Florian Galliot

Refutation of the Bayer-Diaconis-McGrath conjecture for the riffle shuffle card guessing game with feedback

We consider the following card guessing game with feedback, introduced in [BD92]. An initially ordered deck of cards is shuffled via one or several riffle shuffles (or more generally: one a-shuffle).

The player guesses the card on top of the deck, then looks at that card. The player then guesses the next card, looks at that card etc. until there is no card left, and his goal is to get as many correct guesses as possible. The authors detail a simple guessing strategy conjectured to be optimal. We show that this strategy is optimal in the case of a single riffle shuffle but not in general. The present note was sent to

1 Notations used 1.1 General notations for a-shuffles Notation. We denote by P a (n) the set of all possible values p = (p 1 , . . . , p a ) of the multinomial distribution with parameters (n, 1 a , . . . , 1 a ). We see P a (n) as the set of all ways to cut a deck of n cards into a packets. Definition. Let p ∈ P a (n). We call a separation between two packets in p a cut. There are a -1 cuts in total. The location of a cut is the number between 0 and n of the card that is just above the cut (a cut at the very top of the deck is at location 0).

Example. The set P 3 (3) contains 10 elements pictured as follows, with all cuts in red : Notation. We denote by C a (n) the set of all couples (p, σ) where p ∈ P a (n) and σ is p-compatible.

We will exclusively use the maximum entropy description of a-shuffles : let (P, S) be uniform on C a (n), so that S is an a-shuffle.

Our probability space is denoted by (Ω, F , P), and expectation relative to P will be denoted by E.

Specific notations for the card guessing problem with feedback

A deck of n cards is in arrangement S, where S is an a-shuffle. Suppose that m cards have already been revealed, so that we're trying to make the best possible guess for S(m + 1). The "Bayer-Diaconis-McGrath strategy", or "BDM strategy" in short, refers to the strategy described in [START_REF] Bayer | Trailing the dovetail shuffle to its lair[END_REF] where the card guessed is the topmost card of a longest sequence of remaining consecutive cards.

Example. Suppose n = 8, with card 6 revealed first and card 2 revealed second (m = 2). The BDM strategy chooses card 3 next. This situation is pictured as follows, with revealed cards appearing checked : Let i = (i 1 , . . . , i m ) be the vector of all cards that have already been revealed successively.

Notation. We denote by C i a (n) the set of all couples (p, σ) ∈ C a (n) such that σ(1) = i 1 , . . . , σ(m) = i m , and by P i a (n) the set of all p ∈ P a (n) such that there exists a p-compatible permutation σ ∈ S n such that σ(1) = i 1 , . . . , σ(m) = i m .

Notation. Let P

i := P( • | S(1) = i 1 , . . . , S(m) = i m ).
We denote by E i the corresponding expectation.

Remark. Our goal is therefore to identify a card i m+1 that maximizes P i (S(m + 1) = i m+1 ), where (P, S) is uniform on C i a (n) under P i .

Notation. We denote by π = (π 1 , . . . , π q } the ordered partition of {1, . . . , n} \ {i 1 , . . . , i m } consisting of all sequences of remaining consecutive cards from top to bottom. Let r k := |π k |.

Example. In the previous example, we have i = (6, 2), π 1 = {1}, π 2 = {3, 4, 5}, π 3 = {7, 8}.

Definition. A cut is said to be mandatory if P contains a cut at this location P i -a.s. If P contains several cuts at a mandatory location, only one of these cuts is considered as mandatory.

Remark. Note that P i a (n) is the set of all p ∈ P a (n) that contain all mandatory cuts.

Example. Suppose i = (6, 1, 5), then the mandatory cuts are at locations 5 and 4.

Definition. Let p ∈ P i a (n). A cut is said to be in π k if it is adjacent to π k and non-mandatory. This means that if π k = {j, j + 1, . . . , j + r k -1}, then the cuts in π k are all the cuts at locations j -1, j, . . . , j + r k -1 except for the one mandatory cut at location j + r k -1 if there is one. Denoting by b -1 the number of cuts in π k , we say that π k contains b packets. Notation. Let p ∈ P i a (n). We denote by p |π k the restriction of p to π k , i.e. the element of P b (r k ) obtained from p by keeping only π k and the b -1 cuts in π k . Cuts that are not in π k (k fixed) will be denoted by p \ π k , and cuts that are in none of the π k will be denoted by p \ π.

Remark. We will occasionally identify p with an (a -1)-tuple of numbers between 0 and n representing the locations of all cuts in p. The same goes for p \ π k or p \ π.

Notation. Let (p, σ) ∈ C i a (n).
The relative order in σ of the elements of π k , renumbered from 1 to r k in increasing order, will be denoted by σ π k ∈ S r k .

Example. Suppose n = 8, a = 9, i = (3, 7), hence π 1 = {1, 2}, π 2 = {4, 5, 6}, π 3 = {8}. Let p be as follows, where both mandatory cuts appear in black :

p = 8 7 6 5 4 3 2 1 d d d d p |π1 = p |π2 = p |π3 = 8 6 5 4 2 1 ∈ P 2 (2) (π 1 contains 2 packets) ∈ P 5 (3) (π 2 contains 5 packets) ∈ P 3 (1) (π 3 contains 3 packets)
The lone p-compatible permutation for π 1 is 1 2. Any relative order of the cards 4,5,6 is possible, therefore the p-compatible permutations for π 2 are 1 2 3, 1 3 2, 2 1 3, 2 3 1, 3 1 2, 3 2 1. For example, let σ = 3 7 6 1 4 5 2 8, note that (p, σ) ∈ C i 9 (8). The cards of π 1 are in order "1 2", so that σ π1 = 1 2. The cards of π 2 are in order "6 4 5", so that σ π2 = 3 1 2.

Notation. We denote by j k the topmost card of π k . Let p ∈ P a (n) : we denote by h p (j k ) the size of the packet containing j k once the first m cards are removed.

Example. In the previous example, we have h p (j 2 ) = 1. Indeed, the initial packet containing card 4 was {3, 4}, but card 3 has been revealed already.

Remark. We have

P i (S(m + 1) = j k ) = E i (h P (j k )) n -m for all k.

Refutation of the BDM conjecture

It is possible to prove the following :

(i) The optimal card is always the topmost card of one of the π k . In particular, the BDM strategy is optimal for a = 2.

(ii) Choosing the topmost card of a largest π k is not always optimal, as shown by a counterexample with exact computation for a ∈ {3, 4}.

We only detail the proof of (ii) here. More precisely, we prove the following result :

Theorem. For a = 3 (resp. a = 4), and for all n ≥ 12 (resp. n ≥ 13), the BDM strategy is not optimal.

Explicit formulas

The following theorem -apart from the independence part which we don't use -is key to both (i) and (ii) : fixing the number a k of packets in each π k as well as all cuts outside π leads to each π k being arranged following an a k -shuffle.

Theorem 2.1. Let p 0 be a tuple of numbers representing the locations of all cuts that are in none of the π k , and a = (a 1 , . . . , a q ) be a vector of integers. Let A p0,a := {P \ π = p 0 } ∩ q k=1 {π k contains a k packets }.

Then, under P i and given A p0,a : (P |π1 , S π1 ), . . . , (P |πq , S πq ) are uniform on C a1 (r 1 ), . . . , C aq (r q ) respectively, and independent. In particular, under P i and given A p0,a , the S π k are independent a k -shuffles.

Proof.

Let E p0,a := {(p, σ) ∈ C i a (n) | p \ π = p 0 and π k contains a k packets ∀ k} and F a := q k=1 C a k (r k ). Let f : E p0,a -→ F a (p, σ) -→ ((p |π1 , σ π1 ), . . . , (p |πq , σ πq ))
: under P i and given A p0,a , (P, S) is uniform on E p0,a , therefore it suffices to show that all elements of F a have the same number of inverse images by f . Let y = ((p 1 , σ 1 ), . . . , (p q , σ q )) be in F a and x = (p, σ) ∈ E p0,a be an inverse image of y. The choice of p is unique because it is forced by p 0 , p 1 , . . . , p q . Moreover, σ(1) = i 1 , . . . , σ(m) = i m , and the relative order of the cards in π k is forced by σ k for all k : this means that choosing σ comes down to choosing the r k positions of the cards in π k for all k. In conclusion, the number of choices for x is n -m r 1 , . . . , r q , which does not depend on y.

The idea is to compute P i (S(m + 1) = j k ) for all k, using the complete system of events (A p0,a ) p0,a :

P i (S(m + 1) = j k ) = E i (h P (j k )) n -m = 1 n -m p0,a E i (h P (j k ) | A p0,a )P i (A p0,a ) (1)
Proposition 2.3 gives a formula for E i (h P (j k ) | A p0,a ), whereas Proposition 2.4 deals with P i (A p0,a ).

Lemma 2.2. Let (X 1 , . . . , X a ) follow the multinomial distribution with parameters (n, 1 a , . . . , 1 a ). Let

T := min{s ∈ {1, . . . , a} | X s > 0}. Then E(X T ) = n a n a s=1 s n-1 .
Proof.

E(X T ) = a s=1 E(X s 1 {X1=...=Xs-1=0} ) = a s=1 E(X s | X 1 = . . . = X s-1 = 0)P(X 1 = . . . = X s-1 = 0), where E(X s | X 1 = . . . = X s-1 = 0) = n a-s+1 and P(X 1 = . . . = X s-1 = 0) = a-s+1 a n .
Proposition 2.3. With notations as in Theorem 2.1, for all k :

E i (h P (j k ) | A p0,a ) = r k a r k k a k s=1 s r k -1 .
Proof. This follows immediately from Theorem 2.1 and Lemma 2.2, since the packet containing j k in P after the first m cards are revealed is precisely the first non-empty packet in P |π k .

Proposition 2.4. With notations as in the proof of Theorem 2.1 :

|E p0,a | = n -m r 1 , . . . , r q a r1 1 • • • a rq q .
Proof.

Since p \ π is forced (equal to p 0 ), choosing p comes down to choosing how each π k is cut into a k packets. Once p is fixed, choosing σ with σ(1) = i 1 , . . . , σ(m) = i m comes down to choosing positions for the cards of each of the a 1 + . . . + a q packets. This yields :

|E p0,a | = x1,1+...+x1,a 1 =r1 • • • xq,1+...+xq,a q =rq n -m x 1,1 , . . . , x 1,a1 , . . . , x q,1 , . . . , x q,aq = x1,1+...+x1,a 1 =r1 • • • xq,1+...+xq,a q =rq r 1 x 1,1 , . . . , x 1,a1 • • • r q x q,1 , . . . , x q,aq n -m r 1 , . . . , r q = n -m r 1 , . . . , r q   x1,1+...+x1,a 1 =r1 r 1 x 1,1 , . . . , x 1,a1   • • •   xq,1+...+xq,a q =rq r q x q,1 , . . . , x q,aq   = n -m r 1 , . . . , r q a r1 1 • • • a rq q .

Our counterexample for a ∈ {3, 4}

Theorem 2.5. For a = 3, and for all n ≥ 12, the BDM strategy is not optimal.

Proof.

The first counterexample appears at m = 1. Let i := i 1 to simplify notations. If i ∈ {1, n} then (i) shows that the strategy chooses optimally, so suppose 2 ≤ i ≤ n -1. We get q = 2, π 1 = {1, . . . , i -1}, r 1 = i -1, π 2 = {i + 1, . . . , n}, r 2 = n -i. Necessarily P \ π = (i -1) : this is the mandatory cut. Therefore, there are only two terms in the sum (1), corresponding to whether the second and final cut is in π 1 or in π 2 .

-First case : the second cut is in π 1 , i.e. p 0 = (i -1) and a = (2, 1). Proposition 2.4 ensures that there are 2 i-1 n-1 i-1 such combinations. Over all these, the packet containing card i + 1 is always of size n -i, and the average size of the packet containing card 1 is i-1

2 i-1 (1 + 2 i-2
) according to 2.3.

