Lionel Bréhamet

The Ponzano-Regge asymptotic of the supersymmetric 6jS symbols

Keywords: 03.65.Fd Algebraic methods : 04.60.-m Quantum gravity : 11.30.Pb Supersymmetry MSC: 81Q60, 83D05. Keywords. 6j symbol, Quantum Gravity, supersymmetric 6jS symbol. 1

We adapt the Gurau's proof (2008) about the asymptotic limit of Ponzano-Regge formula to supersymmetric 6jS symbols according to their intrinsic parities alpha, beta, gamma. The behaviour at a large scaling shows significant differences depending on these parities. The decay is slowed and the angles of the oscillating parts in cosine are generally shifted or more altered. Our results should be relevant in 3-D Quantum Supergravity and Spin Foam models.

Introduction

Since the growing expansion of quantum gravity theory [START_REF] Rovelli | Quantum Gravity[END_REF] the SU (2) 6j symbols acquired a considerable importance by becoming the basic building blocks of all spin networks. They appear to represent a quantum tetrahedron with quantized edges and even can be viewed as eigenfunctions of a discrete Schrödinger equation [START_REF] Aquilanti | Hamiltonian dynamics of a quantum of space: hidden symmetries and spectrum of the volume operator, and discrete orthogonal polynomials[END_REF]. As well known the classical Ponzano-Regge partition function Z P R [START_REF] Ponzano | Spectroscopic and group theoretical methods in physics[END_REF] was expressed as a sum of products of 6j symbols. In [START_REF] Livine | Three-dimensional Quantum SuperGravity and Supersymmetric Spin Foam Models[END_REF] the occurrence of supersymmetric 6j S symbols [START_REF] Daumens | The super-rotation Racah-Wigner calculus revisited[END_REF][START_REF] Bréhamet | Analytical complements to the parity-independent Racah-Wigner calculus for the superalgebra osp(1|2) Part I[END_REF], related to OSP (1|2), was also studied and led to a similar supersymmetric partition function, called Z sugra . A possible different divergence compared to the classical case was sketched.

Guided by the conceptual approach of Gurau [START_REF] Gurau | The Ponzano-Regge Asymptotic of the 6j Symbol: An Elementary Proof[END_REF], our present task is to give also an elementary proof of the Ponzano-Regge asymptotic of the 6j S supersymmetric symbols. Let us recall the formulation of Gurau:

Under a rescaling of all its arguments by a large k the 6j symbol associated to an Euclidean tetrahedron behaves like

kj 1 kj 2 kj 3 kJ 1 kJ 2 kJ 3 = 1 √ 12πk 3 V cos π 4 + 3 ι=1 kj ι + 1 2 θ jι + kJ ι + 1 2 θ Jι , ( 1
)
where V is the tetrahedron volume and θ jι , θ Jι are the exterior dihedral angles of the tetrahedron corresponding to the edges j ι and J ι respectively.

Tetrahedron of volume V

J 2 J 3 J 1 j 1 j 2 j 3
2 Recalls about 6j and 6j S symbols and notations

We rewrite our formulas [START_REF] Bréhamet | Analytical complements to the parity-independent Racah-Wigner calculus for the superalgebra osp(1|2) Part I[END_REF] with notations as close as possible to those used by Gurau [START_REF] Gurau | The Ponzano-Regge Asymptotic of the 6j Symbol: An Elementary Proof[END_REF] and replace j i ↔ J i , p 1 → v 4 , p 2 → v 2 , p 3 → v 3 , p 4 → v 1 , q → p. Notations used in [START_REF] Gurau | The Ponzano-Regge Asymptotic of the 6j Symbol: An Elementary Proof[END_REF] were

v 1 =j 1 +j 2 +j 3 , v 2 = J 1 +j 2 +J 3 , v 3 = J 1 +J 2 +j 3 , v 4 = j 1 +J 2 +J 3 , (2)
p 1 =j 2 +J 2 +j 3 +J 3 , p 2 = j 3 +J 3 +j 1 +J 1 , p 3 = j 1 +J 1 +j 2 +J 2 . (3)
Diagrammatic representation of a 6j symbol j 1 j 2 j 3 J 1 J 2 J 3 :

the four triangles of any 6j the three columns pairs

v 4 v 2 v 3 v 1 p 1 p 2 p 3
v i is the sum of the values of the three circled spins just above v i in the diagrams. In the same way, p j is the sum of the values of the four circled spins above p j . The variables p,v satisfy the equation

3 j=1 p j = 4 i=1 v i . (4)
As a result any spin is determined by two v i and one p j according to

2j 1 = v 1 + v 4 -p 1 , 2j 2 = v 1 + v 2 -p 2 , 2j 3 = v 1 + v 3 -p 3 , (5)
2J 1 = v 2 + v 3 -p 1 , 2J 2 = v 3 + v 4 -p 2 , 2J 3 = v 2 + v 4 -p 3 , (6)
or by two p j and two v i 'complementary' according to

2j 1 = p 2 +p 3 -v 2 -v 3 , 2j 2 = p 1 +p 3 -v 3 -v 4 , 2j 3 = p 1 +p 2 -v 2 -v 4 , (7)
2J 1 = p 2 +p 3 -v 1 -v 4 , 2J 2 = p 1 +p 3 -v 1 -v 2 , 2J 3 = p 1 +p 2 -v 1 -v 3 . (8)
Formulas for a 6j symbol are well known, however here we shall use an expression [START_REF] Bréhamet | Regge Symmetry of 6-j or super 6-jS Symbols: a Re-Analysis with Partition Properties[END_REF], used elsewhere long ago. That avoids the usual triangle coefficients △ and contains only terms in v i , p j .

j 1 j 2 j 3 J 1 J 2 J 3 = √ R t (-1) t (t+1)! i=4 i=1 (t-v i )! j=3 j=1 (p j -t)! , (9)
where the radical R under square root is the prefactor used by Gurau

R = j=3 j=1 i=4 i=1 (p j -v i )! i=4 i=1 (v i + 1)! . ( 10
)
A short historical review about the supersymmetric 6j S symbols:

The conceptualization of supersymmetric 3j or 6j symbols was done in several years spread over time, with sometimes different notations. It started with Pais and Rittenberg [START_REF] Pais | Semisimple graded Lie algebras[END_REF] in 1975 with a semisimple graded Lie algebra named "graded su(2)". Later in 1977, Scheunert et al. [START_REF] Scheunert | Irreducible representations of the osp(2, 1) and spl(2, 1) graded Lie algebras[END_REF] have computed the super-rotation Clebsch-Gordan coefficients as a product of usual rotation Clebsch-Gordan coefficients with a scalar factor. Finally, in 1981, Berezin and Tolstoy [START_REF] Berezin | The group with Grassmann structure UOSP(1.2)[END_REF] suggested that the (iso)scalar factors form a pseudo-orthogonal matrix. Subsequently these works were materialized in the paper by Daumens et al. [START_REF] Daumens | The super-rotation Racah-Wigner calculus revisited[END_REF] dated 1993 with the introduction of properly defined 6j S symbols. Our definitions of supersymmetric 6j S π symbols [START_REF] Bréhamet | Analytical complements to the parity-independent Racah-Wigner calculus for the superalgebra osp(1|2) Part I[END_REF] of parity π = α, β, γ were published in 2006, more than a decade after the paper of Daumens et al. 1 . These 6j S π symbols were expressed by a single sum formula over an index t [involving monomials Π π (t)] as shown below:

j 1 j 2 j 3 J 1 J 2 J 3 S π = (-1) 4 jιJι R S π t (-1) t t!Ππ (t) i=4 i=1 (t-[vi+ 1 2 ])! j=3 j=1 ([pj+ 1 2 ]-t))! , (11)
where R S π is a supersymmetric prefactor given by

R S π = j=3 j=1 i=4 i=1 [p j -v i ]! i=4 i=1 [v i + 1 2 ]! . ( 12
)
Detailed expressions for R S α , R S γ , R S β are given in Appendix. Notation [ ] means 'integer part of'. In contrast to SU (2), OSP (1|2) triangles v i can be integer or half-integer, however they still satisfy the well known triangular inequalities. Let us recall the definitions of the parities π and monomials Π π (t) of degree 0 or 1 in t:

     π = α if ∀i ∈ [1, 4] v i integer, π = β if ∃ only two distinct v i , v j integer, π = γ if ∀i ∈ [1, 4] v i half-integer.
(13) 1 Ref. [START_REF] Daumens | The super-rotation Racah-Wigner calculus revisited[END_REF] was explicitly used in 2004 by Livine and Oeckl [START_REF] Livine | Three-dimensional Quantum SuperGravity and Supersymmetric Spin Foam Models[END_REF] . On that date our work [START_REF] Bréhamet | Analytical complements to the parity-independent Racah-Wigner calculus for the superalgebra osp(1|2) Part I[END_REF] was not known.

