

d∈D d 0 x d ≥ D(C υ 1 (d 0 ) ⊕ • • • ⊕ C υr(d 0 )
) and Proposition 3.1 gives the existence of a zerosum subsequence, which is a contradiction.

Case 3. One has h(x) < 0, that is to say k(S) = k(G) < k * (G) which is a contradiction.
An interesting special case is the one of finite Abelian groups with rank 2. Indeed, for such groups, all the parameters used to define the polytope P G in the main theorem are known by Theorem 1.1:

D (d,d) (G) = υ 1 (d) + υ 2 (d) -1 and η (d ′ ,d) (G) = 2υ 1 (d ′ , d) + υ 2 (d ′ , d) -2,
and therefore allow us to compute an explicit upper bound for the little cross number k(G) by linear programming methods (see for instance the book [START_REF] Schrijver | Theory of linear and integer programming[END_REF] for an exhaustive presentation of these methods).

Some sequences related to the exponent of a finite Abelian group

Let (α l ) l≥1 and (β l ) l≥1 be the two following sequences of integers, built from the set of prime numbers:

α 1 = 1 and α l = 1 + p l p l -1 α l-1 for all l ≥ 2,
as well as

β l = l i=1 p i -1 p i for all l ≥ 1.
Finally, we define a third sequence (γ l ) l≥1 in the following fashion:

γ l = 3α l -β l for all l ≥ 1.
The first values of (α l ) l≥1 are the following: Since 2l -1 ≤ p l , we can already show, by induction on l, the following statement:

α 1 = 1, α 2 = 2.
α l ≤ 2l, for all l ≥ 1.
Indeed, one has α 1 = 1 ≤ 2, and if the statement is true for l -1, we obtain:

α l = 1 + α l-1 + α l-1 p l -1 ≤ 1 + 2(l -1) + 2(l -1) p l -1 ≤ 2l.
In order to study more precisely the behaviours of α(n) and β(n), we will extensively use a classical lower bound for the l-th prime number, proved by Rosser in [START_REF] Rosser | The n-th prime is greater than n log n[END_REF], and which is the following:

l log l ≤ p l for all l ≥ 1.
We can now prove Lemma 5.1, which gives accurate upper bounds for the sequences (α l ) l≥1 and (γ l ) l≥1 , and Lemma 5.2, which states on the one hand that α l and β l are both equivalent to l when l tends to infinity, and on the other hand that when ω(n) = l is fixed, then both α(n) and β(n) tends to l when P -(n) tends to infinity. (i) For every integer n ∈ N * , with ω(n) = l, we have:

β l ≤ β(n) ≤ α(n) ≤ α l .
(ii) For every integer l ≥ 1, we have:

l ≤ α l ≤ α 9 9
l, where α 9 9 = 166822111 109486080 ≈ 1.5237.

(iii) For every integer l ≥ 1, we have:

5 2 l ≤ γ l ≤ γ 8 8 l, where γ 8 8 = 1784073894563 476759162880 ≈ 3.7421.
Proof. -(i) Let n = q m 1 1 . . . q m l l be an integer with q 1 < • • • < q l . Since for all i ∈ 1, l , one has p i ≤ q i , we obtain the first inequality:

β l = l - l i=1 1 p i ≤ l - l i=1 1 q i = β(n).
The second inequality follows directly from:

β(n) = d∈Dn∩P P -(d) -1 d ≤ d∈Dn P -(d) -1 d = α(n).
We prove the third inequality by induction on the number of distinct prime divisors ω(n) = l of n. For l = 1, the integer n is of the form q m 1 1 and we obtain:

α(q m 1 1 ) = m 1 i=1 q 1 -1 q i 1 = q m 1 1 -1 q m 1 1 ≤ 1 = α 1 .
Assume now that the statement is valid for l -1. Therefore, we have:

α(q m 1 1 . . . q m l l ) = q m l l -1 q m l l + m l i=0 1 q i l α(q m 1 1 . . . q m l-1 l-1 ) ≤ q m l l -1 q m l l + m l i=0 1 q i l α l-1 ≤ 1 + +∞ i=0 1 p i l α l-1 = 1 + p l p l -1 α l-1 = α l ,
which proves the result.

(ii) To start with, it is straightforward that the first inequality l ≤ α l always holds. Concerning the second inequality, one has the following:

α l+1 -α l = 1 + α l p l+1 -1 for all l ≥ 1,
from which we deduce the two following relations:

α l = α 1 + l-1 k=1 (α k+1 -α k ) = l + l-1 k=1 α k p k+1 -1 , as well as α l+1 l + 1 - α l l = 1 l + 1 + α l 1 l + 1 1 + 1 p l+1 -1 - 1 l .
In the remainder of this proof, we will set ε(l

) = α l -l = l-1 k=1 α k p k+1 -1
, for all l ≥ 1.

Using this notation, we obtain the following:

α l l - α 9 9 = l-1 k=9 1 k + 1 + l-1 k=9 α k 1 k + 1 1 + 1 p k+1 -1 - 1 k = l-1 k=9 1 k + 1 - l-1 k=9 k + ε(k) k(k + 1) + l-1 k=9 α k (p k+1 -1)(k + 1) = l-1 k=9 1 k + 1 α k (p k+1 -1) - ε(k) k = l-1 k=9 ε(k + 1) k + 1 - ε(k) k = ε(l) l - ε (9) 9 .
Moreover, using Rosser's lower bound, we obtain for all l ≥ 2:

ε(l) = l-1 k=1 α k p k+1 -1 ≤ l-1 k=1 2k (k + 1) log(k + 1) -1 ≤ 7 + l 2 2dt log t = lf (l),
where we set for all x ∈ R, x ≥ 2:

f (x) = 1 x 7 + x 2 2dt log t .
It is readily seen that this function is non-increasing. Moreover, since:

ε(9) 9 = α 9 -9 9 > 1 2 ,
and since f (l) ≤ 1/2 for all l ≥ 241, we obtain:

α l l ≤ α 9 9
, for all l ≥ 241.

On the other hand, an easy computation allows us to verify that α 9 /9 is also the maximum value taken by (α l /l) l≥1 on 1 ≤ l ≤ 240, which proves the desired result.

(iii) The fact that the first inequality 5l/2 ≤ γ l always holds is straightforward. Moreover, for all l ≥ 1, one has the following equality:

γ l+1 = 3α l+1 -β l+1 = 3 + 3α l + 3α l p l+1 -1 -β l -1 + 1 p l+1 = 2 + γ l + 3α l p l+1 -1 + 1 p l+1 .
Using the inequalities 5l/2 ≤ γ l and α l ≤ α 9 l/9, one can deduce that:

γ l+1 l + 1 - γ l l = 2 l + 1 + γ l 1 l + 1 - 1 l + 3α l (l + 1)(p l+1 -1) + 1 p l+1 (l + 1) ≤ 1 p l+1 (l + 1) - p l+1 2 + α 9 l 3 1 + 1 p l+1 -1 + 1 .
We set, for all x ∈ R, x ≥ 1:

g(x) = - (x + 1) log(x + 1) 2 + α 9 x 3 1 + 1 (x + 1) log(x + 1) -1 + 1.
It is easily seen that this function is non-increasing. Moreover, since a study of g shows that g(l) ≤ 0 for all l ≥ 9333, we obtain:

γ l+1 l + 1 - γ l l ≤ 0, for all l ≥ 9333.
On the other hand, an easy computation allows us to verify that (γ l ) l≥1 is increasing from l = 1 to l = 8 and decreasing from l = 8 to l = 9333, which proves the desired result.