-Second case : the second cut is in π 2 , i.e. p 0 = (i -1) and a = (1, 2). Proposition 2.4 ensures that there are 2 n-i n-1 i-1 such combinations. Over all these, the packet containing card 1 is always of size i -1, and the average size of the packet containing card i + 1 is

n-i 2 n-i (1 + 2 n-i-1 ) according to 2.3.
Thus, we obtain :

E i (h P (1)) = 2 i-1 n-1 i-1 i-1 2 i-1 (1 + 2 i-2 ) + 2 n-i n-1 i-1 (i -1) 2 i-1 n-1 i-1 + 2 n-i n-1 i-1 = (i -1)(2 i-2 + 2 n-i + 1) 2 i-1 + 2 n-i , E i (h P (i + 1)) = 2 i-1 n-1 i-1 (n -i) + 2 n-i n-1 i-1 n-i 2 n-i (1 + 2 n-i-1 ) 2 i-1 n-1 i-1 + 2 n-i n-1 i-1 = (n -i)(2 i-1 + 2 n-i-1 + 1) 2 i-1 + 2 n-i .
• For n even, take i = n 2 + 1. We get r := r 1 = n 2 , r 2 = r -1 < r 1 , and the previous formulas yield :

P i (S(2) = 1) = r(2 r + 1) (n -1)3.2 r-1 , P i (S(2) = n 2 + 2) = (r -1)(2 r + 2 r-2 + 1) (n -1)3.2 r-1 .
It is easily verified that P i (S(2) = 1) < P i (S(2) = n 2 + 2) if and only if r ≥ 6 i.e. n ≥ 12. Since r 1 > r 2 , the BDM strategy chooses card 1 for all n, therefore it is suboptimal if n ≥ 12.

• For n odd, take i = n+3 2 . We get r := r 1 = n+1 2 , r 2 = r -2 < r 1 , and the previous formulas yield :

P i (S(2) = 1) = r(2 r-1 + 2 r-2 + 1) (n -1)5.2 r-2 , P i (S(2) = n+5 2 ) = (r -2)(2 r + 2 r-3 + 1) (n -1)5.2 r-2 .
It is easily verified that P i (S(2) = 1) < P i (S(2) = n+5 2 ) if and only if r ≥ 7 i.e. n ≥ 13.

Example. For n = 12, a = 3, i = (7), the BDM strategy opts for card 1 : given that the top card of the deck is card 7, the next one has a 38.35% chance of being card 8 but only a 36.93% chance of being card 1, despite the sequence of cards starting with 1 being longer. These results are comforted by the following histogram, which shows frequencies for the value of the second card based on 1,000,000 trials of a 3-shuffle on 12 cards conditioned on having card 7 as the top card : Remark. A consequence of this counterexample is that an optimal strategy must depend on a in general, contrary to the BDM strategy. Indeed, if n = 12 and i = (7), then the optimal guess is different if a = 2 (card 1) or if a = 3 (card 8).

Theorem 2.6. For a = 4, and for all n ≥ 13, the BDM strategy is not optimal.

Proof.

The easiest counterexample appears at m = 2, but there also exists some for m = 1 as will be shown in our last section.

• For n even, take i = ( n 2 + 1, n). We get q = 2, π 1 = {1, . . . , n 2 }, r := r 1 = n 2 , π 2 = { n 2 + 2, . . . , n -1}, r 2 = r -2 < r 1 . There are two mandatory cuts here, at locations n 2 and n -1. This is almost identical to the case n odd for a = 3, except that the sum contains a third term corresponding to the possibility that the non-mandatory cut is at location n. We obtain after computation :

E i (h P (1)) = 2 r n-2 r r 2 r (1 + 2 r-1 ) + 2 r-2 n-2 r r + n-2 r r 2 r n-2 r + 2 r-2 n-2 r + n-2 r , E i (h P ( n 2 + 2)) = 2 r n-2 r (r -2) + 2 r-2 n-2 r r-2 2 r-2 (1 + 2 r-3 ) + n-2 r (r -2) 2 r n-2 r + 2 r-2 n-2 r + n-2 r ,
which yields :

P i (S(2) = 1) = r(2 r-1 + 2 r-2 + 2) (n -2)(5.2 r-2 + 1) , P i (S(2) = n 2 + 2) = (r -2)(2 r + 2 r-3 + 2) (n -2)(5.2 r-2 + 1) .
It is easily verified that P i (S(2) = 1) < P i (S(2) = n 2 + 2) if and only if r ≥ 7 i.e. n ≥ 14.

• For n odd, take i = ( n+1 2 , n). We get q = 2, π 1 = {1, . . . , n-1 2 }, r := r 1 = n-1 2 , π 2 = { n+3 2 , . . . , n-1}, r 2 = r -1 < r 1 . A similar reasoning yields :

P i (S(2) = 1) = r(2 r + 2) (n -2)(3.2 r-1 + 1) , P i (S(2) = n+3 2 ) = (r -1)(2 r + 2 r-2 + 2) (n -2)(3.2 r-1 + 1) , hence P i (S(2) = 1) < P i (S(2) = n+3 2 ) if and only if r ≥ 6 i.e. n ≥ 13.

Example. For n = 13, a = 4, i = (7, 13), the BDM strategy opts for card 1 :

Some error regions of the BDM strategy

In this section, we take a look at the case m = 1 and a ∈ {3, 4} to see which values of the first card lead the BDM strategy to be suboptimal for the second card.

• The case m = 1, a = 3. We use our formulas from the proof of Theorem 2.5. Denoting i = (i) where i varies from 2 to n -1, the probabilities P i (S(2) = 1) and P i (S(2) = i + 1) are represented below for n = 52 : hold for any n large enough. For n = 52, the biggest error happens at i = 18 or i = 35 : for i = 18, we have P i (S(2) = 1) ≈ 0.314 and P i (S(2) = 19) ≈ 0.235 despite the BDM strategy recommending card 19.

  Let p ∈ P a (n). A permutation σ ∈ S n is said to be p-compatible if the cards of each packet are in order.

  actually have P i (S(2) = 1) = 65 176 ≈ 0.3693 and P i (S(2) = 8) = 135 352 ≈ 0.3835 :

  actually have P i (S(2) = 1) = 36 97 ≈ 0.3711 and P i (S(2) = 8) = 410 1067 ≈ 0.3843.

P i (S(2) = j 1 )

The coloured region corresponds to values of i for which the BDM strategy chooses the second card suboptimally. We see that this region goes from n 3 to 2n 3 ( n 2 + 1 is the counterexample we used to prove Theorem 2.5). This seems to hold for any n large enough. For n = 52, the biggest error happens at i = 24 or i = 29 : for i = 24, we have P i (S(2) = 1) ≈ 0.444 and P i (S(2) = 25) ≈ 0.283 despite the BDM strategy recommending card 25.

• The case m = 1, a = 4. This is not the case we used for our counterexample in Theorem 2.6, but formulas can be computed just as easily from Proposition 2.3 and Proposition 2.4. Denoting i = (i) again, we obtain :

.

For n = 52, we observe two disjoint error regions this time :

Errors are made between n 4 and 3n 8 , as well as between 5n 8 and 3n 4 symmetrically. This also seems to


Armand Attia

Derivation and role of Kroner's cubic in the calculation of shear moduli of an aggregate of cubic crystals

Introduction 3 Properties of fourth-order tensors 4 Equilibrium 5 Integral equation for strains 6 Role of Eshelby's tensor in integral equation for strains 7 Symmetry properties and model of tensor D 8 Derivation of tensor A for isotropic matrix and isotropic inhomogeneity 9 Self consistent scheme for cubic polycrystals 10 Eshelby's tensor for isotropic aggregate 11 Elastic moduli of a cubic polycrystal 12 Rotational averages 13 Derivation of tensor A for anisotropic inhomogeneity and Kroner's cubic 14 Comparison with experimental data 15 Major symmetries of tensor D 27 16 Conclusions 29 1

Introduction

This is a detailed review of the derivation of Kroner's cubic [START_REF] Kroner | Berechnung der elastischen konstanten des vielkristalls aus den konstanten des einkristalls[END_REF], which is used to calculate the macroscopic shear modulus of an aggregate of cubic crystals. The macroscopic modulus is assumed to apply to an isotropic ensemble of an aggregate of cubic crystals in perfectly randomized orientations.

The first major part of this derivation is based on the treatment of equilibrium by Gubernatis [START_REF] Gubernatis | Macroscopic engineering properties of polycrystalline materials: Elastic properties[END_REF] and the calculation of the Green tensor function as derived by Weinberger et al [START_REF] Weinberger | Lecture notes -elasticity of microscopic structures[END_REF], all leading to an integral equation for strains by Gubernatis [START_REF] Gubernatis | Macroscopic engineering properties of polycrystalline materials: Elastic properties[END_REF].

The second major part is based on Markov's [START_REF] Markov | Heterogeneous Media: micromechanics modeling methods and simulations, chapter "Elementary micromechanics of heteregeneous media[END_REF] treatment for deriving Kroner's cubic from this integral equation. The third major part compares measured shear moduli with the values predicted by Kroner's cubic with Hashin's bounds.

But first, a few words about fourth-order tensors.

Properties of fourth-order tensors

Fourth-order tensors are designated with such script (C), while second-order tensors are designated without such script (C). Markov [START_REF] Markov | Heterogeneous Media: micromechanics modeling methods and simulations, chapter "Elementary micromechanics of heteregeneous media[END_REF] defines the composition of two fourth-order tensors C : B as:

(C : B) ijkl = C ijrs B srkl (1)
For a fourth-order tensor B with only minor symmetry:

B ijkl = B jikl (2)
B ijkl = B jilk (3)
the identity tensor I

I ijkl = 1 2 (δ ik δ jl + δ il δ jk ) (4)
has the property:

B : I = I : B = B (5)
Since:

B ijrs I srkl = 1 2 B ijrs (δ sk δ rl + δ sl δ rk ) = 1 2 (B ijlk + B ijkl ) = B ijkl (6)
I ijrs B srkl = 1 2 (δ ir δ js + δ is δ jr )B srkl = 1 2 (B jilk + B ijkl ) = B ijkl (7)
Consider isotropic fourth-order tensors B with both major and minor symmetries:

B ijkl = B jikl = B ijlk = B klij (8)
Such a tensor can be decomposed as a linear combination of the following orthogonal set of fourth-order tensors I and

I B = b I + b I (9)
where

I ijkl = 1 3 δ ij δ kl (10)
I " ijkl = I ijkl -I ijkl [START_REF] Lubarda | New estimates of the third-order elastic constants for isotropic aggregates of cubic crystals[END_REF] with the properties:

I : I = I (
Note, that because of the above properties of I and I , we find that:

B : I = b I (15) B : I = b I (16)
so that it is clear that b and b are eigenvalues of B with corresponding eigentensors I and I and that with respect to these eigentensors, B is in diagonal form. Therefore, its inverse is a tensor A given in terms of the reciprocals of the eigenvalues (if they do not vanish):

A = B -1 = 1 b I + 1 b I (17)
The scalar product of two fourth-order tensors is defined as:

A :: B = A ijkl B ijkl (18)
Notice that while the tensors I and I are orthogonal, only I is normalized, i.e.: Consider a linear elastic solid of volume V with bounding surface S and elastic coefficients C 1 . This solid is assumed to contain an inclusion of volume W with elastic coefficients C 2 . At equilibrium with no body forces, the stress satisfies:

∂σ ij (x) ∂x j = ∂ ∂x j (C ijkl (x) kl (x)) = 0 (24) (25)
where the strain is related to the displacement via:

kl = 1 2 (u k,l + u l,k ) (26)
and following Gubernatis [START_REF] Gubernatis | Macroscopic engineering properties of polycrystalline materials: Elastic properties[END_REF]:

C(x) = C 0 + [C](x) (27) [C](x) = C 2 (x) -C 1 (x) (28)
in which the elastic modulus C 0 is constant, and the elastic moduli C 1 (x) and C 2 (x) may vary over an infinite region of volume V but differ only over an inclusion of finite volume W . C 1 is associated with the "matrix" region, and C 2 is associated with an inhomogeneity in the inclusion. We will focus on a spherical inclusion and on constant moduli C 1 and C 2 . Therefore, the equilibrium requirement Eq. 24 may be written as:

-C 0 klmn mn,l = ([C] klmn mn ) ,l (29)
Throughout the derivation we will be assuming uniform strain fields both in the matrix and in the inclusion.