For a parity β, both integer triangles shall be denoted by v, v ′ , both other half-integer by v, v ′ . The single integer quadrangle is denoted by p, both other half-integer by p, p ′ . In this case eq. ( 4) transforms into

p + p + p ′ = v + v ′ + v + v ′ . ( 14
)
Thus a parity π just depends on the quality half-integral or integer of a triangle v i . Let us note that only the γ parity may contain supersymmetric symbols whose all the spins are half-integers.

Our monomials Π π (t) were defined in [START_REF] Bréhamet | Analytical complements to the parity-independent Racah-Wigner calculus for the superalgebra osp(1|2) Part I[END_REF] as

Π α (t) =1, (15)
Π β (t) = -t(2j ⋆ β + 1) + (p + 1 2 )(p ′ + 1 2 ) -vv ′ , (16)
Π γ (t) = -t+2(J 1 j 1 +J 2 j 2 +J 3 j 3 )+(J 1 +j 1 +J 2 +j 2 +J 3 +j 3 )+ 1 2 . ( 17
)
All constants Π α (0), Π β (0), Π γ (0) are positive integers.

The special spin j ⋆ β was identified [START_REF] Bréhamet | Analytical complements to the parity-independent Racah-Wigner calculus for the superalgebra osp(1|2) Part I[END_REF] as the vertex common to both half-integer triangles v, v ′ :

2j ⋆ β = p + p ′ -v -v ′ = v + v ′ -p. ( 18
)
Other shorter formulas are available:

J 1 j 1 + J 2 j 2 + J 3 j 3 = 3 i=ι j ι J ι and J 1 + j 1 + J 2 + j 2 + J 3 + j 3 = 1 2 3 j=1 p j . (19)
Oddly enough these 'supersymmetric' quantities are reflected in most parameters of the discriminant necessary for the saddle points computation [START_REF] Gurau | The Ponzano-Regge Asymptotic of the 6j Symbol: An Elementary Proof[END_REF], while the background is that of standard 6j.

An appropriate formulation more adapted to the manipulations of t in [START_REF] Gurau | The Ponzano-Regge Asymptotic of the 6j Symbol: An Elementary Proof[END_REF] is the following

Π β (t) = -(t + 1)(2j ⋆ β + 1) + [(2j ⋆ β + 1) + (p + 1 2 )(p ′ + 1 2 ) -vv ′ ] , (20)
Π γ (t) = -(t + 1) + [2 3 ι=1 j ι J ι + 1 2 3 j=1 p j + 3 2 ]. (21)
The supersymmetric frontal phase (-1) 4 jιJι is worthwhile of attention. From eq. ( 21) in [10]

A = 2 3 ι=1 j ι J ι = i<j v i v j - k<l p k p l , (22)
it can be proved hat

4 j ι J ι =      0 (mod 2) if π = α, 1 + j p j (mod 2) if π = γ, v + v ′ -p (mod 2) if π = β. (23)
As (-1 k 2 N = +1 if k even and (-1) N if k odd, rescaling any spin by k odd leads to

(-1) 4k 2 jιJι |k odd =      +1 if π = α, (-1) (1+ j p j ) if π = γ, (-1) (v+v ′ -p) if π = β. (24)
In the same way if v i is integer, kv i remains integer, whereas if v i is half-integer kv i remains half-integer only if k is odd and the formulas for initial parities β, γ can change. Thus care should be taken of k parity itself. Indeed the following transformations can occur:

j 1 j 2 j 3 J 1 J 2 J 3 S α ×k -→ kj 1 kj 2 kj 3 kJ 1 kJ 2 kJ 3 S α
∀k even or odd, (25)

j 1 j 2 j 3 J 1 J 2 J 3 S β,γ ×k -→ kj 1 kj 2 kj 3 kJ 1 kJ 2 kJ 3 S α if k even, ( 26
)
j 1 j 2 j 3 J 1 J 2 J 3 S β,γ ×k -→ kj 1 kj 2 kj 3 kJ 1 kJ 2 kJ 3 S β,γ if k odd. ( 27
)
The precisions given about 'subleading contributions' in [START_REF] Gurau | The Ponzano-Regge Asymptotic of the 6j Symbol: An Elementary Proof[END_REF] are not altered by the fact that k → ∞ by even or odd integer values. Due to the presence of integer parts, in R S π and t , the Stirling's approximation

n! = √ 2πe (n+ 1 2 ) ln(n)-n = √ 2πe n ln(n)-n+ 1 2 ln(n) , (28)
could give rise to different formulas when all spins are rescaled by a factor k. Namely

[kn]! = (kn)! = √ 2π e [k ln(k)-k]n+kn ln(n)+ 1 2 ln(kn) if n integer. ( 29
)
[kn]! = (kn)! if k even and n half-integer, (kn -1

2 )! if k odd and n half-integer.

(30)

[kn + 1 2 ]! = (kn)!
if k even and n half-integer, (kn + 1 2 )! if k odd and n half-integer.

(

It can be checked that only two approximations are modified for k odd and n half-integer:

[kn]! = √ 2π e [k ln(k)-k]n+kn ln(n) , (32)
[kn + 1 2 ]! = √ 2π e [k ln(k)-k]n+kn ln(n)+ln(kn) . ( 33
)
As a direct consequence of eq. ( 4), we have single out the term e [k ln(k)-k] because in all the summations to carry out with parities α, β, γ the resulting multiplicative factor will yield a null quantity.

Since we look for a dominant behaviour in first order approximation, the equalities are valid up to a multiplicative factor 1 + 1/k. Throughout this paper we will keep the original functions of Gurau H, h, F, f , denoted here by Gothic letters as H, h, F, f. A preliminary analysis shows that, for the three supersymmetric parities α, β, γ, both functions h(j, J) and f(x) remain unchanged as well as the saddle points values x ± . These functions appear under an exponentiated form as e kh(j,J) and e kf(x) . The results were presented as a linear expansion over the six spins as

h(j, J) =j 1 h j 1 + j 2 h j 2 + j 3 h j 3 + J 1 h J 1 + J 2 h J 2 + J 3 h J 3 , (34)
h j 1 = 1 2 ln (j 1 +j 2 -j 3 )(j 1 -j 2 +j 3 )(j 1 +J 2 -J 3 )(j 1 -J 2 +J 3 ) (j 1 +j 2 +j 3 )(-j 1 +j 2 +j 3 )(j 1 +J 2 +J 3 )(-j 1 +J 2 +J 3 )
and so on, (35)

f(x) =ıπx+x ln(x)- (x-v i ) ln(x-v i ) - (p j -x) ln(p j -x). (36)
Equation ( 35) can be rewritten in terms of p, v:

h j 1 = 1 2 ln (p 3 -v 3 )(p 2 -v 2 )(p 3 -v 2 )(p 2 -v 3 ) (p 1 -v 4 )(p 1 -v 1 )v 1 v 4 . (37)
So we have

h α (j, J) = h β (j, J) = h γ (j, J) ≡ h(j, J), (38)
h α j 1 = h β j 1 = h γ j 1 ≡ h j 1 and so on. ( 39
)
f α (x) = f β (x) = f γ (x) ≡ f(x). (40)
Equation ( 18) in [START_REF] Gurau | The Ponzano-Regge Asymptotic of the 6j Symbol: An Elementary Proof[END_REF] is the saddle points equation and remains unchanged, then the same holds for the saddle points solutions x ± lying in the appropriate integration interval. We recall that

f ′ (x) = ıπ ln(x) - ln(x -v i ) + ln(p j -x) = 0. ( 41
)
Changes with respect to the usual 6j symbols can arise either from R S π via e Hπ((j,J) =H((j,J) or from Σ x via e Fπ(x) =F(x) . This will lead to distinct variants of Gurau's proof.