Lemma 5.2. -The two following statements hold:

(i) lim l→+∞ α l l = 1 and lim l→+∞ β l l = 1, (ii) lim
P -(n) → +∞ ω(n) = l α(n) = l and lim P -(n) → +∞ ω(n) = l β(n) = l.
Proof. -(i) Firstly, for all l ≥ 1, one has the following inequality:

l ≤ α l ≤ l + l-1 k=1 2k p k+1 -1 ,
and since the prime number theorem reads as p k ∼ k log k, we can deduce that:

l-1 k=1 k p k+1 -1 ∼ l k=2 1 log k ∼ l log l .
Therefore, when l tends to infinity, we obtain lim l→+∞ (α l /l) = 1.

Secondly, we can deduce from Rosser's lower bound that for every l ≥ 3, one has:

β l ≥ l - 5 6 - l i=3 1 i log i ≥ l -2 -log log l.
Since, on the other hand, one always has β l ≤ l, we obtain lim l→+∞ (β l /l) = 1, which is the desired result.

(ii) The result follows from the very definition of α(n) and β(n).

Upper bounds for the little cross number

As previously stated, the upper bound implied by Theorem 2.1, and given in Corollary 2.2, is expressed as the solution of an integer linear program. Even if this formulation is more precise than any explicit formula derived from Theorem 2.1, one may still like to obtain such a formula in order to interprete the behaviour of the cross number. In the present section, we obtain such a formula in Proposition 6.1. For the proof of this result, we will use the following lemma, which can be found in [START_REF] Geroldinger | Halter-Koch Non-unique factorizations. Algebraic, combinatorial and analytic theory[END_REF], Proposition 5.1.11. Lemma 6.1. -Let H be a finite Abelian group and G ⊆ H a subgroup. Then, one has:

(i) k(G) + k(H/G) exp(G) ≤ k(H).
(ii) If G is a direct summand of H, then:

k(G) + k(H/G) ≤ k(H).
We are now ready to prove the following proposition. We set H = C r n and also:

ϕ(G, H) = k * (H/G) if G is a direct summand of H, k * (H/G)/n otherwise.
Then, one has the following upper bound for the little cross number k(G):

k(G) ≤ d∈Dn min η C r P -(d) , D(C r d ) -1 d -ϕ(G, H).
Proof.

-Since the group G can be injected in the group H = C r n , one obtains, applying Lemma 6.1, the relation k(G) + ϕ(G, H) ≤ k(H). Then, the desired result follows from Theorem 2.1 applied to H.

One can notice that for all r ∈ N * and every p ∈ P, one always has D(C r p ) ≤ η C r p , by definition. Therefore, if we consider an elementary p-group with rank r, we obtain:

k C r p ≤ d∈Dp D(C r d ) -1 d = r(p -1) p = k * C r p .
Let G be a finite Abelian group with r(G) = r and exp(G) = n. Using Theorem 1.1, one obtains that if r = 1, then for all d ∈ D n \P, we have D(C d ) ≥ η C P -(d) . Moreover, when r = 2, then for all d ∈ D n \P, one has the following inequality:

D(C 2 d ) = 2d -1 ≥ 3P -(d) -2 = η C 2 P -(d)
. Yet, as soon as r ≥ 3, and except for special types groups, it becomes more complicated to know exactly, for a given d in D n \P, what is the minimum of η C r P -(d) and D(C r d ). For this reason, Theorem 2.1 and Proposition 6.1 remain, in general, really stronger than Proposition 2.6, which we are going to prove now. Even so, we will see in the next section that Proposition 2.6 implies sharp asymptotical results on the little cross number and the cross number.

Proof of Proposition 2.6. -Applying Proposition 6.1 and Theorem 1.2, we obtain the desired result in the following manner:

k(G) + ϕ(G, H) ≤ d∈Dn min η C r P -(d) , D(C r d ) -1 d ≤ d∈Dn∩P r(P -(d) -1) d + d∈Dn\P c r (P -(d) -1) d = c r α(n) -β(n) + rβ(n).
We can now prove the announced qualitative upper bound.

Proof of Proposition 2.1. -Since, by the definitions of Section 5, one always has the following straightforward inequalities:

ω(n) 2 ≤ β(n) and α(n) ≤ 2ω(n),
we can deduce, by Proposition 2.6 and the inequality r ≤ c r , the following relation:

k(G) ≤ c r α(n) -β(n) + rβ(n) ≤ 3c r + r 2 ω(n),
which gives the desired result.

According to the previous remark, and in the case of r = 1 or 2, η C

k(C n ) ≤ α(n) ≤ α ω(n) ≤ α 9 9 ω(n),
which proves that one can take d 1 = α 9 /9.

(ii) For all m, n ∈ N * with 1 < m|n, one has by Proposition 2.6 applied to G ≃ C m ⊕ C n and Lemma 5.1 (i), (iii):

k(G) ≤ 3α(n) -β(n) -ϕ(G, C 2 n ) ≤ γ ω(n) ≤ γ 8 8 ω(n),
which proves that one can take d 2 = γ 8 /8.

Asymptotical results

In the present section, we will apply the results obtained in Section 6 in order to prove that Conjecture 1.3 holds asymptotically in the two directions of Proposition 2.3 and Proposition 2.5.

Proof of Proposition 2.3. -First, we have: lim

P -(nr) → +∞ Cn 1 ⊕ • • • ⊕ Cn r ∈ E (l 1 ,...,lr ) k * (C n 1 ⊕ • • • ⊕ C nr ) = r i=1 l i ,
and since by the Chinese remainder theorem, every

C n 1 ⊕ • • • ⊕ C nr in E (l 1 ,...,lr) is a direct summand of C r
nr , we obtain using Lemma 6.1 (ii):

k(C n 1 ⊕ • • • ⊕ C nr ) ≤ k C r nr -k * C nr n r-1 ⊕ • • • ⊕ C nr
k(C r n ) ω(n) ≤ c r α(n) -β(n) ω(n) + r β(n) ω(n) ,
which implies, by Lemma 5.2 (i), the following inequalities when ω(n) tends to infinity:

r = lim ω(n) → +∞ k * (C r n ) ω(n) ≤ lim ω(n) → +∞ k(C r n ) ω(n) ≤ r.
The result for K(.) is then deduced from Proposition 1.1 (i).