Integral equation for strains

This concerns the determination of displacement fields in a solid matrix (medium 1) containing an inhomogeneity (medium 2) whose elastic properties differ from those of the surrounding matrix. The goal is to relate the strain in the inhomogeneity to the known strain when there is no inhomogeneity. Only the matrix is assumed to be isotropic. Since the equilibrium equation is based on linear elasticity, Green's (tensor) function can be used to construct the general solution for the displacement. The construction of the Green tensor function follows Weinberger et al [START_REF] Weinberger | Lecture notes -elasticity of microscopic structures[END_REF]. The Green tensor function G mr (x, x ) is defined as the displacement at x in the m th direction in response to a concentrated unit force applied in the r th direction at x . In an infinite homogeneous body, the Green's function depends only on the relative displacement between the points because the underlying differential operator here has constant coefficients and thus is invariant under translation (See Stakgold [START_REF] Stakgold | Boundary value problems of mathematical physics[END_REF] p. 49). Therefore, Green's function may be written as:

G mr (x, x ) = G mr (x -x ) (30)
For a constant point force F acting at x , the displacement at x is given by:

u m (x) = G mr (x -x )F r (x ) (31)
So that the displacement gradient and the strain take the form:

u m,n = G mr,n (x -x )F r (32) mn = 1 2 (u m,n + u n,m ) (33)
and from the constitutive relation, the stress is given by:

σ kl = C 0 klmn mn (34)
because of the symmetry of the linear elastic coefficients:

σ kl = C 0 klmn u m,n (35)
σ kl = C 0 klmn G mr,n (x -x )F r (36)
Now, following Weinberger et al [START_REF] Weinberger | Lecture notes -elasticity of microscopic structures[END_REF], for a given volume V , from the definition of the Dirac delta function, the k component of the force F acting at x can be described by:

V F k δ(x -x )dV (37)
Equilibrium requires that the force F be balanced by tractions acting on the surface S enclosing this volume V , so that:

V F k δ(x -x )dV + S σ kl n l dS = 0 (38) V F k δ(x -x )dV + S C 0 klmn G mr,n (x -x )n l F r dS = 0 (39)
Applying Gauss's theorem:

V F k δ(x -x )dV + V C 0 klmn G mr,nl (x -x )F r dV = 0 (40) V [F r δ kr δ(x -x ) + C 0 klmn G mr,nl (x -x )F r ]dV = 0 (41)
from which:

-C 0 klmn G mr,nl (x -x ) = δ kr δ(x -x ) (42)
For an unbounded isotropic matrix, Weinberger et al [START_REF] Weinberger | Lecture notes -elasticity of microscopic structures[END_REF] derive that Green tensor function for x = 0 in their Eq. (1.79) as:

G mr (x) = 1 16πµ(1 -ν)x (3 -4ν)δ mr + x m x r x 2 (43)
which clearly shows that G mr vanishes as x = |x| approaches infinity, and where µ, ν are the shear modulus and Poisson's ratio of the matrix material. Following Gubernatis [START_REF] Gubernatis | Macroscopic engineering properties of polycrystalline materials: Elastic properties[END_REF], the solution for the displacement is decomposed into:

u i (x) = u 0 i (x) + u 1 i (x) (44)
where u 0 i (x) is any solution of the homogeneous equation:

C 0 klmn mn,l = 0 (45)
and u 1 i (x) satisfies the inhomogeous equation Eq. 29. The solution for the displacement u 1 i (x) is given by applying Stakgold [START_REF] Stakgold | Boundary value problems of mathematical physics[END_REF] Eq. (5.100) to Eqs. 29 and 42 to find the displacement response at x due to the equilibrating "force" at x given by the right hand side of Eq. 29:

u 1 i (x) = V dx G ik (x -x )σ * kl,l (x ) (46)
σ * kl,l (x ) = ∂ ∂x l ([C] klmn (x ) mn (x )) (47)
or,

u 1 i (x) = V dx G ik (x -x ) ∂ ∂x l ([C] klmn (x ) mn (x )) (48)
so that the total displacement is given by:

u i (x) = u 0 i (x) + V dx G ik (x -x ) ∂ ∂x l [C] klmn (x ) mn (x ) (49)
Integrating by parts:

u i (x) = u 0 i (x) + V dx ∂ ∂x l G ik (x -x )[C] klmn (x ) mn (x ) (50)
- V dx ∂ ∂x l G ik (x -x ) [C] klmn (x ) mn (x ) (51)
Applying the divergence theorem and using the fact that the Green function vanishes on the boundary at infinity:

u i (x) = u 0 i (x) - V dx ∂ ∂x l G ik (x -x ) [C] klmn (x ) mn (x ) (52)
Since:

∂ ∂x l G ik (x -x ) = - ∂ ∂x l G ik (x -x ) (53)
The displacement solution becomes:

u i (x) = u 0 i (x) + V dx ∂ ∂x l G ik (x -x ) [C] klmn (x ) mn (x ) (54)
or, defining:

G ik,l (x -x ) = ∂ ∂x l G ik (x -x ) (55)
we can write:

u i (x) = u 0 i (x) + V dx G ik,l (x -x ) [C] klmn (x ) mn (x ) (56)
Now calculate the strains.

ij = 1 2 (u i,j + u j,i ) (57)
Since only the Green tensor function depends on x:

u i,j = u 0 i,j + V dx G ik,lj (x -x ) [C] klmn (x ) mn (x ) (58) u j,i = u 0 j,i + V dx G jk,li (x -x ) [C] klmn (x ) mn (x ) (59) ij (x) = 0 ij (x) + V dx G ijkl (x -x )[C] lkmn (x ) mn (x ) (60)
where:

G ijkl (x) = 1 2 (G ik,lj (x) + G jk,li (x)) (61)
Since the moduli differ only over the inclusion W , Eq. 60 reduces to:

ij (x) = 0 ij (x) + W dx G ijkl (x -x )[C] lkmn (x ) mn (x ) (62)
At this point of the derivation of Kroner's cubic, the assumption of uniform strain mn (x ) in the inclusion becomes important. This is echoed by Wu [START_REF] Wu | The effect of inclusion shape on the elastic moduli of a two-phase material[END_REF].

Let:

P ijkl (x) = - W G ijkl (x -x )dx (63)
The assumption of uniform strain will be consistently supported below. In Section 7, it will be shown that the tensor P is constant, thereby insuring that the strain (on left side of Eq. 62) is also uniform consistently. Of course, 0 is uniform since it satisfies Eq. 45.

Therefore, since the strain is uniform throughout the inclusion:

ij = 0 ij -P ijkl [C] lkmn mn (64)
Now:

I ijmn mn = ij (65)
so that:

(I ijmn + P ijkl [C] lkmn ) mn = 0 ij (66)
Therefore, letting:

B ijmn = I ijmn + P ijkl [C] lkmn (67)
we have:

B ijmn mn = 0 ij (68)
Or, with:

B = I + P : [C] (69)
Markov [START_REF] Markov | Heterogeneous Media: micromechanics modeling methods and simulations, chapter "Elementary micromechanics of heteregeneous media[END_REF] identifies a fourth-order tensor A given by:

A = (I + P : [C]) -1 = B -1 (70)
and Markov [START_REF] Markov | Heterogeneous Media: micromechanics modeling methods and simulations, chapter "Elementary micromechanics of heteregeneous media[END_REF] connects the constant strain 0 with the constant strain in the inclusion:

= A : 0 (71)
The derivation of tensor A will be given in Section 8. The role of 0 will be clarified in Eq. 123 in Section 9.

Role of Eshelby's tensor in integral equation for strains

In order to make further progress from the integral equation above toward Kroner's cubic, we need to derive an important connection between the tensor P and another tensor T related to Eshelby's tensor, as described below.

For this purpose, we will follow Weinberger et al [START_REF] Weinberger | Lecture notes -elasticity of microscopic structures[END_REF]. They introduce the Eshelby tensor S (See [START_REF] Eshelby | Elastic inclusions and inhomogeneities[END_REF] Eq. (3.3) p. 104) written as:

S = T : C (72)
Weinberger et al [START_REF] Weinberger | Lecture notes -elasticity of microscopic structures[END_REF] Eq. (2.13) write:

S ijmn = - 1 2 C lkmn (D iklj + D jkli ) (73)
where, for an ellipsoidal inclusion, Weinberger et al [START_REF] Weinberger | Lecture notes -elasticity of microscopic structures[END_REF] Eq. (2.16) define D ijkl as:

D ijkl (x) = V0 G ij,kl (x -x )dV (x ) (74)
Now by letting:

T ijkl = - 1 2 (D iklj + D jkli ) (75)
we can relate T to P, and from Eqs. 61, 63, 73, 74, and 75, it is clear that:

T = P (76)
7 Symmetry properties and model of tensor D

In order to show that tensor T (and therefore that P) is a constant, we need to establish some symmetry properties of tensor D. Weinberger et al [START_REF] Weinberger | Lecture notes -elasticity of microscopic structures[END_REF] Eq.

(2.61) calculate the D ijkl tensor when x lies inside the ellipsoidal inclusion as:

D ijkl = - abc 4π π 0 2π 0 (zz) -1 ij z k z l sinΦ β 3 dΘdΦ (77)
where, from [START_REF] Weinberger | Lecture notes -elasticity of microscopic structures[END_REF] Eq. (1.70):

(zz) -1 ij = 1 µ 1 δ ij - λ 1 + µ 1 λ 1 + 2µ 1 z i z j (78)
and from [START_REF] Weinberger | Lecture notes -elasticity of microscopic structures[END_REF] Eqs. (2.17), (2.42), (2.43), (2.44):

z 1 = sinΦcosΘ (79) z 2 = sinΦsinΘ (80)
z 3 = cosΦ (81)
For a unit sphere, from [START_REF] Weinberger | Lecture notes -elasticity of microscopic structures[END_REF] Eq. (2.46):

D ijkl = - 1 4πµ 1 π 0 2π 0 (δ ij + 2κ 1 z i z j ) z k z l sinΦdΘdΦ (82)
where, given Lame's constant λ 1 , shear modulus µ 1 , Poisson's ratio ν 1 , and bulk modulus k 1 for the isotropic matrix:

κ 1 = - 1 4 1 1 -ν 1 = - 1 2 λ 1 + µ 1 λ 1 + 2µ 1 (83) ν 1 = 3k 1 -2µ 1 2(3k 1 + µ 1 ) (84)
κ 1 = (- 1 2 ) (3k 1 + µ 1 ) (3k 1 + 4µ 1 ) (85)
From Eq. 82, it is clear that D satisfies the following minor symmetries:

D ijkl = D jikl = D ijlk (86)
However, it is also possible to show that D satisfies the major symmetry:

D ijkl = D klij (87)
This is shown in Section 15 so that we can write:

D ijkl = γδ ij δ kl + ω(δ ik δ jl + δ il δ jk ) (88) γ = - 1 3µ 1 (1 + 2 5 κ 1 ) (89) ω = - 2 15
κ 1 µ 1 ( 90
)
D ijkl = - 1 3µ 1 (1 + 2 5 κ 1 )δ ij δ kl - 2 15
κ 1 µ 1 (δ ik δ jl + δ il δ jk ) (91)
or, in terms of the isotropic basis tensors:

D = - 1 µ 1 (1 + 2 3 κ 1 )I + 4 15 κ 1 I (92)
We now return to the tensor T defined in Eq. 75 with:

D iklj = γδ ik δ lj + ω(δ il δ kj + δ ij δ kl ) (93)
D jkli = γδ jk δ li + ω(δ jl δ ki + δ ij δ kl ) (94)
T ijkl = - 1 2 (D iklj + D jkli ) (95)
T ijkl = 1 3µ 1 2 5 κ 1 δ ij δ kl + 1 2 (1 + 4 5 κ 1 )(δ ik δ lj + δ il δ kj ) (96)
or, in terms of the isotropic basis tensors:

T = t I + t I (97) t = 1 + 2κ 1 3µ 1 (98) t = 1 + 4 5 κ 1 3µ 1 (99)
Using Eq. 85:

t = 1 3k 1 + 4µ 1 (100) t = 3 5µ 1 k 1 + 2µ 1 3k 1 + 4µ 1 (101)
Thus, the tensor T is a constant and so is the tensor P. and isotropic inhomogeneity

Since P = T , we have from the above:

P = p 1 I + p 2 I (102) p 1 = t (103) p 2 = t (104)
For the special case where both the matrix and the inhomogeneity are isotropic, with subscript "1" for the isotropic matrix and subscript "2" for the isotropic inhomogeneity, the corresponding elastic moduli are written as:

C 1 = 3k 1 I + 2µ 1 I (105) C 2 = 3k 2 I + 2µ 2 I
(106) so that:

P : [C] = 3[k]p 1 I + 2[µ]p 2 I (107)
where:

[k] = k 2 -k 1 (108) [µ] = µ 2 -µ 1 (109)
We now proceed to derive the tensor A in Eq. 8. Putting Eq. 107 into Eq. 8 and using Eq. 11 we have:

I + P : [C] ≡ B = b I + b I (110) b = 1 + 3[k]p 1 (111) b = 1 + 2[µ]p 2 (112)
Note, that because of the properties of fourth-order tensors described in Section 3:

A = B -1 = 1 b I + 1 b I (113)
which can be written as:

A = a I + a I (114) a = 1 b = k 1 k 1 + 3[k]k 1 p 1 (115) a = 1 b = µ 1 µ 1 + 2[µ]µ 1 p 2 (116)
Comparing Eqs. 115 and 116 with Markov's [START_REF] Markov | Heterogeneous Media: micromechanics modeling methods and simulations, chapter "Elementary micromechanics of heteregeneous media[END_REF] Eq. (4.58), shown here:

A w = k 1 k 1 + α 1 [k] I + µ 1 µ 1 + β 1 [µ] I (117)
we find that:

3k 1 p 1 = α 1 (118) 2µ 1 p 2 = β 1 (119)
Note that Markov has an error in his Eq. (4.58) for the basis tensor I ijkl which he writes as:

1 2 (δ ij δ kl + δ il δ jk - 2 3 δ ij δ kl ) (120)
which should be corrected to read:

I ijkl = 1 2 (δ ik δ jl + δ il δ jk - 2 3 δ ij δ kl ) (121)
This concludes the derivation of tensor A for isotropic matrix and isotropic inhomogeneity.

Self consistent scheme for cubic polycrystals

The goal is to derive an equation (Kroner's cubic) that relates the shear modulus of the aggregate of cubic crystals to the elastic coefficients of a cubic crystal.

Since the entire aggregate consists only of inclusions, the foregoing development now applies everywhere in this aggregate.

Markov [START_REF] Markov | Heterogeneous Media: micromechanics modeling methods and simulations, chapter "Elementary micromechanics of heteregeneous media[END_REF] assumes that "...this aggregate (polycrystal) is an assembly of monocrystals, homogeneous grains with one and the same elastic properties, defined by the elastic tensor C. The crystallographic axes of each grain vary. Hence the tensor C is rotated in a complicated manner when one moves across the solid and exactly this is what makes the polycrystal heterogeneous". To keep things simple, Markov assumes that "there exist no preferable orientations of grains, i.e. of the crystallographic axes. Hence there is no texture presented so that the polycrystal is macroscopically isotropic, with a tensor of effective moduli ":

C * = 3k * I + 2µ * I ( 122
)
where I and I are given by Eqs. 11, 4, and 10. Markov continues. "Our aim is to develop a certain simple approximate scheme of self-consistent type for evaluating C * by means of the given elastic tensor C for a single grain. Imagine each grain is a sphere, immersed into an unbounded matrix with the effective, but yet unknown, [assumed isotropic] properties C * . Fix one of the grains, W . According to Eq. 71 the strain within such a grain is constant":

gr = A(C, C * ) : ¯ ( 123
)
where ¯ is the "prescribed macrostrain tensor, applied to the polycrytalline representative volume element (RVE)", interpreted here as the strain 0 (see Eq. 71) resulting when the moduli C 1 and C 2 match. Moreover, in virtue of Eqs. 8 and 102, Markov claims that given:

A(C, C * ) ≡ A * (124)
A * = B * (-1) (125) B * = [I + P * : (C -C * )] (126)
so that:

P * = p * 1 I + p * 2 I (127) p * 1 = 1 3k * + 4µ * (128) p * 2 = 3 5µ * k * + 2µ * 3k * + 4µ * (129)
At this point, Markov's argument now is that Eq. 102 applies to both the isotropic matrix and the approximately isotropic aggregate of cubic (anisotropic) crystals in perfectly random orientations. The difference between these two configurations is that in the case of the isotropic aggregate, the elastic modulus C * is unknown.

Markov is expanding the role of tensor A to apply to the case where the inhomogeneity is not isotropic. Note that the tensor P was derived from the Green's function for the isotropic matrix alone. The constitutive tensor C is now allowed to be anisotropic. This is consistent with the previous development.

Markov now describes the critical condition to be satisfied for determining this aggregate effective elastic modulus: "Let us now average Eq. 123 with respect to all possible crystallographic orientations of the axes of the grain. This operation will be denoted by • Ω ". Then:

gr Ω = A * Ω : ¯ (130)
"The key observation now is the identity":

gr Ω = ¯ (131)
"i.e. the strain averaged over all crystallographic orientations in a grain equals the macrostrain. Inserting Eq. 130 into Eq. 131:

A * Ω = I (132)
through which the unknown effective tensor C * can be found". The method for calculating averages over all orientations will be described in Section 12 [START_REF] Bertram | Formulation of anisotropic linear viscoelastic constitutive laws by a projection method[END_REF]

Eshelby's tensor for isotropic aggregate

We can now calculate the Eshelby tensor for the isotropic aggregate by extending Eq. 72 to:

S * = T * : C * (133)
where C * is given in Eq. 122, T * = P * , and where:

P * = p * 1 I + p * 2 I (134)
with the coefficients given in Eqs. 128 and 129. Now for the isotropic aggregate, the bulk modulus is given by:

k * = 2 3 µ * (1 + ν * ) (1 -2ν * ) (135)
From Eqs. 128 and 135:

p * 1 = 1 6µ * (1 -2ν * ) (1 -ν * ) (136)
From Eqs. 129 and 135:

p * 2 = 1 15µ * (4 -5ν * ) (1 -ν * ) ( 137
)
Carrying out the calculation in Eq. 133 subject to the orthogonality of the basis tensors we find:

S * ijkl = a * δ ij δ kl + b * (δ ik δ jl + δ il δ jk ) (138)
where:

a * = k * p * 1 - 2 3 µ * p * 2 (139) b * = µ * p * 2 (140)
Using Eqs. 136 and 137 :

a * = 1 15 (5ν * -1) (1 -ν * ) (141) b * = 1 15 (4 -5ν * ) (1 -ν * ) (142)
This can also be written using:

ν * = c * 12 2(c * 12 + c * 44 ) (143)
so that:

a * = 1 15 (3c * 12 -2c * 44 ) (c * 12 + 2c * 44 ) (144) b * = 1 15 (3c * 12 + 8c * 44 ) (c * 12 + 2c * 44 ) (145)
This is consistent with Kube's result [START_REF] Kube | Elastic constants of polycrystals with generally anistotropic crystals[END_REF] in his Eq. ( 6).

For a cubic polycrystal the calculation of the unknown effective tensor leads to Kroner's cubic for the effective shear modulus. The tensor of the elastic cubic crystal moduli C in this case is given by Markov as:

C = 3kI + 2µ 2 I + 2(µ 1 -µ 2 )O h (146) O h = e 4 1 + e 4 2 + e 4 3 (147)
Markov does not define e 4 i , saying only that O h is the "basic fourth-rank tensor with cubic symmetry (whose axes are along the orthonormal crystallographic basis e i , i = 1, 2, 3)". However, Bertram [START_REF] Bertram | Formulation of anisotropic linear viscoelastic constitutive laws by a projection method[END_REF] defines e 4 i as (no sum on i):

e 4 i = e i ⊗ e i ⊗ e i ⊗ e i (148)
To simplify the calculation of the shear modulus, Markov introduces "three basic fourth-rank tensors with cubic symmetry":

Σ 1 = I (149) Σ 2 = I + I -O h (150) Σ 3 = O h -I (151)
Markov claims that these three tensors are orthogonal, which is true after applying the correction in Eq. 121; that is, for each i, j = 1, 2, 3, i = j:

Σ i : Σ j = 0 (152)
This correction also allows the relation:

Σ 1 + Σ 2 + Σ 3 = I (153)
Finally, this correction also confirms that for each i = 1, 2, 3, with no implied sum on i:

Σ i : Σ i = Σ i ( 154
)
With respect to this basis, Markov's elastic coefficients take the form:

C = 3K Σ 1 + 2µ 2 Σ 2 + 2µ 1 Σ 3 ( 155
)
where:

K = k + 2 3 (µ 1 -µ 2 ) ( 156
)
To interpret Markov's formulation for the elastic coefficients of the cubic crystal in Eq. 146, we introduce the elastic coefficients of the cubic crystal in the form described by Lubarda [START_REF] Lubarda | New estimates of the third-order elastic constants for isotropic aggregates of cubic crystals[END_REF]

C = (c 11 + 2c 12 )Σ 1 + 2c 44 Σ 2 + (c 11 -c 12 )Σ 3 ( 160
)
Comparing Eqs. 155 with 160 shows that:

K = 1 3 (c 11 + 2c 12 ) (161)
µ 1 = 1 2 (c 11 -c 12 ) (162)
µ 2 = c 44 (163)
In the sequel, the symbolic notation:

C = (α, β, γ) ⇔ C = αΣ 1 + βΣ 2 + γΣ 3 (164)
will be useful.

Rotational averages

Before proceeding with the averaging process of the constitutive coefficients and, in particular, of the tensor A, it will be useful to describe the details of calculating rotational averages. According to Eqs. ( 2) and (3) in Andrews [START_REF] Andrews | On three-dimensional rotational averages[END_REF], the rotational average of a fourth order tensor C is given by:

C i1i2i3i4 = Λ i1i2i3i4;λ1λ2λ3λ4 C λ1λ2λ3λ4 (165)
and where Λ is defined by Eq. ( 19) in Andrews [START_REF] Andrews | On three-dimensional rotational averages[END_REF] as:

Λ i1i2i3i4;λ1λ2λ3λ4 = 1 30 δ i1i2 δ i3i4 δ i1i3 δ i2i4 δ i1i4 δ i2i3   4 -1 -1 -1 4 -1 -1 -1 4     δ λ1λ2 δ λ3λ4 δ λ1λ3 δ λ2λ4 δ λ1λ4 δ λ2λ3   (166)
Let's begin with finding the rotational average of tensor I .