3 PARITY α 3.1 Change in the prefactor R S α
Multiplying the spins of the prefactor R S α by a scaling factor k yields

R S α (k) = j=3 j=1 i=4 i=1 k(p j -v i ) ! i=4 i=1 (kv i )! , (3.1)
From eq. ( 29) the numerator and denominator of R S α (k) become

NumPreα =(2π) 6 e [k ln(k)-k](4 j p j -3 i v i )+k[ j,i (p j -v i ) ln(p j -v i )] × e 1 2 j,i ln(p j -v i )+ 1 2 j,i ln(k) =(2π) 6 e [k ln(k)-k](4 j p j -3 i v i )+k[ j,i (p j -v i ) ln(p j -v i )] × e 1 2 j,i ln(p j -v i )+6 ln(k)) . (3.2) DenPreα =(2π) 2 e [k ln(k)-k] i v i +k i v i ln(v i )+ 1 2 i ln(k)+ 1 2 i ln(v i ) =(2π) 2 e [k ln(k)-k] i v i +k i v i ln(v i )) × e 3 2 i ln(v i )-i ln(v i )+2 ln(k) = (2π) 2 Π i (v i ) e [k ln(k)-k] i v i +k i v i ln(v i )+ 1 2 i ln(v 3 i )+2 ln(k) . (3.3) That is R S α (k) =(2π) 4 k 4 Π i (v i ) e k[ j,i (p j -v i ) ln(p j -v i )-i v i ln(v i ) ] × e 1 2 j,i ln (p j -v i ) v 3 i . (3.4)
The similarities with eqs. [START_REF] Daumens | The super-rotation Racah-Wigner calculus revisited[END_REF][START_REF] Bréhamet | Analytical complements to the parity-independent Racah-Wigner calculus for the superalgebra osp(1|2) Part I[END_REF] in [START_REF] Gurau | The Ponzano-Regge Asymptotic of the 6j Symbol: An Elementary Proof[END_REF] are obvious whence

R S α (k) = (2π) 2 k 2 Π i (v i ) e H(j,J)+kh(j,J) (3.5)
3.2 Changes in the factors Σ α and F α (x)

Σ α = min p j max v i (-1) t t! Π i (t -v i )!Π j (p j -t)! . (3.6)
The only difference with [START_REF] Gurau | The Ponzano-Regge Asymptotic of the 6j Symbol: An Elementary Proof[END_REF] lies in the lack of factor e ln(kx+1)) ≈ e

1 2 ln(k 2 x 2 ) whence Σ α = 1 (2π) 3 x=min p j x=max v i G α (x), (3.7)
with

G α (x) = 1 k 3 e 1 2 ln x Π(x-v i )Π(p j -x) × e k{ıπx+x ln(x)-[ i (x-v i ) ln(x-v i )+ j (p j -x) ln(p j -x)]} , (3.8)
so that

Σ α = 1 (2π) 3 x=min p j x=max v i 1 k 3 e Fα(x)+kf(x) . (3.9) with F α (x) = 1 2 ln x Π(x -v i )Π(p j -x) = F(x) -ln(x). (3.10) Accordingly Σ α = 1 (2π) 3 1 k 3 x=min p j x=max v i 1 x e F(x))+kf(x) . (3.11)

Contribution to the saddle points (parity α)

Identifying Σ α of eq. (3.11) as a Riemann sum with 1 k → dx leads to

Σ α = 1 (2π) 3 k 2 min p j max v i dx 1 x e F(x)+kf(x) .
(3.12)

After collecting the three header factors from eqs. (3.5, 3.12) and from the Laplace's approximation [in complex plane] we find

Sdl α (x s ) = √ Πv i √ 2πk 1 x s 1 -f ′′ (x s ) e H(j,J)+F(xs)+k[h(j,J)+f(xs)] , (3.13)
with the same values x s than those used by Gurau. For reducing the equation lengths let us define

ϕ θ k = ι=3 ι=1 (kj ι + 1 2 )θ jι + (kJ ι + 1 2 )θ Jι . (3.14)
Then eq. ( 45) in [START_REF] Gurau | The Ponzano-Regge Asymptotic of the 6j Symbol: An Elementary Proof[END_REF] changes into

Π i=4 i=1 (v i ) √ 48πkV 1 x + e ı π 4 +ıϕ θ k + 1 x - e -ı π 4 -ıϕ θ k = √ Πv i √ 48πkV 1 x + + 1 x - cos π 4 + ϕ θ k + ı 1 x + - 1 x - sin π 4 + ϕ θ k . (3.15)
By using the results and notations of [START_REF] Gurau | The Ponzano-Regge Asymptotic of the 6j Symbol: An Elementary Proof[END_REF] we have

1 x ± = 2A (B ± ı △) = 2A(B ∓ ı △) B 2 + △ = 2A(B ∓ ı △) 4AC = (B ∓ ı △) 2C , (3.16
)

whence 1 x + + 1 x - = B C and 1 x + - 1 x - = - ı √ △ C . (3.17) As C = Πv i and √ △ = 24V we obtain finally 1 √ 48πkV √ Πv i B cos π 4 + ϕ θ k + 24V sin π 4 + ϕ θ k . (3.18)
We recall that coefficient B defined in [START_REF] Gurau | The Ponzano-Regge Asymptotic of the 6j Symbol: An Elementary Proof[END_REF] has the dimension of a volume and writes as

B = 3 i=1 j i J i 3 j=1 p j + 2(j 1 j 2 j 3 + j 1 J 2 J 3 + j 2 J 3 J 1 + j 3 J 1 J 2 ). (3.19)
The square root √ Πv i has the dimension of an area.

4 PARITY γ and k odd

R S γ (k) = j=3 j=1 i=4 i=1 k(p j -v i ) -1 2 )! i=4 i=1 (kv i + 1 2 )! , (4.1)
NumPreγ = (2π) 6 e [k ln(k)-k](4 j p j -3 i v i )+k[ j,i (p j -v i ) ln(p j -v i )] . (4.2) DenPreγ = (2π) 2 e [k ln(k)-k] i v i + i ln(k)+ i ln(v i ) = (2π) 2 k 4 e [k ln(k)-k] i v i + i ln(v i ) . (4.3) Then R S γ (k) = (2π) 2 1 k 2 e 1 2 k[ j,i (p j -v i ) ln(p j -v i )-i ln(v i )] . (4.4)
This means that the prefactor R S γ (k) is given by

R S γ (k) = (2π) 2 1 k 2 e Hγ (i,J)+khγ(i,J) , (4.5)
with H γ (i, J) = 0 and h γ (i, J) ≡ h(i, J).

′ γ = t=min p j t=max v i + 1 2 (-1) t (t + 1)! Π i (t -v i -1 2 )!Π j (p j -t)! , (4.7)
Σ ′′ γ = t=min p j t=max v i + 1 2 (-1) t t! Π i (t -v i -1 2 )!Π j (p j -t)! , (4.8)
so that

Σ γ = -Σ ′ γ + 2 3 i=1 j i J i + 1 2 ( 3 j=1 p j ) + 3 2 Σ ′′ γ . (4.9)
In the first sum Σ ′ γ we carry out simultaneously the variable changes t → kx and v i → kv i , p j → kp j and use eqs. (29-30). The result is

Σ ′ γ = 1 (2π) 3 x=min p j x=max v i + 1 2 G ′ γ (x), (4.10)
where

G ′ γ (x) = e 1 2 ln x 3 Π j (p j -x) +k{ıπx+x ln(x)-[ i (x-v i ) ln(x-v i )+ j (p j -x) ln(p j -x)]} = e F γ ′ (x)+kf(x) , (4.11)
F γ ′ (x) = e 1 2 ln x 3 Π j (p j -x) , (4.12)
so that

G ′ γ (x) = √ x 3 Π j (p j -x)
e kf(x) . (4.13)

The second sum Σ γ ′′ is handled in the same way as the first with slight differences

Σ ′′ γ = 1 (2π) 3 x=min p j x=max v i + 1 2 G ′′ γ (x), (4.14)
with x) . (4.15)

G ′′ γ (x) = e 1 2 ln 1 k 2 x Π j (p j -x) +k{ıπx+x ln(x)-[ i (x-v i ) ln(x-v i )+ j (p j -x) ln(p j -x)]} = 1 k √ x Π j (p j -x) e kf(

Contribution to the saddle points (parity γ)

From eq. (4.13) for ′ γ and the value of -f ′′ (x ± ) found by Gurau, see its equation (33),

-f ′′ (x ± ) = ∓ı √ △ x ± Π j (p j -x ± ) , (4.16)
we derive

2π k(-f ′′ (x ± )) × x 3 ± Π j (p j -x ± ) = 2π k x ± ∓ı √ △ Π j (p j -x ± ) × x 3 ± Π j (p j -x ± ) = 2π k(∓ı √ △) x 2 ± .
(4.17)

The k-dependent header factor comes from eq. (4.5) with (2π) 2 1 k 2 , from eq. (4.10 ) with 1 (2π) 3 and from the equation just above with 2π k what yields finally

1 k 1 √ 2πk 3 . As 1 k → dx the remaining contributions for ′ γ are x 2 ± 2πk 3 (∓ı √ △) e k[h(j,J)+f(x ± )] . (4.18)
For ′′ γ we have the following contribution

1 k x ± 2πk 3 (∓ı √ △) e k[h(j,J)+f(x ± )] , (4.19)
which must be weighted by its multiplicative coefficient

2 3 ι=1 j ι J ι + 1 2 ( 3 j=1 p j ) + 3 2 ≈ k 2 3 ι=1 2j ι J ι . Then ′
γ is no longer dominant that is the only one contribution comes from ′′ γ has the form

3 ι=1 2j ι J ι k 2 k x ± 2πk 3 (∓ı √ △) e k[h(j,J)+f(x ± )] = 3 ι=1 2j ι J ι √ ±ı x ± √ 48πkV e k[h(j,J)+f(x ± )] . (4.20)
As a function of the parameter A defined in eq. ( 21) in [START_REF] Gurau | The Ponzano-Regge Asymptotic of the 6j Symbol: An Elementary Proof[END_REF] for the saddle points equation we obtain