Each of the two previous asymptotical results admits a corollary which may appear more general at first sight. So as to state the first one, we will use the following notation, which recalls the one used for the sets E (l 1 ,...,lr) . For every r, l ∈ N * , we set: With this notation, we obtain the following corollary. Proof. -Every G in E r,l can be injected in the group H ≃ C r exp(G) . Therefore, using Lemma 6.1, we obtain k(G) ≤ k(H) and the desired result follows from Proposition 2.4, applied to the group H. The corresponding statements for K(.) and D(.) are then deduced from Proposition 1.1. Proof. -Every G with rank r(G) ≤ r can be injected in the group H ≃ C r exp(G) . Since we have k(G) ≤ k(H) by Lemma 6.1, the result follows from Proposition 2.5, applied to the group H. The statement for K(.) is then deduced from Proposition 1.1 (i).

  max{k(S)|S zero-sumfree sequence of G}, as well as the cross number of G, denoted by K(G): K(G) = max{k(S)|S minimal zero-sum sequence of G}.

Theorem 1 . 4 .

 14 -Let G be a finite Abelian group with exp(G) = n. Then, for every d ∈ D n , one has the following:

  be a finite Abelian group with exp(G) = n and τ (n) = m. For every zero-sumfree sequence S of G reaching the maximum k(S) = k(G), and being of minimal length regarding this property, the m-tuple x = (|S d 1 |, . . . , |S dm |) is an element of the polytope P G ∩ H G where:

  be a finite Abelian group and d ′ , d ∈ N * be such that d ∈ D exp(G) and d ′ ∈ D d . Then, for every g = [a 1 , . . . , a r ] ∈ G, we have:

  be a finite Abelian group and d ′ , d ∈ N * be such that d ∈ D exp(G) and d ′ ∈ D d . Then, we have the two following equalities:

  5, α 3 = 4.125, α 4 = 5.8125, α 5 = 7.39375 etc.

Lemma 5 . 1 .

 51 -The following statements hold:

Proposition 6 . 1 .

 61 -Let G be a finite Abelian group with r(G) = r and exp(G) = n.

n 1 =.

 1 k C r nr -On the one hand, we have by Lemma 5.2 (ii): lim supP -(nr) → +∞ ω(nr) = lr k(C r nr ) ≤ lim sup P -(nr) → +∞ ω(nr) = lr c r α(n r )β(n r ) + rβ(n r ) = rl r ,on the other hand, since for all i ∈ 1, r , the equality gcd(n i , n r /n i ) = 1 implies ω(n r /n i ) = ω(n r )ω(n i ), we also have: limP -(nr) → +∞ Cn 1 ⊕ • • • ⊕ Cn r ∈ E (l 1 ,...,lr ) lim inf P -(nr) → +∞ Cn 1 ⊕ • • • ⊕ Cn r ∈ E (l 1 ,...,lr ) k(C n 1 ⊕ • • • ⊕ C nr ) ≤ lim sup P -(nr) → +∞ Cn 1 ⊕ • • • ⊕ Cn r ∈ E (l 1 ,...,lr ) k(C n 1 ⊕ • • • ⊕ C nr ) ≤ rl r -r i=1 (l rl i ) = r i=1 l i .The corresponding statements for K(.) and D(.) are then deduced from Proposition 1.1.Since, as mentioned in Section 2, Proposition 2.4 is an immediate corollary of Proposition 2.3 by specifying n 1 = • • • = n r , we now prove an asymptotical result of an other type.Proof of Proposition 2.5. -First, we have:lim ω(n) → +∞ k * (C r n ) ω(n) = r,Moreover, by Proposition 2.6 applied to C r n , we obtain:

E

  r,l = {G finite Abelian group | r(G) = r, ω(exp(G)) = l}.

Corollary 7 . 1 .

 71 -For all integers r, l ∈ N * the three following statements hold: (i) lim supP -(exp(G)) → +∞ G ∈ E r,l k(G) ≤ rl,(ii) lim supP -(exp(G)) → +∞ G ∈ E r,l K(G) ≤ rl,

Corollary 7 . 2 .

 72 -For all integers r ∈ N * , the two following statements hold:

  Let K be a nonempty subset of {1, . . . , m}. Then, the sum k∈K [a k,1 , . . . , a k,r ] is an element of G d/d ′ if and only if:

	. . .	. . .	1 , . . . , nr gcd(d,nr) a ′ 1,r , . . .
	g m = [a m,1 , . . . , a m,r ] =	n 1 gcd(d,n 1 ) a ′ m,1 , . . . , nr gcd(d,nr) a ′ m,r .

  r P -(d) and D(C r d ) are known and easy to compare. Therefore, we can prove Proposition 2.2. Proof of Proposition 2.2. -Applying Theorem 1.1, one can choose c 1 = 1 and c 2 = 3.

(i) For every n ∈ N * , one has by Proposition 2.6 and Lemma 5.1 (i), (ii):

Acknowledgments

I am grateful to my Ph.D. advisor Alain Plagne for his help during the preparation of this article. I would like also to thank Wolfgang Schmid for his remarks on a preliminary version of this paper.


Huiping Shen

Chengqi Wu

Damien Chablat
email: damien.chablat@cnrs.fr

Guang Lei Wu

Ting-Li Yang
email: yangtl@126.com

TOPOLOGICAL DESIGN OF AN ASYMMETRIC 3-TRANSLATIONAL PARALLEL MECHANISM WITH ZERO COUPLING DEGREE AND MOTION DECOUPLING

In this paper a new asymmetric 3-translational (3T) parallel manipulator, i.e., RPa(3R) 2R+RPa, with zero coupling degree and decoupled motion is firstly proposed according to topology design theory of parallel mechanism (PM) based on position and orientation characteristics (POC) equations. The main topological characteristics such as POC, degree of freedom and coupling degree are calculated. Then, the analytical formula for the direct and inverse kinematic are directly derived since coupling degree of the PM is zero. The study of singular configurations is simple because of the independence of the kinematic chains.

INTRODUCTION

In many industrial production lines, process operations require pure translation movements only. Therefore, the 3-DOF translational parallel mechanism (TPM) has a significant potential value since it is a relatively simple structure and easily controlled [START_REF] Liu | Kinematic and workspace analysis of a new type of spatial 3-DOF parallel mechanism[END_REF].

Many scholars have being studied the TPM. For example: original design of 3-DOF TPM is the Delta Robot which was presented by Clavel [START_REF] Clavel | A Fast Robot with Parallel Geometry[END_REF]. The structure manipulators of TPM have been developed [START_REF] Stock | Optimal Kinematic Design of Spatial Parallel Manipulators: Application to Linear Delta Robot[END_REF][START_REF] Bouri | The Linear Delta: Developments and Applications // Robotics[END_REF][START_REF] Kelaiaia | Multiobjective optimization of a linear Delta parallel robot[END_REF]. Tsai et al [START_REF] Tsai | Kinematics of a Novel Three DOF Translational Platform[END_REF] presented the 3-DOF TPM, the moving actuators are prismatic joints and the subchain is 4R parallelogram mechanism (P is prismatic joint and R is revolute joint). The same architecture was optimized to have isotropic posture in the center of its workspace with 3 or 5 degrees of freedom in [START_REF] Chablat | Architecture Optimization of a 3-DOF Parallel Mechanism for Machining Applications, the Orthoglide[END_REF][START_REF] Caro | Kinematic Analysis and Trajectory Planning of the Orthoglide 5-Axis[END_REF]. Li et all developed a 3-UPU parallel mechanism (U is universal joint) [START_REF] Li | Kinematics of a Special 3-DOF 3-UPU Parallel Manipulator[END_REF] and analyzed the instantaneous kinematics performance of the TPM. In [START_REF] Zhao | Kinematics analysis of a three dimensional mobile parallel platform mechanism[END_REF], the authors suggested a 3-RRC TPM and developed the forward and inverse solution equation (C is cylindrical joint).