In the following, the subscripts h and Ω are understood. Following Andrews, we have:

I ijkl = Λ ijkl;αβγω I αβγω ( 167
)
where we must have the correspondence:

(i 1 , i 2 , i 3 , i 4 ) ⇔ (i, j, k, l) (168) (λ 1 , λ 2 , λ 3 , λ 4 ) ⇔ (α, β, γ, ω) (169)
and where

I αβγω = 1 3 δ αβ δ γω (170)
Carrying out the above calculations results in:

I = I (171)
Similarly, with the correspondence:

(i 1 , i 2 , i 3 , i 4 ) ⇔ (i, k, j, l) (172) (λ 1 , λ 2 , λ 3 , λ 4 ) ⇔ (α, γ, β, ω) (173)
we find:

δ ik δ jl = δ ik δ jl (174)
and, with the correspondence:

(i 1 , i 2 , i 3 , i 4 ) ⇔ (i, l, j, k) (175) (λ 1 , λ 2 , λ 3 , λ 4 ) ⇔ (α, ω, β, γ) (176)
we find:

δ il δ jk = δ il δ jk (177)
so that:

I = I (178)
and, therefore:

I = I (179)
Let's now find the rotational average of tensor O h . Following Andrews, we have:

O ijkl = Λ ijkl;αβγω O αβγω ( 180
)
From Eqs. 147 and 148:

e 4 i = δ αi e α ⊗ δ βi e β ⊗ δ γi e γ ⊗ δ ωi e ω (181)
So that:

O αβγω = 3 i=1 (δ αi δ βi δ γi δ ωi ) (182)
and from Eqs. 166, 180, and 182 we obtain:

O ijkl = 1 30 (δ αβ δ γω A ijkl + δ αγ δ βω B ijkl + δ αω δ βγ C ijkl )O αβγω (183)
A ijkl = 4δ ij δ kl -δ ik δ jl -δ il δ jk (184)
B ijkl = -δ ij δ kl + 4δ ik δ jl -δ il δ jk (185)
C ijkl = -δ ij δ kl -δ ik δ jl + 4δ il δ jk (186)
Now:

δ αβ δ γω O αβγω = 3 (187)
δ αγ δ βω O αβγω = 3 (188)
δ αω δ βγ O αβγω = 3 (189)
so that:

O ijkl = 1 10 (A ijkl + B ijkl + C ijkl ) = 1 5 (δ ij δ kl + δ ik δ jl + δ il δ jk ) (190)
We now define H (following Markov [START_REF] Markov | Heterogeneous Media: micromechanics modeling methods and simulations, chapter "Elementary micromechanics of heteregeneous media[END_REF] in his Eq. (4.56))as:

H ijkl = δ ij δ kl + δ ik δ jl + δ il δ jk (191)
so that:

O = 1 5 H (192)
Note that from Eqs. 10, 11, and 191:

H = 5I + 2I (193)
We now prove the following identities which we will need in the sequel:

Σ 2 = 3 5 I (194)
Σ 3 = 2 5 I (195)
From 150, 171, 179,190, and 193:

Σ 2 = I + I" -O (196)
Σ 2 = I + I" - 1 5 H (197)
Σ 2 = I + I" - 1 5 (5I + 2I") (198)
Σ 2 = 3 5 I" (199)
Eq. 195 is shown is a similar fashion. From Eqs. 150 and 151:

Σ 2 + Σ 3 = I (200)
so that:

Σ 1 = I = Σ 1 (201)
Σ 2 = 3 5 I" = 3 5 (Σ 2 + Σ 3 ) (202)
Similarly, from Eq. 151:

Σ 3 = O -I (203)
Σ 3 = 1 5 H -I (204)
Σ 3 = 1 5 (5I + 2I ) -I (205)
Σ 3 = 2 5 I (206)
Σ 3 = 2 5 (Σ 2 + Σ 3 ) (207)
13 Derivation of tensor A for anisotropic inhomogeneity and Kroner's cubic

For the tensor C of a single grain, see Eq. 155, we have:

C = (α, β, γ), α = 3K , β = 2µ 2 , γ = 2µ 1 (208)
Due to the orthogonal properties of the Σ i s, inversion and multiplication take the forms:

C -1 = ( 1 α , 1 β , 1 γ ) (209)
With C = (α , β , γ ) and C = (α , β , γ ):

C : C = (α α , β β , γ γ ) (210)
The most important formula, however, concerns the tensors of the form given by Eq. 164, averaged over all orientations, using Eqs. 201, 202, 207:

C Ω = (α, β, γ) Ω = (α, β, β) (211) β = 1 5 (3β + 2γ) (212)
From Eqs. 126 and 208 the quantity:

B * = I + P * : (C -C * ) (213)
has the form:

B * = (1, 1, 1) + (p * 1 , p * 2 , p * 2 ) • [(3K , 2µ 2 , 2µ 1 ) -(3k * , 2µ * , 2µ * )] (214)
which reduces to:

B * = (1 + 3p * 1 (K -k * ), 1 + 2p * 2 (µ 2 -µ * ), 1 + 2p * 2 (µ 1 -µ * )) (215)
But Eq. 215 is the inverse of Eq. 124, so that:

A * = ( 1 1 + 3p * 1 (K -k * ) , 1 1 + 2p * 2 (µ 2 -µ * ) , 1 1 + 2p * 2 (µ 1 -µ * ) ) (216)
whose average follows Eqs. 211 and 171:

A * Ω = (α * , β * , β * ) ( 217
)
α * = 1 1 + 3p * 1 (K -k * ) (218)
β * = 3 5 1 1 + 2p * 2 (µ 2 -µ * ) + 2 5 1 1 + 2p * 2 (µ 1 -µ * ) (219)
Finally, the average of A * must satisfy Eq. 132 subject to Eq. 153, so that:

α * Σ 1 + β * (Σ 2 + Σ 3 ) = I = I + I (220) Σ 2 + Σ 3 = I (
β * = 1 (226)
from which:

k * = K (227)
and substituting Eq. 129 into Eq. 219 and applying Eq. 226 gives a cubic equation for the aggregate shear modulus µ * in the form:

(µ * ) 3 + a(µ * ) 2 -bµ * -c = 0 (228) a = 9k * +
These coefficients agree exactly with Kube's [START_REF] Kube | Elastic constants of polycrystals with generally anistotropic crystals[END_REF] coefficients in his Eq. (16b) (after a bit of algebra) as well as with Kroner's [START_REF] Kroner | Berechnung der elastischen konstanten des vielkristalls aus den konstanten des einkristalls[END_REF] Eq. ( 22) which shows the cubic for the aggregate shear modulus G as:

G 3 + ᾱG 2 + βG + γ = 0 (235) ᾱ = (3 K + 4ν) 8 (236) β = -( K + 12ν) μ 8 (237) γ = -K μ ν 4 (238)

Comparison with experimental data

Hashin and Shtrikman [START_REF] Hashin | On some variational principles in anisotropic and nonhomogeneous elasticity[END_REF], [START_REF] Hashin | A variational approach to the theory of the elastic behaviour of polycrystals[END_REF] have derived bounds for the isotropic shear modulus of aggregates of cubic crystals. Gschneider [START_REF] Gschneider | Solid State Physics, chapter Physical properties and interrelationships of metallic and semimetallic elements[END_REF] provides experimental data, and Simmons [START_REF] Simmons | Single crystal elastic constants and calculated aggregate properties: a handbook[END_REF] provides cubic crystal elastic coefficients in Mbar. Sisodia [START_REF] Sisodia | Shear moduli of polycrystalline cubic elements[END_REF] compares some experimental data with calculated shear modulus and reports the formulation of the Hashin-Shtrikman bounds, G * 1 and G * 2 in terms of the single crystal cubic coefficients c ij as:

G * 1 = G 1 + 3 5 G 2 -G 1 -4β 1 -1 (248)
G * 2 = G 2 + 2 5 G 1 -G 2 -6β 2 -1 (249)
where

β 1 = - 3 5 K + 2G 1 G 1 (3K + 4G 1 ) (250)
β 2 = - 3 5 K + 2G 2 G 2 (3K + 4G 2 ) (251) K = c 11 + 2c 12 3 (252) G 1 = c 11 -c 12 2 (253)
G 2 = c 44 (254)
An anisotropy index à is defined by:

à = G 2 G 1 = 2c 44 c 11 -c 12 (255)
which reduces to unity for isotropic material. Table 1 shows the cubic coefficients and the anisotropy index for several materials. Notice that Tungsten (W) and Molybdenum (Mo) come closest to being isotropic.

To compare the shear modulus determined by Kroner's cubic with experimental values, we use a simple iterative Newton's method, beginning with a trial value at the arithmetic average of the Hashin-Shtrikman's bounds. Table 1 shows these bounds and compares this calculated value with the experimental value. Notice that for elements which are not in Group IV-A, the error is under 20%, while for Group IV-A, the error varies from 40% to 74%, except for diamond for which Klein [START_REF] Klein | Anisotropy of Young's modulus and Poisson's ratio in diamond[END_REF] gives excellent agreement with the experimental value using Kroner's cubic. The values for c 11 , c 12 , c 44 were taken from Table 1 in Klein's article [START_REF] Klein | Anisotropy of Young's modulus and Poisson's ratio in diamond[END_REF], based on Brillouin scattering data from monocrystalline data for Young's modulus and two Poisson's ratios. The experimental value for the isotropic shear modulus was calculated from experimental values for Young's modulus and Poisson's ratio on a large sample subject to load indentation.

Sisodia [START_REF] Sisodia | Shear moduli of polycrystalline cubic elements[END_REF] mentions that if the anisotropic index à > 1, then G * 2 > G * 1 and vice versa. Notice also from Table 1, for those (most) elements where the anisotropy index exceeds unity, it is the case that G * 2 > G * 1 . For Molybdenum and Tungsten which are practically isotropic, the bounds coincide. Finally, Niobium is the only element with à < 1, and there G * 2 < G * 1 .

Units:

Gschneider's experimental values are given in units of kg f cm 2 × 10 6 . This is converted to GPa by multiplying by gravitational acceleration as 9.80665 m s 2 , so that Gschneider's units are converted as follows:

kg f cm 2 × 10 6 × 10 4 cm 2 m 2 × 9.80665 m s 2 × N kg m s 2 × P a N m 2 × GP a 10
D ijkl = - 1 4πµ (M ijkl + 2κ 1 N ijkl ) (257)
M ijkl = π 0 2π 0 (δ ij z k z l sinΦ) dΘdΦ (258)
N ijkl = π 0 2π 0 (z i z j z k z l sinΦ) dΘdΦ (259)
It is clear that N satisfies both major and minor symmetries. It remains to show that M ijkl satisfies major symmetry. Clearly, if i = j, then M ijkl = 0. It can also be shown that if k = l, then M ijkl = 0. For this purpose, since M ijkl satisfies minor symmetries, it suffices to show that:

M ij12 = M ij13 = M ij23 = 0 (260)
Putting Eqs. 79,80,81 into Eq. 258 as needed to form M ij12 , M ij13 , M ij23 , we find that in each case the Θ-integral vanishes. For M ij12 , the Θ-integral involves:

2π 0 (cosΘsinΘ)dΘ = 1 4 2π 0 (sin2Θ)d(2Θ) = (- 1 4 )[cos2Θ] 2π 0 = 0 (261)
For M ij13 , the Θ-integral involves:

2π 0 cosΘdΘ = [sinΘ] 2π 0 = 0 (262)
For M ij23 , the Θ-integral involves:

2π 0 sinΘdΘ = -[cosΘ] 2π 0 = 0 (263)
From the above, it is clear that the only non-zero values of M ijkl are obtained only if i = j and k = l. With i = j, M ijkl reduces to the second order quantity:

M kl = π 0 2π 0 (z k z l sinΦ) dΘdΦ (264)
With k = l = 1, 2, 3 and putting Eqs. 79,80,81 into Eq. 264 as needed, we find that:

M ijkl = 4 3 π (269)
These results clearly show that, given all the non-zero terms, M ijkl does have major symmetry as well as minor symmetry. Now that we have shown that both N ijkl and M ijkl have both major and minor symmetry, we can construct the following model for D ijkl which assumes both major and minor symmetry:

D ijkl = γδ ij δ kl + ω(δ ik δ jl + δ il δ jk ) ( 271
)
where γ is given by any member of the list in Eq. 269, say 1122 for which:

so that:

ω = D 1313 = - 2κ 1 4πµ 1 N 1313 = - 2κ 1 4πµ 1 4π 15 = - 2 15
κ 1 µ 1 (287)
giving our model as:

D ijkl = - 1

Conclusions

This paper provides an introduction to some early work in the modeling of polycrystals. Comparison with experimental data shows remarkable success in the calculation of the effective isotropic shear modulus of polycrystals with cubic symmetry. In the carbon family, Kroner's cubic does very well for diamond but poorly for Germanium, Lead, and Silicon. These discrepancies require further research, especially for more experimental data.