Sdl γ (x ± ) = A x ± √ 48πkV e ±ı π 4 +k[h(j,J)+f(x ± )] . (4.21)
From eq. ( 44) in [START_REF] Gurau | The Ponzano-Regge Asymptotic of the 6j Symbol: An Elementary Proof[END_REF] the real part of

e k[j 1 (h j 1 +f j 1 )+j 2 (h j 2 +f j 2 )+j 3 (h j 3 +f j 3 )+J 1 (h J 1 +f J 1 )+J 2 (h J 2 +f J 2 )+J 3 (h J 3 +f J 3 )]
is zero, then its remains only the imaginary part (the

h j i ,J i being real) e ±ık[j 1 θ j 1 +j 2 θ j 2 +j 3 θ j 3 +J 1 θ J 1 +J 2 θ J 2 +J 3 θ J 3 ] whence Sdl γ (x ± ) =A x ± √ 48πkV e ±ı π 4 × e ±ık[j 1 θ j 1 +j 2 θ j 2 +j 3 θ j 3 +J 1 θ J 1 +J 2 θ J 2 +J 3 θ J 3 ] . ( 4
v, v ′ v, v ′ 1 2 integer j * β p integer v, v ′ integer p, p ′ 1 2 integer v 4 , v 1 j 1 p 1 v 2 , v 3 p 2 , p 3 v 2 , v 1 j 2 p 2 v 3 , v 4 p 3 , p 1 v 3 , v 1 j 3 p 3 v 4 , v 2 p 1 , p 2 v 2 , v 3 J 1 p 1 v 4 , v 1 p 2 , p 3 v 3 , v 4 J 2 p 2 v 2 , v 1 p 3 , p 1 v 4 , v 2 J 3 p 3 v 3 , v 1 p 1 , p 2
Let us define J * β as the column companion of the spin j * β and represent it by an empty oval. An appropriate diagrammatic representation looks as follows j * β represented by •, J * β by an oval, only both half-integer v, v ′ are drawn

j * β = j 1 J * β = J 1 j * β = j 2 J * β = J 2 j * β = j 3 J * β = J 3 j * β = J 3 J * β = j 3 j * β = J 2 J * β = j 2 j * β = J 3 J * β = j 3 5.1 Change in the prefactor R S β
Multiplying the spins of the prefactor R S β by a scaling odd factor k yields

R S β (k) = (k(p -v))!(k(p -v ′ ))!k((p -v) - 1 2 )!(k(p -v ′ ) - 1 2 )! × (k(p -v))!(k(p -v ′ ))!(k(p -v) - 1 2 )!(k(p -v ′ ) - 1 2 )! × (k(p ′ -v))!(k(p ′ -v ′ ))!(k(p ′ -v) - 1 2 )!(k(p ′ -v ′ ) - 1 2 )! × kv!kv ′ ! -1 kv + 1 2 )!(kv ′ + 1 2 )! -1 . (5.1)
From eqs. (32-33) the numerator of R S β (k) becomes

NumPreβ = (2π) 6 e [k ln(k)-k](4 j p j -3 i v i )+k[ j,i (p j -v i ) ln(p j -v i )] ×e 1 2 [ln(p-v)+ln(p-v ′ )+ln(p-v)+ln(p-v ′ )+ln(p ′ -v)+ln(p ′ -v ′ )]+3 ln(k) . (5.2)
and the denominator is

DenPreβ = (2π) 2 e [k ln(k)-k](v+v ′ +v+v ′ ) × e k[v ln(v)+v ′ ln(v ′ )+v ln(v)+v ′ ln(v ′ )] × e 1 2 [ln(kv)+ln(kv ′ )]+ln(kv)+ln(kv ′ ) = (2π) 2 e [k ln(k)-k] i v i +k[ i v i ln(v i )] × e 3 ln(k) e 1 2 {ln(v)+ln(v ′ )}+ln(v)+ln(v ′ ) . (5.3) Note the lack of frontal factor in 1 k m in R S β (k). Whence R S β (k) = (2π) 4 e k[ j,i (p j -v i ) ln(p j -v i )-i v i ln(v i )] × e 1 2 ln (p-v)(p-v ′ )(p-v)(p-v ′ )(p ′ -v)(p ′ -v ′ ) (vv ′ )(v 2 v ′2 )
.

(5.4)

Thanks to the correlation table and the definition of the spin J * β it can be proved that

e 1 2 ln (p-v)(p-v ′ )(p-v)(p-v ′ )(p ′ -v)(p ′ -v ′ ) (vv ′ )(v 2 v ′2 ) = (p -v)(p -v ′ ) v v ′ e h J * β , (5.5)
where the header fraction is dimensionless. Then

R S β (k) =(2π) 4 e k[ j,i (p j -v i ) ln(p j -v i )-i v i ln(v i )] × (p -v)(p -v ′ ) v v ′ e h J * β . (5.6) Finally R S β (k) =(2π) 2 (p -v)(p -v ′ ) v v ′ e 1 2 h J * β +kh(i,J) .
(5.7)

Changes in the factors Σ β and F β (x)

From eqs.( 11) and (20) we define two sums Σ ′ β , Σ ′′ β to be gathered to form a a Σ β :

Σ ′ β = tmax t min (-1) t (t+1)! (t-v)!(t-v ′ )!(t-v-1 2 )!(t-v ′ -1 2 )!(p-t)!(p+ 1 2 -t)!(p ′ + 1 2 -t)! , (5.8)
Σ ′′ β = tmax t min (-1) t t! (t-v)!(t-v ′ )!(t-v-1 2 )!(t-v ′ -1 2 )!(p-t)!(p+ 1 2 -t)!(p ′ + 1 2 -t)! , (5.9) so that Σ β = -(2j ⋆ β + 1)Σ ′ β + [(2j ⋆ β + 1) + (p + 1 2 )(p ′ + 1 2 ) -vv ′ ] Σ ′′ β , (5.10)
with

t min = max (v, v ′ , v+
(-1) kx (kx+1)! (k(x-v))!(k(x-v ′ ))!(k(x-v)-1 2 )!(k(x-v ′ )-1 2 )!(k(p-x))!(k(p-x)+ 1 2 )!(k(p ′ -x)+ 1
2 )! .

(5.12) By using factorial approximations for large k we write Σ ′ β in detail as

Σ ′ β = 1 (2π) 3 x=t max x=t min G ′ β (x), (5.13) with G ′ β (x) = e [k ln(k)-k]x+k{ıπx+x ln(x)}+ 1 2 ln(k 3 x 3 ) × e -1 2 ln k(x-v)+[k ln(k)-k](x-v ′ )+k(x-v ′ ) ln(x-v ′ )+ 1 2 ln k(x-v ′ ) × e -[k ln(k)-k](x-v)+k(x-v) ln(x-v) × e -[k ln(k)-k](x-v)+k(x-v) ln(x-v)+[k ln(k)-k](x-v ′ )+k(x-v ′ ) ln(x-v ′ ) × e -[k ln(k)-k](p-x)+k(p-x) ln(p-x)+ 1 2 ln k(p-x) × e -[k ln(k)-k](p-x)+k(p-x) ln(p-x)+ln k(p-x) × e -[k ln(k)-k](p ′ -x)+k(p ′ -x) ln(p ′ -x)+ln k(p ′ -x) .
(5.14)

All rearrangements done it remains

G ′ β (x) = 1 k 2 e 1 2 ln x 3 Π(x-v i )Π(p j -x) + 1 2 ln (x-v)(x-v ′ ) (p-x)(p ′ -x)
× e k{ıπx+x ln(x)-(x-v i ) ln(x-v i )-(p j -x) ln(p j -x)} .

(5.15)

It results in Σ ′ β (k) = 1 (2π) 3 x=tmax x=t min 1 k 2 e 1 2 ln (x-v)(x-v ′ ) (p-x)(p ′ -x)] +F β ′ (x)+kf β ′ (x) , (5.16)
with

F β ′ (x) = 1 2 ln x 3 Π(x -v i )Π(p j -x) ≡ F(x), (5.17)
f β ′ (x) ≡ f(x). (5.18)
In the same way we derive

Σ ′′ β (k) = 1 (2π) 3 x=tmax x=t min 1 k 3 e 1 2 ln (x-v)(x-v ′ ) (p-x)(p ′ -x)] +F β ′′ (x)+kf β ′′ (x) , (5.19)
with

F β ′′ (x) = 1 2 ln x Π(x -v i )Π(p j -x) ≡ F(x) -ln(x), (5.20)
so that e F β ′′ (x) = 1 x e F(x) and f β ′′ (x) ≡ f(x).