Considering the anisotropy of kinematics, Zhao et al [START_REF] Zhao | Dimensional synthesis of a three translational degrees of freedom parallel robot while considering kinematic anisotropic property[END_REF] analyzed the dimension synthesis and kinematics of the 3-DOF TPM based on the Delta PM. Zeng et.al [START_REF] Zeng | Tri-pyramid Robot: Design and kinematic analysis of a 3-DOF translational parallel manipulator[END_REF][START_REF] Qiang | Tri-pyramid Robot: stiffness modeling of a 3-DOF translational parallel manipulator[END_REF][START_REF] Lee | Error modeling for sensitivity analysis and calibration of the tri-pyramid parallel robot[END_REF] introduced a 3-DOF TPM called as Tri-pyramid robot and presented a more detailed analytical approach for the Jacobi matrix. G. Bhutani et al [START_REF] Bhutani | Novel design solution to high precision 3 axes translational parallel mechanism[END_REF] established a new design for the 3-UPU mechanism by taking into account mathematical models. Gao et al [START_REF] Li | A three-DOF translational manipulator with decoupled geometry[END_REF] developed a TPM with decoupled motion. However, the most previous TPMs generally suffer from two major problems: i) the degree of coupling κ of the mechanism is greater than zero, which means its forward position solution is generally not analytical, and ii) the mechanism does not have input-output decoupling characteristics [START_REF] Shen | Kinematic decoupling design method and application of parallel mechanism[END_REF], which leads to the complexity of motion control and path planning.

According to topology design theory of PM based on position and orientation characteristics (POC) equations [START_REF] Yang | Topology design of robot mechanism[END_REF], a new TPM is proposed in this paper. The TPM is designed with simple structure and features zero coupling degree, which makes it possible to produce analytical models for the direct and inverse kinematic model. In addition, TPM's motion control and trajectory planning is easier thanks to its partially decoupled motion characteristics.

STRUCTURAL DESIGN

The 3T parallel manipulator proposed in this paper is illustrated in Fig. 1. The base platform 0 is connected to the moving platform 1 by two hybrid simple opened chain (HSOCs).

To illustrate the structure of the manipulator, the CAD design of the hybrid chain is shown in Fig. 1

Analysis of the POC set

The POC set equations for serial and parallel mechanisms are expressed respectively as follows:

1 bi Ji m i MM   I (1) 1 Pa bi n i MM   I (2)
where Ji M -POC set generated by the i-th joint; 3 bi M -POC set generated by the end link of i-th branched chain; Pa M -POC set generated by the moving platform of PM.

The topological architecture of the two HSOCs of the mechanism can be denoted as:

 )} || ( ) || || ( ) ( R { 13 12 1 2 22 3 2 4S 4S 4S 4S 11 1 R R R R R R R P P HSOC          (4 ) 2 31 32 33 () R P HSO R CR R PP .
The POC sets of the end of the two HSOCs are determined according to Eqs.( 1) and (2) as follows:

        1 1 1 1 11 1 1 1 11 2 2 3 23 12 1 1 1 21 12 12 ( ) ( ( )) ( ) ( ( )) ( ) ( ) ( ) ( ) ( ) HSOC t R t abcd t ab M r R r ab r bd t R t R t r R r R r R                                             P PP P P P 2 1 1 1 1 31 32 33 32 1 1 1 0 31 32 33 3 1 31 ( ) ( ) ( ) () ( ) ( ) ( ) () HSOC t R t R t R tR M r R r R r R r t rR                             

Determining the DOF

The general and full-cycle DOF formula for PMs proposed in author's work [START_REF] Yang | Topology design of robot mechanism[END_REF] is given:

11 m i ij Lj v Ff    (3a) 1 1 ( 1) dim i j L v j i j b b j MM             I (3b)
where, F -DOF of PM. f i -DOF of the i th joint. m -number of all joints of the PM. v -number of independent loops (v=m-n+1, n -number of links).

Lj

 -number of independent equations of the j th loop. The mechanism can be broken down into two independent loops, and their constraint equations are calculated as follows:

 The first independent loop is consisted of branch I and II, the HSOC 1 is deduced as:

  21 22 23 4S 4S 4S 4S 11 1 || || ( R R R R R P P R HSOC    
Thus, as obtained from Eq. ( 2), the POC set of the sub PM is

2 22 21 (1 2) 1 20 21 d ( ) im ) ( Pa I II t tt M M M r r R r R                            P (4)
In accordance with Eq. (3), the independent constraint equation numbers 1 L  and the DOF, respectively, can be obtained as follows:

 

( ) ( ) dim dim ( ) ( ) ( ) dim 5 ( ( )) , L III t R t R t M r R r t R R r R rR                                                                 P P P P   1 11 8 6 (6 5) 3 j m iL ij F f           .

Determining the coupling degree

Definition of coupling degree

According to the composition principle of mechanism based on SOC units, any PM can be decomposed into a series of Assur kinematics chains (AKCs), and an AKC with v independent loops can be broken down into v single-openchains (SOC). The constraint of the j th SOC is defined by 0 1 5, 4, 2, 1 0 1, 2, 3,...

j j j m j i j L j i j fI                           (4)
where, m j -number of joints contained in the j th SOC j . i f -DOF of the i th joints; J I -number of actuated joints in the j th SOC j ; j L  -number of independent equations of the j th loop.

Then, the coupling degree of AKC is

           v j j in 1 m 2 1 (5)
The physical meaning of the coupling degree κ can be explained as follows:

 The coupling degree κ reflects the correlation and dependence between kinematic variables of each independent loop of the mechanism. It has been proved that the higher κ, the greater the complexity of the kinematic and dynamic problems of the mechanism will  For 0   , the motion of each loop can be obtained independently, and we can finally obtain the solution of the direct kinematic model; If 0   , this means that the direct kinematic model must be solved using several constraint equations.

Calculation of coupling degree

The independent displacement equations ( 1, 2) Li i

 

of loop 1 and loop 2 have been calculated in the previous section, i.e., Both z and w axis are determined by the right hand Cartesian coordinate rule, as shown in Fig. 3(A). For ease of comprehension, the 3T PM is redesigned in 3D view as shown in Fig. 3(B).

The length of each actuated arms 2, 5 and 12 is

1 1 2 2 3 3 3 3 1 with B B A B l A l Al    
The length of driven links 6 and the length of longer links of the both parallelograms 4 and 11 are

1 1 2 2 3 3 4 C B C B B Cl   
The shorter links of the parallelogram are 2l 5 and the point B i and C i (i=1, 2, 3) are the midpoint of the short edge. Thus, the parameters of the other links are respectively defined below. C Dl  , 1 1

3 3 1 1 6 D D E C Fl C   , 1 1 7 D El  .
The three input angles are defined as θ 1 , θ 2 and θ 3 , as shown in The coordinates of each end-point of the three actuated arms 2, 5 and 12, i.e., B 1 , B 2 and B 3 are easily calculated as

1 1 3 1 3 1 [0 cos sin ] T B l l l   , 2 1 3 2 3 2 [ 0 sin ] T l cos l Bl    , 3 1 3 3 3 3 [ 0 sin ] T l co l B s l   
As stated in Eq. ( 4), the output link 3' of the can only produce two dimensional translational motions in the xoz plane, i.e.