  12) I : I = I (13) so that: I : I = I : (I -I ) = I : I -I : I = 0

  an isotropic fourth-order tensor B with major and minor symmetries, if we write B = b I + b I (21) Then to solve for b and b , we have: b = B :: I (22) since I is normalized, but: b = B :: I I :: I (23) since I is not normalized.

  221) α * I + β * I = I + I (222) (α * -1)I + (β * -1)I = 0

K = c 1111

 1111 cubic symmetry, c 66 = c 44 , so that Kroner's coefficients ᾱ, β, γ are consistent with the above coefficients a, b, c, namely:+ 2c 12 )(c 11 -c 12 ) = -c(247)as required, thus completing the derivation of Kroner's cubic.

  ijkl = 1111, 1122, 1133, 2211, 2222, 2233, 3311, 3322, 3333

sin 2

 2 Φcos 2 Θsin 2 Φsin 2 ΘsinΦ dΘdΦ (represents any D ijkl term for which M ijkl = 0 but N ijkl = 0, that is, for the following list of (ijkl) indices for which (i = j and k = l) but for which (i = k and j = l) or (i = l and j = k): 1212, 2121,1313, 3131, 2323, 3232, 1221, 2112, 1331, 3113, 2332, 3223 (281) For D 1313 , M 1313 = 0 and:

  in Eq. (3.1) (Note Lubarda's sign error: +2c 44 should be -2c 44 ):C ijkl = c 12 δ ij δ kl + 2c 44 I ijkl + (c 11 -c 12 -2c 44 ) Âijkl

		(157)
	which is equivalent in Markov's notation to:
	C = 3c 12 I + 2c 44 (I + I ) + (c 11 -c 12 -2c 44 )O h	(158)
	or,
	C = (3c 12 + 2c 44 )I + 2c 44 (I ) + (c 11 -c 12 -2c 44 )O h	(159)
	but, in terms of Markov's basis:


J I B Camargo
email: camargo@obs.u-bordeaux1.fr

C Ducourant

R Teixeira

J.-F Le Campion

M Rapaport

P Benevides-Soares

Extension of the ICRF for selected areas down to the 16th magnitude -II

Keywords:

, aiming at the extension of the ICRF at optical wavelengths in regions of special astronomical interest, using observations from the Bordeaux and Valinhos meridian circles. Along with the new fields, the main differences, when compared to the first release, are: a much larger sky coverage, the replacement of the AC2000 by its upgraded version AC2000.2 as one of the first epoch astrometrical sources, inclusion of Tycho-2 and 2MASS photometry when available, and the correction for a magnitude equation on the Valinhos right ascension system as well. The resulting catalogue contains 678 828 entries with positional external precisions, on both coordinates, ranging from 50-60 mas (V ≤ 13.5) to 70-140 mas (13.5 < V ≤ 16.0). For the proper motions, precisions range from 3 mas/year to about 15 mas/year, depending on magnitude and declination.

Introduction

One of the most important tasks of astrometry rests upon the adoption of a concept and the materialisation of a celestial reference system, from which one can derive the positions of heavenly bodies as well as study their variations as a function of time.

The corresponding celestial reference frame is given by a catalogue containing the coordinates of reference points along with other parameters, so that it is possible for the user to materialise the reference system at any given epoch.

Three characteristics are important for such a reference frame. They are inertiality, rigidity and accessibility. Briefly speaking, inertiality can be understood as non-rotation of the coordinate axes, rigidity relates to the agreement between the coordinate axes materialised by any subset of the reference frame, and accessibility is achieved by the appropriate availability of reference points in the sky, so that the user can use a suitable number of such points in a sky region. This work is concerned mainly with this last topic.

A major breakthrough in astronomical reference systems was achieved with the adoption by IAU, as from 1st of January 1998, of the ICRS (International Celestial Reference System) [START_REF] Esa Sp Feissel | The HIPPARCOS and Tycho Catalogues[END_REF]Ma et al. 1998;Johnston & de Vegt 1999), as the celestial reference system to replace the FK5 [START_REF] Fricke | Fifth Fundamental Catalogue -Basic Fundamental Stars[END_REF]). The ICRS is materialised by a set of 212 extragalactic compact radio sources listed in the ICRF (International Celestial Reference Frame) (Ma et al. 1998). The positions of these sources have been precisely determined by VLBI techniques to accuracies better than 0.5 mas.

In optical wavelengths, the ICRS is realised by the HIPPARCOS catalogue [START_REF] Esa Sp Feissel | The HIPPARCOS and Tycho Catalogues[END_REF] with astrometrical parameters for about 118 000 objects, typically brighter than V = 10.0 (completeness up to V = 8.0), distributed all over the sky. This gives a star density of about 3 objects per square degree, which is insufficient to provide a desirable accessibility to the ICRS. The improvement given by the Tycho-2 catalogue notwithstanding (∼50 stars per square degree and limiting magnitude V = 12.5 -completeness up to V = 11.0), small field astrometry has still its density needs unfulfilled.

A worldwide effort is in progress to extend the ICRF in optical wavelengths from ground-based observations, supported by the IAU Commission 8 new working-group The "Future Development of Ground-Based Astrometry". Examples of such an effort are the USNO-A2.0 catalogue [START_REF] Monet | USNOA-V2[END_REF], UCAC project (Zacharias et al. 2000), A Catalogue of Faint Reference Stars in 398 Fields of Extragalactic Radio Reference Frame Sources -ERLcat (de Vegt et al. 2001), Valinhos Meridian Circle Catalogue -VMCC [START_REF] Camargo | [END_REF], and M2000 [START_REF] Rapaport | A Catalog of Astrometric Standards[END_REF] among others.

The present work is part of this effort and upgrades its previous version, the VMCC. The main differences are: a) observational data now provides 511 525 objects from the Bordeaux CCD meridian circle (hereinafter BMC) and 167 303 objects from the Valinhos CCD meridian circle (hereinafter VMC) -only BMC data was retained whenever common objects between both instruments were detected. VMC data from [START_REF] Camargo | [END_REF], as will be discussed in Sect. 3.2, was reanalysed for this present work; b) Tycho-2 B and V photometry have been included in the catalogue, as well as c) J, H, K photometry from 2MASS; d) correction for a magnitude equation present in the Valinhos right ascension system, and e) replacement of AC2000 by its upgraded version AC2000.2 (Urban et al. 2001) as one of the first epoch sources for proper motion measurements.

Also, in our catalogue (hereinafter BVMCcat) we provide: mean J2000 positions and respective epochs, proper motions, V 1 magnitude from the meridian observations, internal precisions for the astrometric parameters, time interval for proper motion determination, number of meridian observations as well as the involved current and first epoch sources.

The meridian circles

The main characteristics of the BMC and VMC have already been described by various papers (Viateau et al. 1999; 1 In this work, no check for variability was performed. Dominici et al. 1999;[START_REF] Camargo | [END_REF][START_REF] Rapaport | A Catalog of Astrometric Standards[END_REF], and details about both of them can be found in Viateau et al. (1999).

Their locations are, respectively, Floirac (λ = 00 • 31 39 (east), φ = +44 • 50 07 ) in France, and Valinhos (λ = 46 • 58 03 (west), φ = -23 • 00 06 ) in Brazil. Both instruments use CCDs operating in drift scan mode, and the bandpass, 5200 Å to 6800 Å, makes the resulting photometric band close to the Johnson's V band system (Dominici et al. 1999).

The limiting magnitude is about V = 17.0 for the BMC and one magnitude brighter for the VMC, due to the larger CCD of the BMC. The internal precision of the mean positions, as a function of magnitude, is shown in Fig. 1. From those panels, it is possible to notice the better performance (about 20 mas) of the BMC compared to that of the VMC. This is explained by the different size of the CCDs, allowing a longer integration time for the BMC, as well as slight differences in the reduction parameters used in each one (ponderation of observations, eliminations). It is also possible to notice a difference (about 10 mas) between the precisions of the right ascension and declination systems presented by the VMC, the origin of which is under investigation. Left and right panels of Fig. 1 are given within the optimal range of magnitude for the instrumental measurements (Viateau et al. 1999).

Astrometric data

Current epoch observations and reduction

Our observational data comprises 242 meridian strips (see Table 1 and Fig. 2), whose declination heights are 28 for Bordeaux and 13 for Valinhos, most of them centered on extragalactic radio sources, to support programs devoted to detecting their photometric variability as well as the determination of their optical position from larger telescopes. Also, strips Fig. 2. Distribution of the observations in the sky as given by Bordeaux (left) and Valinhos (right). All overlapping zones were previously treated, in order to eliminate redundant entries from the BVMCcat. The adopted procedure was to retain only the BMC observation whenever an object was observed by both BMC and VMC.

containing pre-main sequence stars observed in the great southern star-forming regions (Chamaeleon, Lupus and Upper Scorpius -Ophiuchus) (Teixeira et al. 2000) and other strips containing regions of young stars, open clusters and Pluto were also included in this data set. All meridian images have been reduced (Viateau et al. 1999), for astrometry and photometry, with the Tycho-2 catalogue.

Magnitude equation

The presence of a magnitude equation was found on the right ascension system of the Valinhos observations, and corrected according to the curve depicted in Fig. 3, upper panel, before the determinations of the results presented here, that is, before the necessary treatment to build the BVMCcat.

This feature should be compared to the colour dependence found in Bordeaux [START_REF] Rapaport | A Catalog of Astrometric Standards[END_REF], the cause of both effects probably being the same. It is a parallax error arising from out-of-focus images caused by the axial spread of the focal plane as a function of the wavelength. The consequent errors are essentially colour dependent. Magnitude dependence arises through the centering algorithms, which include the maximum possible number of pixels, so that bright objects will reflect the influence of the out-of-focus chromatic halo, which is not detected in the case of faint objects [START_REF] Benevides-Soares | [END_REF].

Further colour dependence effects on the meridian positions were not corrected since colour indexes were not available for all stars. No magnitude equation was found in the BMC observations. VMC observations) with the aid of almost the same ancient epoch astrometric catalogues presented in [START_REF] Camargo | [END_REF], and following the same procedures. The main differences are the use of the AC2000.2 instead of its previous version, the AC2000 (Urban et al. 1998), and the elimination of those observational data with standard deviations larger than 250 mas in either coordinates. When a given object observed by either the BMC or the VMC had its set of counterparts identified from the first epoch material, no elimination was done on this set. This is also different from the adopted procedure in [START_REF] Camargo | [END_REF]. In this work, AC2000.2 precisions were taken from the individual plate solutions (Urban et al. 1998). For referencing, the other employed catalogues are presented as follows: SERC-J plates measured with the MAMA measuring machine [START_REF] Guibert | Proc. of the 78th Colloq. of the IAU[END_REF], PPM South [START_REF] Bastian | The PPM-South Catalogue[END_REF], PPM North (Röser & Bastian 1998), USNO-A2.0 [START_REF] Monet | USNOA-V2[END_REF], TAC-2 (Zacharias & Zacharias 1999), and CPC-2 (Zacharias et al. 1999).