(5.21)

Contribution to the saddle points (parity β)

Identifying Σ ′ β of eq. ( 5.16) as a Riemann sum with 1 k → dx leads to

Σ ′ β (k) = 1 (2π) 3 k tmax t min dx e 1 2 ln (x-v)(x-v ′ ) (p-x)(p ′ -x)] +F(x)+kf(x) .
(5.22) Equation (5.10), with j ⋆ β → kj ⋆ β , gives the following dominant contribution

-(2kj ⋆ β + 1)Σ ′ β (k) = - 2j ⋆ β (2π) 3 tmax t min dx e 1 2 ln (x-v)(x-v ′ ) (p-x)(p ′ -x)] +F(x)+kf(x) .
(5.23)

In the same way

Σ ′′ β (k) = 1 (2π) 3 k 2 tmax t min dx 1 x e 1 2 ln (x-v)(x-v ′ ) (p-x)(p ′ -x)] +F(x)+kf(x) .
(5.24)

Its coefficient in (5.10), [(2j ⋆ β + 1) + (p + 1 2 )(p ′ + 1 2 ) -vv ′ ], is rescaled into [(2kj ⋆ β + 1) + (kp + 1 2 )(kp ′ + 1 2 ) -k 2 vv ′ ] = k 2 [p p ′ -vv ′ ] + k[2j ⋆ β + 1 2 (p + p ′ )] + 3 4 . (5.25)
The Σ ′′ β (k) contribution is also dominant and can be written down as

1 x [p p ′ -vv ′ ] (2π) 3 tmax t min dx e 1 2 ln (x-v)(x-v ′ ) (p-x)(p ′ -x)] +F(x)+kf(x) . (5.26)
Finally the integral approximation has the form

Σ β = -2j ⋆ β + [p p ′ -vv ′ ] x 1 (2π) 3 x=tmax x=t min e 1 2 ln (x-v)(x-v ′ ) (p-x)(p ′ -x)] +F(x)+kf(x) .
(5.27)

Writing f(x) as the equation ( 27) used by Gurau for the saddle points

f(x ± ) = j 1 f j 1 (x ± )+j 2 f j 2 (x ± )+j 3 f j 3 (x ± )+J 1 f J 1 (x ± )+J 2 f J 2 (x ± )+J 3 f J 3 (x ± ), (5.28)
using its result, see eqs. (26-27-44) in [START_REF] Gurau | The Ponzano-Regge Asymptotic of the 6j Symbol: An Elementary Proof[END_REF],

ℜ(f j * β (x + )) = ℜ(f j * β (x -)) = -h j * β (not depending on x + or x -), (5.29)
our correlation table for j * β shows that the following identification holds

ℜ ln (x ± -v)(x ± -v ′ ) (p -x ± )(p ′ -x ± ) ≡ ℜ{f j * β (x ± )} = -h j * β .
(5.30)

In contrast the imaginary part is

ℑ f j * β (x ± ) = ±θ j * β . (5.31) It follows that Σ β (x ± ) = e -1 2 h j * β -2j ⋆ β + [p p ′ -vv ′ ] x ± 1 (2π) 3 e ± ı 2 θ j * β +F(x ± )+kf(x ± ) .
(5.32)

After the Laplace's approximation the final header factor is simply

1 √ 2πk .
Ultimately from eq. (5.7) the saddle points contributions for parity β have the form

Sdl β (x ± ) = 1 √ 2πk (p -v)(p -v ′ ) v v ′ e 1 2 h J ⋆ β × e -1 2 h j * β F(x ± ) -f ′′ (x ± ) -2j ⋆ β + [p p ′ -vv ′ ] x ± × e ± ı 2 θ j * β +k[h(i,J)+f(x ± )] .
(5.33)

From eqs. (32-33) in [START_REF] Gurau | The Ponzano-Regge Asymptotic of the 6j Symbol: An Elementary Proof[END_REF] clearly

-f ′′ (x -) = +ı √ △ x -Π((p j -x -)
.

(5.34)

From eq. ( 35) in [START_REF] Gurau | The Ponzano-Regge Asymptotic of the 6j Symbol: An Elementary Proof[END_REF] we obtain

Sdl β (x ± ) = 1 √ 2πk (p -v)(p -v ′ ) v v ′ e 1 2 h J ⋆ β × e k[j 1 (h j 1 +f j 1 )+j 2 (h j 2 +f j 2 )+j 3 (h j 3 +f j 3 )] × e k[J 1 (h J 1 +f J 1 )+J 2 (h J 2 +f J 2 )+J 3 (h J 3 +f J 3 )] × e 1 2 [f j 1 +f j 2 +f j 3 +f J 1 +f J 2 +f J 3 ] × e -1 2 h j * β 1 ∓ı √ △ -2j ⋆ β + [p p ′ -vv ′ ] x ± e ± ı 2 θ j * β . (5.35)
From △ = (24) 2 V 2 and 1 √ ∓ı = e ±ı π 4 we derive

1 ∓ı √ △ = 1 √ 24V e ±ı π 4 .
(5.36)

The result for Sdl β (x ± ) becomes the complex expression

Sdl β (x ± ) = 1 √ 48πkV (p -v)(p -v ′ ) v v ′ e 1 2 h J ⋆ β × e k[j 1 (h j 1 +f j 1 )+j 2 (h j 2 +f j 2 )+j 3 (h j 3 +f j 3 )+J 1 (h J 1 +f J 1 )+J 2 (h J 2 +f J 2 )+J 3 (h J 3 +f J 3 )] × e 1 2 [f j 1 +f j 2 +f j 3 +f J 1 +f J 2 +f J 3 ] × e -1 2 h j * β e ±ı π 4 -2j ⋆ β + [p p ′ -vv ′ ] x ± e ± ı 2 θ j * β .
(5.37)

From eq. ( 44) in [START_REF] Gurau | The Ponzano-Regge Asymptotic of the 6j Symbol: An Elementary Proof[END_REF] the real part of e k[j 1 (h j 1 +f j 1 )+j 2 (h j 2 +f j 2 )+j 3 (h j 3 +f j 3 )+J 1 (h J 1 +f J 1 )+J 2 (h J 2 +f J 2 )+J 3 (h J 3 +f J 3 )] is zero, then its remains only the imaginary part (the h j i ,J i being real) which is e ±ık[j 1 θ j 1 +j 2 θ j 2 +j 3 θ j 3 +J 1 θ J 1 +J 2 θ J 2 +J 3 θ J 3 ] whence

Sdl β (x ± ) = 1 √ 48πkV (p -v)(p -v ′ ) v v ′ e 1 2 h J ⋆ β × e ±ık[j 1 θ j 1 +j 2 θ j 2 +j 3 θ j 3 +J 1 θ J 1 +J 2 θ J 2 +J 3 θ J 3 ] × e ±ı 1 2 [θ j 1 +θ j 2 +θ j 3 +θ J 1 +θ J 2 +θ J 3 ] × e 1 2 ℜ[f j 1 +f j 2 +f j 3 +f J 1 +f J 2 +f J 3 ] × e -1 2 h j * β e ±ı π 4 -2j ⋆ β + [p p ′ -vv ′ ] x ± e ± ı 2 θ j * β .
(5.38)

That is

Sdl β (x ± ) = 1 √ 48πkV (p -v)(p -v ′ ) v v ′ × e -1 2 [h j 1 +h j 2 +h j 3 +h J 1 +h J 2 +h J 3 +h j * β -h J ⋆ β ] × [v + v ′ -p -p ′ ] + [p p ′ -vv ′ ] x ± e ±ı( π 4 +ϕ θ k β ) , (5.39)
where a new angle ϕ θ k β depending on θ j * β is defined via

ϕ θ k β = ι=3 ι=1 [(kj ι + 1 2 )θ jι + (kJ ι + 1 2 )θ Jι ] + 1 2 θ j * β . (5.40)
We have 1

x + = (B -ı △) 2C and 1 x - = (B + ı △) 2C , (5.41)
whence adding the contribution + and -, taken into account C = Π(v i ), √ △ = 24V , we obtain

1 Π(v i ) 2Π(v i )[v+v ′ -p-p ′ ]+B[p p ′ -vv ′ ] cos( π 4 +ϕ θ k β )+24V [p p ′ -vv ′ ] sin( π 4 +ϕ θ k β ) .
A preliminary result is then

kj 1 kj 2 kj 3 kJ 1 kJ 2 kJ 3 S β ≈ (-1) (v+v ′ -p) √ 48πkV Π(v i ) (p -v)(p -v ′ ) v v ′ × e -1 2 [h j 1 +h j 2 +h j 3 +h J 1 +h J 2 +h J 3 ]-1 2 [h j * β -h J ⋆ β ] × 2Π(v i )[v+v ′ -p-p ′ ]+B[p p ′ -vv ′ ] cos( π 4 +ϕ θ k β )+24V [p p ′ -vv ′ ] sin( π 4 +ϕ θ k β
) .