            1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 2 2 2 2 4 C B C B C B C B C B C B x x y y z z l x x y y z z l                  (6)
Equation ( 6) leads to:

11 1 1 1 CC a z b xc   ( 7
)
Where 2 [START_REF] Bhutani | Novel design solution to high precision 3 axes translational parallel mechanism[END_REF] 2( 2)

B x a l  , ) ( 2 1 2 B B 1 z z b   , 2 2 1 1 2 2 2 2 15 2 () B B B B c l z y x z      If a 1 =0 and b 1 =0, then 0 -2 B 1 1   y c . But 1 B
y could not be zero. Hence, a 1 and b 1 are not zero at the same time, and we can have two cases as follows:

 1 0 a  . In such a case, we have

1 1 1 1 1 11 2 2 2 4 / () C C B C B cb l z z y z x            (8a)
 1 0 a  . In such a case, we have d ab ,

1 2 1 1 1 1 2( ) B b c z a e  , 11 2 2 2 2 2 1 1 4 1 () BB a y z l c f     .

Direct kinematics of AKC 2

From the Fig. 3, the coordinates of 1      

D , 1 E , 1 F , 3 D ,
3 3 3 3 3 3 2 2 2 2 4 B C B C B C x y z x l yz       (9)
Equation ( 9) leads to

22 cos b a   (10) With   13 2 7 2 2 2 DB a x x l l   , 1 3 1 3 2 2 2 2 2 4 7 2
( 2 ) ( )

D B D B b x x l z l z l      
After  is obtained by Eq. ( 10), the coordinate of 3 D and 1

F can be obtain. Thus, the coordinates of P on the platform is

      31 31 31 /2 /2 /2 DF DF DF xx yy zz x y z              (11)

Inverse kinematics

To solve the inverse kinematics, we compute the values of θ 1 , θ 2 and θ 3 as a function of the coordinate P of the moving platform. We have also to evaluate the value of β, which is a passive variable form the Eq. [START_REF] Zhao | Kinematics analysis of a three dimensional mobile parallel platform mechanism[END_REF].

Solution for

For a position of the moving platform, there are two positions for points [START_REF] Liu | Kinematic and workspace analysis of a new type of spatial 3-DOF parallel mechanism[END_REF] C and 2 C with Then, due to the link length constraints defined by

1 1 2 2 4 CC B B l  , i.e.             1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 2 2 2 2 4 C B C B C B C B C B C B x x y y z z l x x y y z z l                  (12)
Then, Eq. ( 12) leads to:

  22 33 2 arctan (2 4 )/ i i i i i i l z l h g zg    , for i=1, 2 (13)
where

1 1 C z z  , 2 2 C z z  , 11 2 2 2 2 1 1 3 4 () CC h l l x z l      , 11 2 2 2 2 1 1 3 4 () CC g l l x z l      , 1 1 2 2 2 2 2 2 1 3 4 1 3 ( ) 2 ( ) C C C h l l x z x l ll       , 1 1 2 2 2 2 2 2 1 3 4 1 3 ( ) 2 ( ) C C C g l l x z x l ll      .

Solution for θ 3

Similarly, the coordinates of 3 D and 3 C can be easily obtained

  32 ,, T D x l y z   ,   3 2 6 ,, T C x l y z l    .
Due to the link length constraints defined by 3 3 4 B Cl  , the constraint equation is established as below.



    

3 3 3 3 3 3 2 2 2 2 4 C B C B C B x x y y z z l       (14)
From Eq. ( 14), we can evaluate 3  as following   We can conclude that this robot has 16 solutions to the inverse kinematic model, twice as many as the Delta robot. This is due to the mobile platform which is made with two parts. For the same position of P , there are two values for  and for each  value, there are two values for θ 1 and θ 2 . However, there are only two θ 3 values in total. The number of solutions in the inverse kinematic model is the product of 4 by 4 by 2, i.e. 16 solutions.

SINGULARITY ANALYSIS

The singularity analysis of parallel robots has been welldocumented in the literature. We can find the parallel and serial Jacobian matrix, named A and B respectively [START_REF] Gosselin | Singularity analysis of closedloop kinematic chains[END_REF][START_REF] Wenger Ph | Definition sets for the direct kinematics of parallel manipulators[END_REF], by differentiating the constraint equations with respect to time.

Then we obtain the parallel and serial singularities by studying det(A) and det(B) , respectively [START_REF] Chablat | Working modes and aspects in fully-parallel manipulator[END_REF].

CONCLUSIONS

In this article, a new asymmetrical new parallel mechanism of pure parallel translation with a degree of zero coupling and complete motion decoupling has been proposed. The inverse and direct kinematic models are obtained. The mechanism is simple in structure and easy to manufacture, which can be used in transportation, positioning and other operations of the manufacturing industry. The study of singular configurations is simple because of the independence of the kinematic chains.

  (A). Two HSOCs composed of series of links are shown in Fig. 1(B). The first HSOC includes the branches I and II: The shorter link 3 of a parallelogram composed of four spherical pairs ( a S , b S , c S and d S ) is connected at point a, by actuated arm 2, to the base 0 by a revolute joint 11 R . The extended part of the opposite link 3' of the parallelogram is connected in parallel with a sub-chain composed of two links(6 and 5) and three parallel revolute joints (3R, i.e., two links (7 and 8) and two parallel revolute joints (2R, i.e., 12 13 || RR ) are connected in series with the link 3', which are connected to the moving platform 1. Because the two revolute joints(2R, i.e.part of which is part of the moving platform 1 and produce three translations and one rotation. The second HSOC includes the branch III: The parallelogram composed of four revolute joints ( e R , f R , g R and h R ) is connected in series with two parallel revolute joints (2R, i.e., 32 31 || RR ), and is connected to the moving platform which produce three translations and one rotation. It is obvious that the HSOC has the same branch as the typical Delta (but the Delta contains three such complex HSOCs). (A) 3D CAD DESIGN (B) KINEMATIC STRUCTURE FIGURE 1. 3D CAD DESIGN AND KINEMATIC STRUCTURE OF 3T ROBOT TOPOLOGICAL ANALYSIS

  moving platform 1 of the PM produces a pure translation motion.

  generated by the sub-PM formed by the former j branches;( 1) bj M  -POC set generated by the end link of j+1sub-chains.



  By Eq.(5), it can be seen that the mechanism contains two AKCs with zero coupling degree (κ 1 =0, κ 2 =0), hence, the closed-form forward position solution of the PM can be easily obtained by AKC 1 and AKC 2 . , the mechanism shown in Fig.1is developed under a planar view in the Fig. 2. The base 0 and the moving platform 1 are square shaped, and the sizes of their sides are 1 2l and 2 2l , respectively. The three revolute joints (3R, i.e., 11 R , 21 R and 31 R ) of the base 0 are distributed at the midpoint of each edge.