First and intermediate epoch material

The BVMCcat

The BVMCcat contains 678 828 stars brighter than V ∼ 16.0, observed at least 3 times with either the BMC or the VMC, whose distribution as a function of magnitude can be seen in Fig. 4.

A weighted least squares procedure was used to derive positions and proper motions. The adopted weights were taken from the precisions given by the employed material. On average, these values are: 250 mas for the USNO-A2.0, 100 mas for the TAC-2, 300 mas for the AC2000.2, 270 mas for the PPM-North, 110 mas for the PPM-South, 50 mas for the CPC-2 and 300 mas for the SERC-J plates. For the meridian positions, 50 mas was used [START_REF] Rapaport | A Catalog of Astrometric Standards[END_REF]Teixeira et al. 2000), which is a good compromise between the internal precisions of the mean positions for the Bordeaux and Valinhos instruments.

Figure 5 depicts average internal precisions of the BVMCcat for mean positions (upper panel) and proper motions (lower panel) as a function of magnitude.

The profile presented by the upper panel of Fig. 5 reflects the decreasing number of participant catalogues for the position/proper motion determination and the larger (5 times) weight given to the meridian positions as compared to the one given to those from USNO-A2.0 and the SERC-J plates. Theoretically, they should be in accordance with external verifications (presented later in the text) at least within the optimal magnitude range for the BMC and VMC astrometry (Fig. 1). To the faintest magnitudes, mean positions and their respective epochs will be close to their meridian counterparts.

The lower panel of Fig. 5 depicts a different behaviour of proper motion accuracies, given by a declination threshold of δ = -17 • . As explained in [START_REF] Camargo | [END_REF], this is due to the USNO-A2.0, which is the only source of first epoch positions for objects fainter than V ∼ 13.0. In USNO-A2.0 the mean epoch for objects with δ ≤ -17 • is 1980, and 1955 for those with δ > -17 • .

Table 3 explains the role of each column of the BVMCcat, given in Table 2.

External verifications of the BVMCcat

Three catalogues were employed to check the BVMCcat: the HIPPARCOS and TYCHO-2 catalogues, for the BVMCcat stars with V ≤ 12.0, and UCAC1 (Zacharias et al. 2000), when available, for the fainter ones. All differences were obtained after transferring these reference positions, through their respective proper motions, to the epoch of the BVMCcat ones. Data derived from both meridian instruments are homogeneous after correction for the magnitude equation on the VMC right ascension system, so that no separate treatment will be provided.

Comparison with HIPPARCOS

The BVMCcat contains 2119 HIPPARCOS objects, out of which 1167 were selected by the criterion that eliminated those with V < 8.0 (in order to avoid saturated observations from the meridian instruments), with HIPPARCOS MultFlag2 flag set (in order to avoid binary/multiple systems), as well as some few remaining idiosyncratic points that failed to pass a 3σ filter in position, that is, whose modulus of the positional difference were greater than 3 times the root mean square (rms) of the differences in either coordinates.

Figures 6 and 7 depict the differences in the sense BVMCcat minus HIPPARCOS as a function of magnitude at the epoch of the BVMCcat, and Table 4 shows related quantities obtained from the rms of these differences. The quantities presented in the aforementioned table show an overall good agreement between the BVMCcat data and that of the HIPPARCOS catalogue.

The propagated errors3 of HIPPARCOS at the epoch of the BVMCcat positions, about 10 mas, were considered negligible, so that we conclude that the BVMCcat external positional precision is about 50 mas within the magnitude interval 8.0 ≤ V ≤ 12.0. The comparison of the BVMCcat proper motions with those from HIPPARCOS shows that, for objects with the AC2000.2 as the first epoch, the external error for the measurements in the BVMCcat is 3 mas/year.

Comparison with Tycho-2

The BVMCcat contains 45 394 objects in common with Tycho-2, whose differences in positions (at the epoch of the BVMCcat) and proper motions are depicted by Figs. 8 and9 as a function of magnitude, after a 3σ filter in position.

Poor positional history within the employed catalogues and possible cross-identification difficulties contribute to the largest differences depicted in Fig. 9.

In Table 5, a more detailed comparison of position is shown, where a 3σ filter within each magnitude bin was applied. For the same objects presented in the aforementioned table, related quantities for the rms of the differences in proper motions are given in Table 6.

As in the comparison with HIPPARCOS, an overall good agreement between the positions is shown by Table 5, which corroborates the value of 50 mas for the BVMCcat external positional precision (lines 6 and 7 of Table 5) when 8.0 ≤ V ≤ 11.5. External precision was obtained by subtracting from the σ (∆αcosδ,∆δ) values, as evaluated by the rms of the differences, the formal positional error of the Tycho-2 positions at the comparison epoch, as calculated from [START_REF] Høg | [END_REF].

For V > 11.5, one can notice a degradation of the external precision of the BVMCcat, which may result partly from Tycho-2 underestimated errors in its faintest branch, as pointed out in [START_REF] Rapaport | A Catalog of Astrometric Standards[END_REF] and supported later in the text (Sect. 5.3), by the comparison BVMCcat and UCAC1. V magnitude 8.0-8.5 8.5-9.0 9.0-9.5 9.5-10.0 10. 0-10.5 10.5-11.0 11.0-11.5 11.5-12 V magnitude 8.0-8.5 8.5-9.0 9.0-9.5 9.5-10.0 10.0-10.5 10.5-11.0 11.0-11.5 11.5-12.0 From Table 6, we evaluate the BVMCcat external errors on proper motions as ∼2 mas/year within the interval 8.0 ≤ V ≤ 12.0.

σ ∆µαcosδ 3 3 2 2 2 2 2 3 σ ∆µ δ 3 3 2 2 2 2 2 2 < ∆µ α cosδ > -0.2 -0.3 -0.4 -0.2 -0.1 -0.1 -0.2 -0.2 < ∆µ δ > 0

Comparison with UCAC1

The UCAC1 catalogue was employed to probe the BVMCcat results on fainter magnitudes, aiming at the objects fainter than V = 12.0. We should stress that, in this verification, only the data derived from the VMC was employed, that is, the predominantly southern declinations, since UCAC1 is only available in that hemisphere.

Table 7 provides a detailed view of this comparison. In it, as in Table 5, all differences in position were binned in intervals of 0.5 mag, and those above a 3σ filter in the respective bin were eliminated. V magnitude 11.0-11.5 11.5-12.0 12.0-12.5 12.5-13.0 13.0-13.5 13.5-14.0 14.0-14.5 14.5-15.0 15. 0-15.5 V magnitude 11.0-11.5 11.5-12.0 12.0-12.5 12.5-13.0 13.0-13.5 13.5-14.0 14.0-14.5 14.5-15.0 15. 0-15.5 External precision is evaluated to 50-60 mas up to V = 13.5, well within the BMC and VMC optimal magnitude interval for astrometry (Fig. 1). This external precision reaches 140 mas for the fainter objects (V ≥ 15.0).

The external precisions presented for the magnitude interval 11.5 ≤ V ≤ 12.0, when compared to those in the same interval of Table 5, support a possible underestimation of the Tycho-2 errors for right ascensions within this magnitude interval.

To the same objects presented in Table 7, the rms of the differences in proper motions (Table 8) is 7 mas/year for both coordinates (11.0 ≤ V ≤ 16.0), and no systematic deviation was detected. Such an estimation, however, reflects that both BVMCcat and UCAC1 use the same first epoch positions (USNO-A2.0) to derive most of these quantities.

Conclusions

The BVMCcat extends the ICRF to the optical domain in zones of special astronomical interest, and the overall agreement between the BVMCcat data and that of well-known astrometric material enables it to serve as a reference catalogue for small field astrometry as well as to provide kinematical parameters for astrophysical studies within the limits of ground-based astrometry.

The collaboration of northern (Bordeaux) and southern (Valinhos) Observatories allowed a more efficient sky coverage and a larger number of entries for the BVMCcat, as compared to our previous work [START_REF] Camargo | [END_REF].

The external comparisons showed that no significant systematic trend was found either in positions or proper motions. The external errors of the BVMCcat are 50-60 mas for positions on both coordinates when V ≤ 13.5 and 70-140 mas for fainter magnitudes (13.5 ≤ V < 16.0). For proper motions, external errors are 3 mas/year when an AC2000.2 position is present as first epoch (V < ∼ 13.0) reaching probably 15 mas/year for the faintest objects at southern declinations, as suggested by the internal error degradation (Fig. 5, lower panel).

Acknowledgements. The authors wish to express their thanks to the computer service of IAG/USP (Department of Astronomy) and of the Bordeaux Observatory, for their valuable help and patience during data-processing. The authors are also indebted to the observers of both Brazilian and French meridian circles. J. I. B. Camargo expresses his thanks to Dr. C. L. Dal Ri Barbosa and Dr. G. M. Tanco, for their help with the 2MASS data. The authors are thankful to Dr. G. Daigne, Dr. C. Soubiran and J. P. Périé for suggestions and contributions. A partial financial support from FAPESP, CAPES, PRONEX and CNRS is gratefully acknowledged. This work has made use of the NASA's Astrophysics Data System Abstract Service. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. The authors express their thanks to the referee, Dr. C. Ma, for fruitful suggestions and corrections. J. I. B. Camargo is supported by CNPq -Brazil.

Fig. 1 .

 1 Fig. 1. Internal precisions of the mean positions of the Bordeaux and Valinhos meridian observations.

  Positions and proper motions presented here were derived from the combination of the present epoch data(BMC and

Fig. 3 .

 3 Fig. 3. Positional differences in the sense Valinhos minus Bordeaux as a function of magnitude.

Fig. 4 .

 4 Fig. 4. Object distribution per visual magnitude in the BVMCcat.

Fig. 5 .

 5 Fig. 5. Internal astrometric precisions of the BVMCcat as a function of magnitude.

Fig. 6 .

 6 Fig. 6. Positional differences in the sense BVMCcat minus HIPPARCOS, as a function of magnitude, at the epoch of the BVMCcat. Upper panel: right ascension. Lower panel: declination.

Fig. 7 .

 7 Fig. 7. Proper motion differences in the sense BVMCcat minus HIPPARCOS as a function of magnitude. Upper panel: right ascension. Lower panel: declination.

  are in [mas/year]. The same objects from Table 5 are used.

Fig. 8 .

 8 Fig. 8. Positional differences in the sense BVMCcat minus Tycho-2, as a function of magnitude, at the epoch of the BVMCcat. Upper panel: right ascension. Lower panel: declination.

Fig. 9 .

 9 Fig. 9. Proper motion differences in the sense BVMCcat minus Tycho-2 as a function of magnitude. Upper panel: right ascension. Lower panel: declination.

Table 4 .

 4 Statistical characteristics of the differences BVMCcat minus HIPPARCOS.

	Objects	1167
	< ∆αcosδ >	4
	< ∆δ >	-4
	< ∆µ α cosδ > -0.4
	< ∆µ δ >	-0.1
	σ ∆αcosδ	50
	σ ∆δ	50
	σ ∆µαcosδ	3
	σ ∆µ δ	3
	All units are in [mas] or [mas/year]. Comparisons were done at the
	epoch of the BVMCcat.

Table 5 .

 5 Statistical properties of the positional differences BVMCcat minus Tycho-2.

Table 6 .

 6 All astrometric units are in[mas]. External precisions are calculated by subtracting from the σ (∆αcosδ,∆δ) values the respective propagated Tycho-2 error at the BVMCcat epoch. Statistical properties of the proper motion differences BVMCcat minus Tycho-2.