(5.42)

Factor 1 2 h J ⋆ β is compensatory, i.e. 1 2 h J ⋆ β term vanishes whereas 1 2 h J ⋆ β becomes h J ⋆ β .
Unfortunately this evidence will be more or less lost in the general formulas we try to explicit below.

Computation of e

h J * β -h j * β yields e h J * β -h j * β = e 1 2 ln (p-v)(p-v ′ )(p ′ -v)(p ′ -v ′ )(p-v)(p-v ′ ) v v ′ (p-v)(p-v ′ )(p-v)(p-v ′ (p ′ -v)(p ′ -v ′ ) v v ′ , (5.43) whence e -1 2 [h J * β -h j * β ] = 4 (p -v)(p -v ′ )(p -v)(p -v ′ (p ′ -v)(p ′ -v ′ ) v v ′ (p -v)(p -v ′ )(p ′ -v)(p ′ -v ′ )(p -v)(p -v ′ ) v v ′ .
(5.44) Besides from eqs. (8-9-11) in [START_REF] Gurau | The Ponzano-Regge Asymptotic of the 6j Symbol: An Elementary Proof[END_REF] we can write e -H(i,J) = e

-1 2 3 j=1 (h j j +h J j ) = e -1 4 ln Π i,j (p j -v i ) v 3 i = 4 Π i,j [v 3 i /(p j -v i )]
(5.45)

We conclude that

kj 1 kj 2 kj 3 kJ 1 kJ 2 kJ 3 S β ≈ (-1) (v+v ′ -p) √ 48πkV Π(v i ) (p -v)(p -v ′ ) v v ′ × 4 Π i,j [v 3 i /(p j -v i )] 4 (p -v)(p -v ′ )(p -v)(p -v ′ (p ′ -v)(p ′ -v ′ ) v v ′ (p -v)(p -v ′ )(p ′ -v)(p ′ -v ′ )(p -v)(p -v ′ ) v v ′ × 2Π(v i )[v+v ′ -p-p ′ ]+B[p p ′ -vv ′ ] cos( π 4 +ϕ θ k β )+24V [p p ′ -vv ′ ] sin( π 4 +ϕ θ k β ) = (-1) (v+v ′ -p) √ 48πkV 4 Π i,j [(p j -v i )v i ] × (p-v)(p-v ′ ) v v ′ 4 (p-v)(p-v ′ )(p-v)(p-v ′ (p ′ -v)(p ′ -v ′ ) v v ′ (p-v)(p-v ′ )(p ′ -v)(p ′ -v ′ )(p-v)(p-v ′ ) v v ′ × 2Π(v i )[v+v ′ -p-p ′ ]+B[p p ′ -vv ′ ] cos( π 4 +ϕ θ k β )+24V [p p ′ -vv ′ ] sin( π 4 +ϕ θ k β
) .

(5.46) Note that denominator 4 Π i,j [(p jv i )v i ] has the dimension of an area like √ Πv i in the formula (3.18) for approximating the supersymmetric symbol 6j S α . The following line with √ and 4 √ is dimensionless.

Results for the supersymmetric limits

Prior to properly present results as similar as possible with standard 6j symbols we will use a formula to shift the angular arguments, namely:

a cos(x) + b sin(x) = N cos(x -ψ), (6.1)
where

N = a 2 + b 2 and tan(ψ) = b a . (6.2)
Then for each parity α, γ, β, we get

N α = B 2 + (24V ) 2 , ψ α = arctan 24V B , (6.3
)

N γ = N α , ψ γ = -ψ α , (6.4
)

N β = √ (2Π(v i )[v+v ′ -p-p ′ ]) 2 +(24V [p p ′ -vv ′ ]) 2 , (6.5
)

ψ β = arctan 24V [p p ′ -vv ′ ] 2Π(v i )[v+v ′ -p-p ′ ] . (6.6)
We recall that V is the tetrahedron volume and B which also has the dimension of a volume is given by

B = 3 ι=1 j ι J ι 3 j=1 p j + 2(j 1 j 2 j 3 + j 1 J 2 J 3 + j 2 J 3 J 1 + j 3 J 1 J 2 ). (6.7)
Parity α, : Under a rescaling of all its arguments by a large k a supersymmetric 6j S α symbol behaves like

kj 1 kj 2 kj 3 kJ 1 kJ 2 kJ 3 S α = 1 √ 48πkV √ Πv i N α cos π 4 + ϕ θ k -ψ α , (6.8)
where the angle ϕ θ k is defined by

ϕ θ k = ι=3
(v+v ′ -p) √ 48πkV 4 Π i,j [(p j -v i )v i ] (p -v)(p -v ′ ) v v ′ × 4 (p-v)(p-v ′ )(p-v)(p-v ′ (p ′ -v)(p ′ -v ′ ) v v ′ (p-v)(p-v ′ )(p ′ -v)(p ′ -v ′ )(p-v)(p-v ′ ) v v ′ N β cos π 4 + ϕ θ k β -ψ β , (6.12)
where a new angle depending on θ j * β is defined as This means that only one term among the six (kj ι + 1 2 )θ jι , (kJ ι + 1 2 )θ Jι transforms into (kj j * β +1)θ j * β . Denominator in 4 √ has the dimension of an area like √ Πv i .

ϕ θ k β = ι=3 ι=1 [(kj ι + 1

Conclusion

Compared to a standard 6j symbol [SU (2)] a major difference with the supersymmetric 6j S π [OSP (1|2)] lies in a slower decay as a function of the scaling parameter k. Indeed it becomes 1 k instead of 1 k 3 . The standard term in cos contains always angular arguments depending on the six tetrahedral angles θ jι , θ Jι . However all the angles are shifted from their standard values by an angle different according to each parity π. For parity α the usual expressions like (kj ι + 1

2 )θ jι are unchanged. For parity γ (and k odd) the terms in 1 2 vanish so that it remains (kj ι )θ jι and so on. For parity β (and k odd) the angular dependence takes a form where a special angle θ j * β modifies the standard formula into (kj j * β + 1)θ j * β + ι (kj ι + 1 2 )θ jι + (kj ι + 1 2 )θ Jι | jι,Jι =j * β .

If k is even all supersymmetric 6j S π have the same asymptotic behaviour, ie that of parity α. Pertinent interpretations are clearly within the Quantum Supergravity framework.

Appendix: Expression of supersymmetric prefactors

R S α = j=3 j=1 i=4 i=1 (p j -v i )! i=4 i=1 (v i )! , (A.1) R S γ = j=3 j=1 i=4 i=1 (p j -v i -1 2 )! i=4 i=1 (v i + 1 2 )! , (A.2) R S β =(p -v)!(p -v ′ )!(p -v - 1 2 )!(p -v ′ - 1 2 )! × (p -v)!(p -v ′ )!(p -v - 1 2 )!(p -v ′ - 1 2 )! × (p ′ -v)!(p ′ -v ′ )!(p ′ -v - 1 2 )!(p ′ -v ′ - 1 2 )! × v!v ′ ! -1 (v + 1 2 )!(v ′ + 1 2 )! -1 (A.3)

4. 1

 1 Change in the prefactor R S γ Multiplying the spins of the prefactor R S γ by a scaling odd factor k yields

(4. 6 ) 4 . 2

 642 Changes in the factors Σ γ and F γ (x) From eqs. (11) and (21) let be two sums Σ ′ γ , Σ ′′ γ to be gathered to form a Σ γ : Σ

θbehaves like kj 1 kj 2 kj 3 kJ 1 kJ 2 kJ 3

 3 jι , θ Jι are the exterior dihedral angles of the tetrahedron corresponding to the edges j ι and J ι respectively. Parity γ: [If k is even, refer to formula (6.8)] Under a rescaling of all its arguments by a large k (odd) a supersymmetric 6j S γ ι θ jι + J i θ Jι ). (6.11)Parity β: [If k is even, refer to formula (6.8)] Under a rescaling of all its arguments by a large k (odd) a supersymmetric 6j S β behaves like kj 1 kj 2 kj 3 kJ 1 kJ 2 kJ 3

  similar to that of the standard sum Σ analyzed by Gurau whereas Σ ′′ β looks like our Σ α explicited in sect. 3.2. With all spins multiplied by the factor k, and changes t → kx, p, v → kp, kv, Σ

	1 2 , v ′ + 1 2 ) and t max = min (p, p+ 1 2 , p ′ + 1 2 ) .	(5.11)
	Computation of Σ
	the sum x=tmax x=t min over the quotient below	′ β (k) rewrites as

′

β is very


Marie-Camille Caumon

Alexandre Tarantola

SPECTRA OF AQUEOUS FLUID INCLU- Raman

-C M Caumon

RAMAN SPECTRA OF AQUEOUS FLUID INCLUSIONS: EFFECT OF MINERAL BIREFRINGENCE AND METASTABILITY ON SALINITY MEASUREMENT

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction: Crustal fluids play a major role in ore deposits, basin diagenesis and metamorphic reactions, among others. The knowledge of the chemical composition of individual fluid inclusions is essential for the understanding of past fluid transport and circulations. Microthermometry, by the observation of specific phase transitions as a function of temperature, is the general approach to determine salinity in aqueous fluid inclusions [START_REF] Roedder | Fluid inclusions[END_REF]. However in some cases, this method cannot be carried out because of, for instance, difficulties encountered in carrying optical observations, metastability, or if the thermodynamic system is not enough constraint. Besides, some minerals are highly cleavable, particularly carbonates, and consequently fluid inclusions might easily stretch, leak or decrepitate during freezing or heating on the heating-cooling stage, because of the volume expansion of the content with temperature.