FIGURE 2 .

 2 FIGURE 2. A PLANAR VIEW OF THE 3 ROBOTS Without losing the generality, the frame coordinate system O-xyz is established on the base 0. The x-axis is parallel to the axis of 11 R , and the y-axis is perpendicular to the axis of 11 R . On the moving platform 1, the moving coordinate system uvw is established at P. The u axis and the v axis are perpendicular and parallel to the axis of the 33 R respectively.

Fig. 3 (

 3 Fig. 3(a). That is, the angle between the vectors 11A B and the

  can be deduced as below.

For 2 or 3 / 2 

 232 the mechanism studied, the singularities are similar to those of a Delta robot. A new singular configuration exists because the mobile platform is made in two parts. When β is equal to /, the robot admit a new singular configuration because the determinant of the parallel Jacobian matrix A vanished. This defines in the workspace, a vertical plane passing through the origin and parallel to the plane (xz).CASE STUDYTo illustrate this study, we refer to the dimension parameters of the ABB robot 14R, i.e, 1 300 l  l  . Its parallel and serial singularity can be computed with the Siropa library[START_REF] Siropa | Algebraic and robotic functions[END_REF]. The determinant of matrix A can be written base elimination methods, these can be plotted in the Cartesian space. Figure4depicts the singular configurations in the Cartesian space and Fig.5the serial singularities with a different color for each term of the Eqs (15) and (16).

FIGURE 4 .

 4 FIGURE 4. PARALLEL SINGULARITIESOF THE ROBOT

ACKNOWLEDGMENTS:

This research is sponsored by the NSFC (Grant No. 51475050 and No. 51375062) and Jiangsu Key Development Project (No.BE2015043).


Sylvain Gandon
email: sylvain.gandon@cefe.cnrs.fr

Sepideh Mirrahimi
email: sepideh.mirrahimi@math.univ-toulouse.fr

A Hamilton-Jacobi method to describe the evolutionary equilibria in heterogeneous environments and with non-vanishing effects of mutations

In this note, we characterize the solution of a system of elliptic integro-differential equations describing a phenotypically structured population subject to mutation, selection and migration. Generalizing an approach based on Hamilton-Jacobi equations, we identify the dominant terms of the solution when the mutation term is small (but nonzero). This method was initially used, for different problems from evolutionary biology, to identify the asymptotic solutions, while the mutations vanish, as a sum of Dirac masses. A key point is a uniqueness property related to the weak KAM theory. This method allows to go further than the Gaussian approximation commonly used by biologists and is an attempt to fill the gap between the theories of adaptive dynamics and quantitative genetics.

Résumé

Méthode Hamilton-Jacobi pour décrire des équilibres évolutives dans les environnements hétérogènes et avec des mutations non-évanescentes. Dans cette note, nous étudions un système d'équations intégrodifférentielles elliptiques, décrivant une population structurée par trait phénotypique soumise à des mutations, à la sélection et à des migrations. Nous généralisons une approche basée sur des équations de Hamilton-Jacobi pour détérminer les termes dominants de la solution lorsque les effets des mutations sont petits (mais non-nuls). Cette méthode était initialement utilisée, pour différents problèmes venant de la biologie évolutive, pour identifier les solutions asymptotiques, lorsque les effets des mutations tendent vers 0, sous forme des sommes de masses de Dirac. Un point clé est une propriété d'unicité en rapport avec la théorie de KAM faible. Cette méthode nous permet d'aller au-delà des approximations Gaussiennes habituellement utilisées par les biologistes et contribue ainsi à relier les théories de la dynamique adaptative et de la génétique quantitative.

Introduction

During the last decade, an approach based on Hamilton-Jacobi equations with constraints has been developed to describe the asymptotic evolutionary dynamics of phenotypically structured populations, in the limit of vanishing mutations. The mathematical modeling of such phenomena leads to parabolic (or elliptic for the steady case) integro-differential equations, whose solutions tend, as the diffusion term vanishes, to a sum of Dirac masses, corresponding to dominant traits. These asymptotic solutions can be described using the Hamilton-Jacobi approach. There is a large literature on this method. We refer to [START_REF] Diekmann | The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach[END_REF][START_REF] Perthame | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF][START_REF] Mirrahimi | Concentration phenomena in PDEs from biology[END_REF] for the establishment of the basis of this approach for problems from evolutionary biology. Note that related tools were already used in the case of local equations (for instance KPP type equations) to describe the propagation phenomena (see for instance [START_REF] Freidlin | Geometric optics approach to reaction-diffusion equations[END_REF][START_REF] Evans | A PDE approach to geometric optics for certain semilinear parabolic equations[END_REF]).

In almost all the previous works, the Hamilton-Jacobi approach has been used to describe the limit of the solution, corresponding to the population's phenotypical distribution, as the mutations' steps vanish. However, from the biological point of view, it is sometimes more relevant to consider non-vanishing mutation's steps. A recent work [START_REF] Mirrahimi | Uniqueness in a class of Hamilton-Jacobi equations with constraints[END_REF] has pointed out that such tools can also be used, for a simple model with homogeneous environment, to characterize the solution while mutations' steps are small but nonzero. In this note, we show how such results can be obtained in a more complex situation with a heterogeneous environment.

Our purpose in this note is to study the solutions to the following system, for z ∈ R,

       -ε 2 n ′′ ε,1 (z) = n ε,1 (z)R 1 (z, N ε,1 ) + m 2 n ε,2 (z) -m 1 n ε,1 (z), -ε 2 n ′′ ε,2 (z) = n ε,2 (z)R 2 (z, N ε,2 ) + m 1 n ε,1 (z) -m 2 n ε,2 (z), N ε,i = R n ε,i (z)dz, for i = 1, 2, (1)
with

R i (z, N i ) = r i -g i (z -θ i ) 2 -κ i N i , with θ 1 = -θ and θ 2 = θ.
(2) This system represents the equilibrium of a phenotypically structured population under mutation, selection and migration between two habitats. For more details on the modeling and the biological motivations, see Section 2.

Note that the asymptotic behavior, as ε → 0 and along subsequences, of the solutions to this system, under the assumption m i > 0, for i = 1, 2, and for bounded domains, was already studied in [START_REF] Mirrahimi | Migration and adaptation of a population between patches[END_REF]. In the present work, we go further than the asymptotic limit along subsequences and we obtain uniqueness of the limit and identify the dominant terms of the solution when ε is small but nonzero.