	.0

Table 7 .

 7 Statistical properties of the positional differences BVMCcat minus UCAC1.

Table 8 .

 8 Statistical properties of the proper motion differences BVMCcat minus UCAC1.

	15.5-16.0

  All

											15.5-16.0
	σ ∆µαcosδ	5	7	7	6	7	5	5	6	8	8
	σ ∆µ δ	6	8	6	7	8	6	6	7	8	9
	< ∆µ α cosδ >	-0.2	0.0	-0.1	0.0	0.1	-0.4	-0.2	-0.1	0.1	0.1
	< ∆µ δ >	0.5	0.6	0.1	-0.2	-0.3	-0.3	-0.4	-0.4	-0.4	-0.6

astrometric units are in [mas/year]. The same objects from Table 7 are used.

Table 1 .

 1 List of strips observed by the meridian circles.

	Target	#Objects σαcosδ σ δ [mas]	α [J2000]	δ [J2000]	mag [V]	Nobs
	Valinhos zones
	NGC 7772	292	075 084	00 00 05.0 +16 16 01		14
	C 0001-302	220	081 105	00 01 56.5 -30 06 38		05
	ICRF J001031.0+105829	261	113 117	00 10 31.0 +10 58 30 15.4	10
	ICRF J004959.4-573827	209	094 101	00 49 59.5 -57 38 27 18.5	13
	VV96 J005334.9+124136	447	080 114	00 53 34.9 +12 41 36 15.7	10
	ICRF J011205.8+224438	660	070 091	01 12 05.8 +22 44 39 20.0	13
	ICRF J013305.7-520003	176	065 088	01 33 05.8 -52 00 04		06
	UV Cet	137	113 118	01 36 27.0 -17 57 30		05
	ICRF J015002.6-072548	156	122 118	01 50 02.7 -07 25 48 15.6	06
	ICRF J023951.2+041621	273	110 127	02 39 51.3 +04 16 21 18.5	08
	ICRF J024008.1-230915	552	098 104	02 40 08.2 -23 09 16 16.6	10
	ICRF J024104.7-081520	382	075 092	02 41 04.8 -08 15 21 12.3	06
	J032021.2-192631	85	117 143	03 20 21.2 -19 26 31		04
	HD 23158	289	081 126	03 47 15.4 +24 15 57		04
	NGC 1778	104	137 157	05 00 48.1 +36 59 28		05
	ICRF J053850.3-440508	467	076 094	05 38 50.4 -44 05 09 16.5	08
	NGC 2323	214	110 118	07 03 51.6 -08 15 58		04
	ICRF J073807.3+174218	1002	074 099	07 38 07.4 +17 42 19 16.2	09
	NGC 2506	1062	076 100	08 03 54.6 -10 47 08		07
	YZ Cnc	292	101 117	08 09 39.8 +28 08 20		06
	ICRF J081108.8-492943	1580	076 097	08 11 08.8 -49 29 43		06
	XY Pup	1454	093 097	08 11 53.5 -11 59 05		05
	C 0810-277	5534	082 094	08 12 29.0 -28 12 35		04
	RX Pup	819	068 098	08 13 37.8 -41 42 28		06
	AC Cnc	197	110 127	08 43 59.5 +12 50 40		10
	NGC 2682	938	082 094	08 51 56.7 +11 59 40		07
	WY Cnc	308	073 096	08 59 28.6 +26 41 01		07
	ICRF J091552.4+293324	222	075 091	09 15 52.4 +29 33 24 16.4	05
	ICRF J092129.3-261843	948	082 097	09 21 29.4 -26 18 34 18.4	08
	C 0939-536	6004	086 094	09 45 18.7 -53 51 44		08
	Chamaeleon	1094	090 095	09 49 35.8 -76 49 12		14
	C 0949-529	3789	078 101	09 55 42.5 -53 13 37		09
	AL Leo	95	105 128	09 56 59.4 +18 17 30		04
	ICRF J100159.9-443800	1196	094 099	10 01 59.9 -44 38 01 17.0	04
	ICRF J101353.4+244916	264	094 115	10 13 53.4 +24 49 16 16.6	10
	IC 2581	4211	034 041	10 24 51.3 -57 39 33		10
	ICRF J103502.1-201134	535	065 076	10 35 02.2 -20 11 34 19.0	09
	ICRF J103716.0-293402	1010	070 091	10 37 16.1 -29 34 03 16.5	11
	TW Hya	1041	041 048	10 55 25.1 -34 43 26		20
	ICRF J110331.5-325116	253	073 087	11 03 31.5 -32 51 17 16.3

Table 1 .

 1 continued.

	Target	#Objects σαcosδ σ δ [mas]	α [J2000]	δ [J2000]	mag [V]	Nobs
	0310+0122	225	107 123	03 53 05.9 +01 34 16		04
	ICRF J040748.4-121136	1101	059 064	04 05 29.1 -12 13 41 14.9	18
	ICRF J033930.9-014635	1385	074 087	04 07 39.3 -01 33 36 18.4	04
	PP 13	460	096 105	04 08 31.4 +38 07 05		06
	BP Tau	1728	077 083	04 22 50.0 +29 06 42		06
	DI Tau	290	085 093	04 30 07.4 +26 32 16		06
	ICRF J043311.0+052115	5626	061 069	04 45 46.7 +05 21 24 15.1	25
	DS Tau	1125	083 087	04 50 34.1 +29 24 49		06
	V836 Tau	2070	076 082	05 05 28.7 +25 22 33		07
	LkHA 333	1983	068 083	05 05 42.5 -03 19 48		04
	UX Ori	1791	061 071	05 22 27.4 -03 47 44		06
	PQ Ori	2761	059 071	05 43 56.5 -02 21 11		12
	V523 Ori	1638	081 100	05 45 25.2 -01 22 35		03
	V350 Ori	2279	056 066	05 49 33.9 -09 43 35		09
	ICRF J060752.6+672055	592	107 088	06 09 15.4 +67 20 29 20.6	04
	RNO 66	1822	063 074	06 09 18.0 -06 55 53		07
	V625 Ori	3298	074 080	06 12 52.7 +08 56 43		03
	W84	9249	071 081	06 30 19.7 +09 40 26		06
	0647+250	19993	058 057	06 30 50.3 +25 04 18		23
	MO Mon	4684	073 083	06 35 18.4 +09 21 06		06
	Z CMa	2420	068 079	07 00 24.6 -11 28 26		06
	ICRF J073807.3+174218	6928	045 049	07 37 24.9 +17 40 55 16.2	38
	1ES0806+524	1006	073 066	07 50 06.6 +52 20 04		18
	ICRF J075706.6+095634	3775	057 060	07 56 41.3 +09 55 48 15.0	16
	ICRF J082057.4-125859	3376	074 086	08 23 41.5 -13 00 30 15.0	04
	OJ287	7473	043 047	08 27 18.7 +20 03 59		53
	ICRF J091552.4+293324	1633	065 069	09 31 53.0 +29 33 21 16.4	16
	1011+496	1565	064 056	09 57 10.8 +49 27 44		23
	ICRF J095820.9+322402	957	077 067	09 59 25.1 +32 24 16 15.8	09
	ICRF J110427.3+381231	1728	058 053	11 13 44.5 +38 12 50 12.9	54
	1ES1113+432	233	084 081	11 30 40.7 +42 58 18		08
	ICRF J115019.2+241753	609	059 065	11 59 16.2 +24 17 56 15.7	14
	ICRF J115931.8+291443	459	064 070	11 59 31.6 +29 15 12 14.4	21
	ICRF J123049.4+122328	2485	071 072	12 30 03.2 +12 28 08 12.9	13
	ON231	404	094 098	12 37 24.1 +28 13 29		03
	ICRF J130533.0-103319	779	090 096	12 59 07.1 -10 26 51 15.2	06
	1ES1255+244	362	074 092	13 00 48.0 +24 12 17		03
	ICRF J125614.2+565225	353	084 067	13 07 06.7 +56 52 39 13.8	13
	ICRF J131028.6+322043	1631	057 060	13 19 29.6 +32 21 03 15.2	19
	ICRF J135704.4+191907	1712	060 065	13 45 24.4 +19 18 53 16.0	38
	1400+162	2653	059 063	13 59 57.3 +16 02 11

Table 1 .

 1 continued.

	Target	#Objects σαcosδ σ δ [mas]	α [J2000]	δ [J2000]	mag [V]	Nobs
	ICRF J140700.3+282714	926	072 081	14 05 52.7 +28 26 37 15.4	09
	RXJ1412.2	1327	070 085	14 11 20.8 -16 30 12		06
	RXJ1419.3	552	091 107	14 15 49.4 -23 22 44		04
	ICRF J141946.5+542314	1211	069 058	14 25 14.9 +54 22 53 15.7	25
	ICRF J142700.3+234800	1242	059 068	14 33 31.1 +23 47 40 15.0	06
	1517+656	1664	075 054	15 17 23.3 +65 24 40		59
	1553+113	5335	042 050	15 54 59.1 +11 08 15		49
	ICRF J164258.8+394836	2855	055 053	16 52 00.8 +39 47 05 16.0	39
	1722+119	15521	047 054	17 30 03.1 +11 52 21		66
	LkHA 122	8607	082 092	18 00 50.8 -22 53 58		07
	BD-10 4662	5012	076 088	18 20 31.5 -10 11 19		06
	VV Ser	5535	067 080	18 25 25.8 +00 02 39		07
	Coku Ser G6	5257	061 072	18 27 41.0 +00 29 29		07
	LkHA 118	29348	080 092	18 42 46.3 -24 18 40		08
	AS 310	13588	061 072	18 50 32.8 -04 58 25		16
	FH Aql	28283	056 067	19 01 02.8 -05 36 50		15
	WW Vul	15200	059 066	19 07 03.4 +21 11 45		15
	Par 21	31525	061 071	19 21 41.1 +09 37 48		13
	LHA 483-41	14181	080 086	19 22 50.6 +23 52 36		07
	V536 Aql	27734	068 080	19 41 14.6 +10 29 34		08
	1548C27	15796	064 071	19 57 22.8 +23 23 15		11
	1ES1959+650	5259	078 060	19 58 42.6 +65 08 33		72
	V1685 Cyg	10044	059 058	20 27 39.9 +41 20 37		20
	2032+107	7621	055 066	20 32 25.9 +10 51 53		18
	LkHA 168	7222	069 068	20 45 32.3 +44 31 11		09
	LkHA 191	9099	067 065	20 55 35.1 +43 50 05		14
	LkHA 321	4004	081 076	20 55 59.7 +49 50 50		06
	BD+41 3731	10758	069 063	21 02 44.0 +42 14 28		16
	LkHA 324	9583	078 075	21 15 26.9 +50 18 31		08
	V1082 Cyg	13809	071 076	21 30 36.5 +43 20 50		06
	LkHA 349	5889	083 068	21 35 22.6 +57 31 48		10
	BD+46 3471	15319	066 064	21 46 04.2 +47 21 50		08
	ICRF J220314.9+314538	7586	052 057	21 59 20.6 +31 46 08 15.6

HIPPARCOS' Double/Multiple Systems flag. For detailed information, please refer to ESA (1997).

According to relations 1.5.23, from Sect. 1.5 -"Transformation of Astrometric Data and Associated Error Propagation". For detailed information, please refer to ESA (1997).

Online Material

= Mean, in units of milliarc seconds to both coordinates, for the positional precision of the objects computed in the 2nd column; [B/V, A, U, P, T, C, E] employed astrometric catalogues for mean position and proper motion derivation 1 An * present right after the Tycho-2 number indicates that the object is not contained in the Tycho-1. 2 B to BMC or V to VMC; A = AC2000.2; U = USNO-A2.0; P = PPM; T = TAC-2; C = CPC-2; E = ESO plates.