In replacement of microthermometry, salinity can be determined by Raman spectroscopy using a method based on the change of the shape of the water stretching vibration band with salinity. This change is due to the decrease of the strength of the hydrogen bonds in water by dissolved anions, often chloride [START_REF] Walrafen | [END_REF]3]. An increase of the chloride concentration results in a decrease of the intensity of the OH stretching vibration band of water around 3200 cm -1 . Different methods of treatment of the raw spectrum correlate the deformation of the OH stretching vibration band of water to chlorinity [4][5][6][7][8][9].

The Raman signal of water is also sensitive to the fluid density and the birefringence of the host crystal. Birefringence is critical in quartz and carbonates, and more generally in all highly birefringent minerals. Depending on the crystallographic orientation of the sample, the value obtained for salinity in quartz samples displayed variations higher than 50 % [5,[7][8][9]. It has been observed that the "true" value is measured when the crystal is at its extinction position [6][7][8][9]. However, the optical phenomena behind the effect observed on the Raman spectra are not elucidated. Moreover, only the case of quartz was studied. Consequently, there is no general method at this time to determine salinity from the Raman signal of water in birefringent minerals. Metastability may also affect the OH stretching vibration band of liquid water, the decrease of the fluid density affecting the strength of hydrogen bonds.

Objectives: The purpose of the present study is to measure the effect of (i) metastability and (ii) mineral birefringence on the Raman spectra of water in aqueous fluid inclusions, in order to define a protocol to obtain accurate salinity measurement in any minerals, as an alternative to microthermometry. Some examples of fluid inclusions in 1. Kyanite (Zambia [10]), 2. Calcite [11], 3. Celestite, 4. Siderite [12].

Material and methods: Several minerals (quartz, calcite, siderite, barite, kyanite, celestite, and sphalerite) with various birefringence properties are selected. The Raman spectra of water in fluid inclusions are acquired as a function of the orientation of the optical axis toward laser polarization direction (Ar + laser) using a LabRAM or LabRAM HR microspectrometer (Horiba Jobin-Yvon). The salinity is compared to the one determined by microthermometry.

Phase diagram of water in the metastability range.

The effect of metastability on the Raman signal of water is measured during a heating-cooling cycle below homogenization temperature of a fluid inclusion in a quartz sample (Alpes, 2.0 mass% NaCl). The aqueous fluid inclusion is first heated up to Th (207°C). The Raman spectra of metastable water in the OH stretching vibration region are recorded by step of 5°C decreasing Abstract for 11 th GeoRaman International Conference, June 15-19, 2014, St. Louis, Missouri, USA temperature down to bubble nucleation (165°C). The fluid inclusion is then heating up to Th recording the spectra of water at equilibrium by step of 5°C. The spectra of metastable and equilibrated water are compared by measuring the intensities ratio between 3260 and 3425 cm -1 to determine the variations in hydrogen bonding [9].

Results: Birefringence. By elucidating the roles of the polarization properties of the grating, of the birefringence and optical system of the mineral, and of the polarization of the OH stretching vibrations of liquid water, a method is proposed to ensure reliable measurements of salinity by Raman spectroscopy not only in quartz but in any minerals. The experimental results help to define a protocol for the determination of salinity in aqueous fluid inclusions by Raman spectroscopy in case of birefringence of the host mineral.

Metastability. The Raman spectrum of water shows small differences when liquid water is metastable or at equilibrium with vapor (Fig. below). The intensity ratio is higher in metastable water between 170 and ~190°C, in agreement with a decrease of the strength of hydrogen bonding in metastable water.

Intensity ratio in the Raman spectrum of water as a function of temperature and heating-cooling cycle. (1) Metastable water is cooled down to 170°C. ( 2) Bubble nucleates at 165°C.

(3) Water at equilibrium with vapor heated up to 205°C. Th=207°C, 2.0 mass% NaCl.

Conclusion:

In some cases, microthermometry cannot be used to determine salinity in aqueous fluid inclusions. Raman microspectrometry is a good alternative technique if properly used. In case of birefringence of the host mineral, the section must be properly oriented in relation to the polarization direction of the laser. If the fluid inclusion is metastable, the effect of the physical state of the fluid must be considered in order to avoid overestimation of salinity.


Adrien Roussel
email: adrien.roussel@inria@ifpen.fr

Comparaison de moteurs exécutifs pour la parallélisation de solveurs linéaires itératifs

Keywords: moteur exécutif, calcul parallèle, solveur linéaire, méthode itérative

Dans le domaine de la simulation de réservoir, un grand système linéaire creux doit être résolu à chaque itération d'une méthode de Newton, ce qui est très consommateur en temps. Des méthodes numériques itératives sont utilisées pour résoudre ces équations. Elles rejouent la même séquence d'opérations jusqu'à converger vers la solution. Pour des systèmes non-structurés comme ceux que nous rencontrons, les méthodes itératives de Krylov comme le BiCGStab sont très utilisées. La parallélisation que nous proposons repose ici sur un modèle à base de tâches avec dépendances de données. Nous expérimentons notre implémentation au dessus de plusieurs moteurs exécutifs tels que : OpenMP, OmpSs, X-Kaapi et HARTS. Nous mettons en avant l'impact, parfois néfaste, que peut avoir la gestion des tâches. Un surcoût trop élevé peut freiner l'exécution, mais également un mauvais placement de tâche sur une architecture hiérarchique. La répétition à chaque itération de ces éléments peuvent impacter l'exécution.

Introduction

Dans le domaine de la simulation de réservoirs de pétrole, les modèles d'écoulements en milieux poreux conduisent à la discrétisation de systèmes d'équations aux dérivées partielles (EDP) selon un schéma aux volumes finis, qui amènent à la résolution d'un système nonlinéaire avec une méthode itérative de Newton. À chaque itération, un système linéaire creux non structuré doit être résolu. La méthode du gradient biconjugué (BiCGStab) [START_REF] Saad | Iterative Methods for Sparse Linear Systems[END_REF] est un solveur linéaire fréquemment utilisé pour ce type de systèmes. Les moteurs exécutifs sont des outils qui permettent d'exploiter efficacement les architectures hétérogènes émergentes. Chacun d'eux possède ses propres mécanismes internes qui sont plus ou moins efficaces selon les propriétés algorithmiques de l'application. Beaucoup de moteurs exécutifs récents reposent sur un modèle de programmation par tâches. Ces outils se doivent de porter une attention particulière aux méthodes itératives, notamment aux surcoûts qui se cumulent au cours des itérations.

Nous menons ici une étude comparative de différents moteurs exécutifs reposant sur un modèle de tâches avec dépendances de données, afin d'évaluer leurs performances sur des méthodes itératives d'algèbre linéaire. Même si certains moteurs exécutifs permettent d'exécuter des applications sur des architectures à mémoire distribuée, dans cet article, nous nous restreignons à l'exploitation de processeurs multi-coeurs à mémoire partagée afin de montrer comment profiter pleinement des performances au sein d'un noeud de calcul sur machine homogène. Nous avons implémenté une méthode itérative à destination des architectures NUMA multi-coeurs avec différents moteurs exécutifs. Nous comparons chaque implémentation parallèle, afin de mettre en évidence les impacts que peuvent avoir ces outils sur les performances d'une méthode itérative. Nous définissons plusieurs critères de comparaison que nous appliquons pour aider à analyser les résultats d'expériences sur une architecture NUMA. À l'exécution, nous cherchons à évaluer aussi bien le surcoût par itération que le placement des tâches selon la topologie de la machine. Dans un premier temps, nous détaillons chacun des moteurs exécutifs utilisés dans cette étude. Nous présentons ensuite un moyen de décrire de manière unique un algorithme itératif, indépendamment des moteurs exécutifs, grâce au développement d'une API d'algèbre linéaire. Nous nous consacrons par la suite à l'évaluation de performances de notre méthode. Enfin, nous concluons ces travaux dans la dernière partie de ce papier.