The main elements of the method:

To describe the solutions n ε,i (z) we use a WKB ansatz

n ε,i (z) = 1 √ 2πε exp u ε,i (z) ε .
Note that a first approximation that is commonly used in the theory of 'quantitative genetics' (a theory in evolutionary biology that investigates the evolution of continuously varying traits [START_REF] Rice | Evolutionary theory: mathematical and conceptual foundations[END_REF]-chapter 7), is a gaussian distribution of the following form

n ε,i (z) = N i √ 2πεσ exp -(z -z * ) 2 εσ 2 = 1 √ 2πε exp -1 2σ 2 (z -z * ) 2 + ε log Ni σ ε .
Here, we try to go further than this a priori gaussian assumption and to approximate directly u ε,i . To this end, we write an expansion for u ε,i in terms of ε:

u ε,i = u i + εv i + ε 2 w i + O(ε 3 ). (3)
We prove that u 1 = u 2 = u is the unique viscosity solution to a Hamilton-Jacobi equation with constraint. The uniqueness of the viscosity solution to such Hamilton-Jacobi equation with constraint is related to the uniqueness of the Evolutionary Stable Strategy (ESS), see Section 3 for definition and for the result on the uniqueness of the ESS, and to the weak KAM theory [START_REF] Fathi | Weak Kam Theorem in Lagrangian Dynamics[END_REF]. In section 4, we compute explicitly u which indeed satisfies max R u(z) = 0, with the maximum points attained at one or two points corresponding to the ESS points of the problem.

We then notice that, while u(z) < 0, n ε,i (z) is exponentially small. Therefore, only the values of v i and w i close to the zero level sets of u matter, i.e. the ESS points. In section 5, we provide the main elements to compute formally v i and hence its second order Taylor expansion around the ESS points and the value of w i at those points. Then, we show, in section 6, that these approximations together with a fourth order Taylor expansion of u around the ESS points are enough to approximate the moments of the population's distribution with an error of order ε 2 . The mathematical details of our results will be provided in [START_REF] Mirrahimi | A Hamilton-Jacobi approach to characterize the evolutionary equilibria in heterogeneous environments[END_REF]. The biological applications will be detailed in [START_REF] Mirrahimi | The equilibrium between selection, mutation and migration in spatially heterogeneous environments[END_REF].

Model and motivation

The solution of (1) corresponds to the steady solution to the following system, for (t,

z) ∈ R + × R,      ∂ t n i (t, z) -ε 2 ∂ 2 ∂z 2 n i (t, z) = n i (t, z)R i (z, N i (t)) + m j n j (t, z) -m i n i (t, z), i = 1, 2, j = 2, 1, N i (t) = R n i (t, z)dz, for i = 1, 2. (4)
This system represents the dynamics of a population that is structured by a phenotypical trait z, and which lives in two habitats. We denote by n i (t, z) the density of the phenotypical distribution in habitat i, and by N i the total population's size in habitat i. The growth rate R i (z, N i ) is given by [START_REF] Day | Competition and the effect of spatial resource heterogeneity on evolutionary diversification[END_REF], where r i represents the maximum intrinsic growth rate, g i is the strength of the selection, θ i is the optimal trait in habitat i and κ i represents the intensity of the competition. The constants m i are the migration rates between the habitats. In this note we assume that there is positive migration rate in both directions, i.e.

m i > 0, i=1 ,2. (5)
However, the source and sink case, where for instance m 2 = 0, can also be analyzed using similar tools. We refer to [START_REF] Mirrahimi | The equilibrium between selection, mutation and migration in spatially heterogeneous environments[END_REF] for the analysis of this case. We additionally assume that

max(r 1 -m 1 , r 2 -m 2 ) > 0. ( 6
)
This guarantees that the population does not get extinct.

Such phenomena have already been studied by several approaches. A first class of results are based on the adaptive dynamics approach, where one considers that the mutations are very rare such that the population has time to attain its equilibrium between two mutations and hence the population's distribution has discrete support (one or two points in a two habitats model) [START_REF] Meszéna | Adaptive dynamics in a 2-patch environment: a toy model for allopatric and parapatric speciation[END_REF][START_REF] Day | Competition and the effect of spatial resource heterogeneity on evolutionary diversification[END_REF][START_REF] Fabre | Evolution of a structured population in a heterogeneous environment[END_REF]. A second class of results are based on an approach known as 'quantitative genetics', which allows more frequent mutations and does not separate the evolutionary and the ecological time scales . A main assumption in this class of works is that one considers that the population's distribution is a gaussian [START_REF] Hendry | Population mixing and the adaptive divergence of quantitative traits in discrete populations: a theoretical framework for empirical tests[END_REF][START_REF] Ronce | When sources become sinks: migration meltdown in heterogeneous habitats[END_REF] or, to take into account the possibility of dimorphic populations, a sum of one or two gaussian distributions [START_REF] Yeaman | Predicting adaptation under migration load: the role of genetic skew[END_REF][START_REF] Débarre | Quantifying the effects of migration and mutation on adaptation and demography in spatially heterogeneous environments[END_REF].

In our work, as in the quantitative genetics framework, we also consider continuous phenotypical distributions. However, we don't assume any a priori gaussian assumption. We compute directly the population's distribution and in this way we correct the previous approximations. To this end, we also provide some results in the framework of adaptive dynamics and in particular, we generalize previous results on the identification of the ESS to the case of nonsymetric habitats. Furthermore, our work makes a connection between the two approaches of adaptive dynamics and quantitative genetics.

The adaptive dynamics framework

In this section, we introduce some notions from the theory of adaptive dynamics that we will be using in the next sections [START_REF] Meszéna | Adaptive dynamics in a 2-patch environment: a toy model for allopatric and parapatric speciation[END_REF]. We also provide our main result in this framework.

Effective fitness: The effective fitness W (z; N 1 , N 2 ) is the largest eigenvalue of the following matrix:

A(z; N 1 , N 2 ) =   R 1 (z; N 1 ) -m 1 m 2 m 1 R 2 (z; N 2 ) -m 2   (7)
This indeed corresponds to the effective growth rate associated with trait z in the whole metapopulation when the total population's sizes are given by (N 1 , N 2 ). Demographic equilibrium: Consider a set of points Ω = {z 1 , • • • z m }. The demographic equilibrium corresponding to this set is given by (n 1 (z), n 2 (z)), with the total population's sizes (N 1 , N 2 ), such that

n i (z) = m j=1 α i,j δ(z -z j ), N i = m j=1 α i,j , W (z j , N 1 , N 2 ) = 0,
and such that (α 1,j , α 2,j ) T is the right eigenvector associated with the largest eigenvalue W (z j , N 1 , N 2 ) = 0 of A(z j ; N 1 , N 2 ). Evolutionary stable strategy: A set of points

Ω * = {z * 1 , • • • , z * m } is called an evolutionary stable strategy (ESS) if W (z, N * 1 , N * 2 ) = 0, for z ∈ A and, W (z, N * 1 , N * 2 ) ≤ 0, for z ∈ A, where (N * 1 , N * 2 )
is the total population's sizes corresponding to the demographic equilibrium associated with the set Ω * .

Since, there are only two habitats we expect that at most two distinct traits coexist at the evolutionary stable equilibrium. We prove indeed the following: Theorem 3.1 Assume (5)- [START_REF] Fabre | Evolution of a structured population in a heterogeneous environment[END_REF]. There exists a unique set of points Ω * which is an evolutionary stable strategy. Such set has at most two elements.

We call an evolutionary stable strategy which has one (respectively two) element, a monomorphic (respectively dimorphic) ESS. We can indeed give a criterion to have monomorphic or dimorphic ESS, and we can identify the dimorphic ESS in the general case (see [START_REF] Mirrahimi | A Hamilton-Jacobi approach to characterize the evolutionary equilibria in heterogeneous environments[END_REF] for more details).

How to compute the zero order terms u i

The identification of the zero order terms u i is based on the following result. Note that the part (ii) of the theorem below is a variant of Theorem 1.1 in [START_REF] Mirrahimi | Migration and adaptation of a population between patches[END_REF].

Theorem 4.1 Assume (5)-( 6).

(i) As ε → 0, (n ε,1 , n ε,2 ) converges to (n * 1 , n * 2 )
, the demographic equilibrium of the unique ESS of the model. Moreover, as ε → 0, N ε,i converges to N * i , the total population's size in patch i corresponding to this demographic equilibrium. (ii) As ε → 0, both sequences (u ε,i ) ε , for i = 1, 2, converge along subsequences and locally uniformly in R to a continuous function u ∈ C(R), such that u is a viscosity solution to the following equation

   -|u ′ | 2 = W (z, N * 1 , N * 2 ), in R, max z∈R u(z) = 0. (8)
Moreover, we have the following condition on the zero level set of u:

supp n * 1 = supp n * 2 ⊂ {z | u(z) = 0} ⊂ {z | W (z, N * 1 , N * 2 ) = 0}. (iii
) There exists constants (λ i , ν i ), for i = 1, 2, which can be determined explicitly from m 1 , m 2 , g 1 , g 2 , κ 1 , κ 2 and θ, such that, under the condition

r 2 = λ 1 r 1 + ν 1 , r 1 = λ 2 r 2 + ν 2 , (9)
we have

supp n * 1 = supp n * 2 = {z | u(z) = 0} = {z | W (z, N * 1 , N * 2 ) = 0}. ( 10
)
The solution of (8)-( 10) is unique and hence the whole sequence (u ε,i ) ε converge locally uniformly in R to u.

Note that a Hamilton-Jacobi equation of type [START_REF] Freidlin | Geometric optics approach to reaction-diffusion equations[END_REF] in general might admit several viscosity solutions. Here, the uniqueness is obtained thanks to [START_REF] Lions | Generalized solutions of Hamilton-Jacobi equations[END_REF] and a property from the weak KAM theory, which is the fact that the viscosity solutions are completely determined by one value taken on each static class of the Aubry set ([10], Chapter 5 and [START_REF] Contreras | Action potential and weak kam solutions[END_REF]). In what follows we assume that (9) and hence (10) always hold. We then give an explicit formula for u considering two cases:

(i) Monomorphic ESS : We consider the case where there exists a unique monomorphic ESS z * and the corresponding demographic equilibrium is given by (N * 1 δ(z * ), N * 2 δ(z * )). Then u is given by

u(z) = - z z * -W (x; N * 1 , N * 2 )dx . (11)
(ii) Dimorphic ESS : We next consider the case where there exists a unique dimorphic ESS (z * a , z * b ) with the demographic equilibrium: n i = ν a,i δ(zz * a ) + ν b,i δ(zz * b ), and ν a,i + ν b,i = N * i . Then u is given by

u(z) = max -| z z * a -W (x; N * 1 , N * 2 )dx|, -| z z * b -W (x; N * 1 , N * 2 )dx| .

How to compute the next order terms

In this section, we give the main elements to compute formally v i and the value of w i at the ESS point, with v i and w i the correctors introduced by (3), in the case of monomorphic population. For the details of the computations for both monomorphic and dimorphic populations, we refer the interested reader to [START_REF] Mirrahimi | A Hamilton-Jacobi approach to characterize the evolutionary equilibria in heterogeneous environments[END_REF].

We consider the case of monomorphic population where the demographic equilibrium corresponding to the monomorphic ESS is given by (N * 1 δ(zz * ), N * 2 δ(zz * )). One can compute, using [START_REF] Meszéna | Adaptive dynamics in a 2-patch environment: a toy model for allopatric and parapatric speciation[END_REF], a Taylor expansion of order 4 around the ESS point z * :

u(z) = - A 2 (z -z * ) 2 + B(z -z * ) 3 + C(z -z * ) 4 + O(z -z * ) 5 .
To provide an approximation of the moments of the population's distribution, we have to compute constants D i , E i and F i such that

v i (z) = v i (z * ) + D i (z -z * ) + E i (z -z * ) 2 + O(z -z * ) 3 , w i (z * ) = F i .
A first element of the computations is obtained by replacing the functions u, v i and w i by the above approximations to compute N ε,i = R n ε,i (z)dz. This leads to

v i (z * ) = log N * i √ A , N ε,i = N * i + εK * i + O(ε 2 ), with K * i = N * i 3C A 2 + E i A + F i .
Note also that writing (1) in terms of u ε,i we obtain

   -εu ′′ ε,1 (z) = |u ′ ε,1 | 2 + R 1 (z, N ε,1 ) + m 2 exp u ε,2 -u ε,1 ε -m 1 , -εu ′′ ε,2 (z) = |u ′ ε,2 | 2 + R 2 (z, N ε,2 ) + m 1 exp u ε,1 -u ε,2 ε -m 2 . (12)
A second element, is obtained by keeping the zero order terms in the first line of (12) and using [START_REF] Freidlin | Geometric optics approach to reaction-diffusion equations[END_REF] to obtain

v 2 (z) -v 1 (z) = log 1 m 2 W (z, N * 1 , N * 2 ) -R 1 (z, N * 1 ) + m 1 . (13)
The last element, is derived from keeping the terms of order ε in ( 12) which leads to

-u ′′ = 2u ′ v ′ i -κ i K * i + m j exp(v j -v i )(w j -w i ), for {i, j} = {1, 2}. (14)
The functions v i and the coefficients D i , E i and F i can be computed by combining the above elements.

Approximation of the moments

The above approximations of u, v i and w i the ESS points allow us to estimate the moments of the population's distribution. In the monomorphic case these approximations are given below:

                     N ε,i = n ε,i (z)dz = N * i (1 + ε(F i + E i A + 3C A 2 )) + O(ε 2 ), µ ε,i = 1 N ε,i zn ε,i (z)dz = z * + ε( 3B A 2 + D i A ) + O(ε 2 ), σ 2 ε,i = 1 N ε,i (z -µ ε,i ) 2 n ε,i (z)dz = ε A + O(ε 2 ), s ε,i = 1 σ 3 ε,i N ε,i (z -µ ε,i ) 3 n ε,i (z)dz = 6B A 3 2 √ ε + O(ε 3
2 ).

One can obtain similar approximations in the case of dimorphic ESS. To compute the above integrals, replacing the approximation (3) in the integrals, a natural change of variable is to take zz * = √ εy. Therefore each term zz * can be considered as of order √ ε in the integration. This is why, to obtain a first order approximation of the integrals in terms of ε, it is enough to have a fourth order approximation of u(z), a second order approximation of v i (z) and a zero order approximation of w i (z), in terms of z around z * .

Acknowledgements

S. Mirrahimi has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 639638), and from the french ANR projects KIBORD ANR-13-BS01-0004 and MODEVOL ANR-13-JS01-0009.