Moteurs exécutifs

L'utilisation d'environnements de programmation est un moyen efficace pour programmer les architectures parallèles. Certains embarquent leur propre compilateur (OpenMP, OpenACC, . . .) avec un moteur exécutif associé, alors que d'autres fournissent uniquement des API du moteur exécutif sous-jacent. Les moteurs exécutifs permettent de faire le lien entre le matériel et l'application afin de programmer ces architectures de manière transparente et efficace. Nous détaillons dans cette partie différents moteurs exécutifs basés sur des modèles de programmation par tâches avec dépendances de données. Parmi eux, le standard de programmation OpenMP [START_REF]OpenMP application program interface version 4[END_REF], OmpSs [START_REF] Ayguadé | Extending openmp to survive the heterogeneous multi-core era[END_REF], X-Kaapi [START_REF] Gautier | X-kaapi : a multi paradigm runtime for multicore architectures[END_REF] et HARTS [START_REF] Gratien | An abstract object oriented runtime system for heterogeneous parallel architecture[END_REF].

OpenMP

OpenMP [START_REF]OpenMP application program interface version 4[END_REF] est devenu avec les années un standard de programmation très répandu dans les modèles à mémoire partagée. A l'aide de directives de pré-compilation, il permet d'écrire des applications parallèles sans avoir à réécrire intégralement le code. Un pool de threads est instancié à l'entrée d'une région parallèle, et le travail est ensuite distribué entre les threads. Le modèle par tâche est entré en vigueur depuis la version 3.0 du standard grâce à la directive #pragma omp task, qui a ensuite été étendu par l'ajout des dépendances de données dans la version 4.0. L'ordonnancement des tâches prêtes dépend de l'implémentation du compilateur, et il n'existe aucune information à ce sujet.

OmpSs

OmpSs [START_REF] Ayguadé | Extending openmp to survive the heterogeneous multi-core era[END_REF] fournit un modèle de programmation par tâches utilisé aussi bien pour les architectures homogènes (cluster composé de processeurs multi-coeurs) que hétérogènes (multi-GPU). OmpSs est implémenté au dessus du compilateur Mercurium, et du moteur exécutif Nanos++. Il permet d'annoter un code de la même manière que OpenMP, en spécifiant les dépendances sur les données à l'instanciation d'une tâche. Plusieurs politiques d'ordonnancement sont fournies, qui peuvent prendre en compte aussi bien la topologie NUMA de l'architecture que des politiques de priorité des tâches.

Cas d'étude : les solveurs linéaires

Nous avons développé une API nous permettant de décrire chaque algorithme indépendamment du moteur exécutif utilisé. Dans cette partie, nous présentons notre API et détaillons comment sont partitionnées les données selon la topologie de l'architecture.

API abstraite d'algèbre linéaire parallèle

Les algorithmes de solveurs linéaires, en particulier les méthodes itératives de Krylov, peuvent être vus comme des séquences successives d'opérations algébriques qui sont exécutées à chaque itération. Ces opérations peuvent aussi bien être des opérations de type BLAS (niveau 1 ou 2), que des produits matrice vecteur creux ou l'application d'un préconditionneur. Nous proposons ici une autre manière d'implémenter des méthodes itératives parallèles en utilisant une programmation par tâche, nous permettant par la suite de viser différents moteurs exécutifs. Notre API abstraite nous permet de cacher aux utilisateurs les spécificités des moteurs exécutifs, comme l'instanciation des tâches, les outils de partitionnement des données et la construction du DAG de tâches. Une des séquences du solveur BiCGStab [START_REF] Saad | Iterative Methods for Sparse Linear Systems[END_REF] teurs (comme OmpSs et XKaapi) permettent de placer les tâches en fonction de la localité des données ou simplement sur un noeud de calcul spécifique. Pour XKaapi, une simple clause dans la définition de la tâche suffit. Pour OmpSs, il faut appeler une fonction interne au moteur et spécifier où sera exécuter la tâche.

Partitionnement et localité des données

Les techniques de parallélisation de solveurs itératifs sont décrites dans [START_REF] Saad | Iterative Methods for Sparse Linear Systems[END_REF] et reposent sur le partitionnement du graphe d'adjacence de la matrice, G A = (V, E). L'ensemble des sommets V représente les lignes ou les colonnes de la matrice, tandis que E désigne ses valeurs non nulles. Les algorithmes de partitionnement permettent de diviser ce graphe en sous-graphes, où chaque partition k est responsable d'un sous-ensemble de G A noté G k = (V k , E k ). Une tâche reçoit alors un pointeur vers la sous-matrice ainsi formée par ce sous-ensemble. Nous exécutons en parallèle des tâches d'allocation et d'initialisation pour chaque sous-ensemble V k . De cette manière, les matrices sont allouées sur les bancs NUMA où sont exécutées ces tâches. Cette initialisation parallèle nous permet de pouvoir bénéficier de certaines optimisations dans le cas où le moteur exécutif prend en compte la localité des données au moment d'ordonnancer les tâches. Dans ce cas, on annote la tâche pour qu'elle s'exécute sur le même banc NUMA où a été initialisée la matrice.

Étude comparative : de la description des tâches jusqu'à l'exécution

Dans cette partie, nous comparons les comportements des moteurs exécutifs présentés en section 2 sur un solveur linéaire BiCGStab [START_REF] Saad | Iterative Methods for Sparse Linear Systems[END_REF] préconditionné. Les systèmes linéaires proviennent soit d'un ensemble de matrices M R extraites de simulations réelles de réservoir, soit d'un autre ensemble de matrices M Lp venant de la discrétisation aux volumes finis d'un problème de Laplace 2D sur un cube unitaire. Nous notons N le nombre de lignes de la matrice. La machine utilisée pour nos expérience est composée de 2 bancs NUMA reliés par un bus QPI, où chacun possède 32Gb de mémoire et un processeur 8 coeurs Intel Xeon E5-2670 cadencé à 2.60GHz. Dans un premier temps, nous cherchons à mettre en avant les différences de performances entre les différentes exécutions sur des systèmes aux tailles différentes. Nous cherchons par la suite à extraire les surcoûts engendrés par les moteurs exécutifs. Enfin, nous cherchons à voir les mécanismes en place prenant en compte la topologie des architectures NUMA. Dans les expériences présentées, T seq désigne le temps séquentiel pour exécuter une séquence d'opérations sans moteur exécutif, et T p le temps pour exécuter celle-ci sur p processeurs. Nous utilisons ici la version OpenMP 4.0 implémentée par le compilateur Gnu GCC 4.9.0, OmpSs avec les versions 0.10a de Nanos++ et 1.99.9 de Mercurium et enfin X-Kaapi en version 3.1.0 rc10 1 .

Travaux similaires

Bien que des bibliothèques proposent des implémentations parallèles pour les opérations d'algèbre linéaire, aucune ne propose un modèle de programmation par tâches. Nous pouvons notamment citer Hypre [START_REF]Hypre : High performance preconditioners[END_REF] qui propose une implémentation à travers un modèle de programmation par passage de message avec MPI, ou MUMPS [START_REF] Amestoy | Shared memory parallelism and low-rank approximation techniques applied to direct solvers in FEM simulation[END_REF] qui repose sur une implémentation hybride MPI-OpenMP. Les auteurs de [START_REF] Olivier | Comparison of openmp 3.0 and other task parallel frameworks on unbalanced task graphs[END_REF] comparent plusieurs moteurs exécutifs OpenMP 3.0 à d'autres bibliothèques ou langages, tels que Cilk et IntelTBB, sur des graphes de tâches déséquilibrés, qui différent des opérations que nous avons évaluées ici. Les résultats maintenant dépassés ont pu montrer d'importantes variations de performances entre les différentes applications. Dans [START_REF] Lacoste | Taking advantage of hybrid systems for sparse direct solvers via task-based runtimes[END_REF], deux moteurs exécutifs (StarPU [START_REF] Augonnet | Starpu : A unified platform for task scheduling on heterogeneous multicore architectures[END_REF] et PARSEC [START_REF] Bosilca | A generic distributed dag engine for high performance computing[END_REF]) sont proposés pour remplacer l'ordonnanceur spécialisé du solveur linéaire parallèle PaStiX2 . Bien que les moteurs exécutifs sont efficaces, les auteurs ont pu observer des différences de performances entre les outils.

Conclusion

Nous avons implémenté ici un solveur linéaire itératif en utilisant 4 moteurs exécutifs tels que OpenMP 4.0, OmpSs, X-Kaapi et HARTS. Nous avons pu testé chaque implémentation sur des systèmes venant de divers horizons, afin d'étudier le comportement de chacun vis-à-vis de la structure itérative des algorithmes sur des machines parallèles. Nous avons également pu étudier les stratégies adoptées sur des architectures fortement NUMA. Sur un socket, les différences entre les moteurs exécutifs se ressentent dûes aux différentes implémentations des mécanismes internes. Ces différences s'élargissent dès lors que l'on passe sur plusieurs bancs NUMA. Nous avons montré l'intérêt de la persistance de tâches et de DAGs qui sont réutilisés à chaque itération des méthodes itératives telles que les solveurs linéaires. Nous avons également mis en avant l'importance de prendre en compte la localité des données à l'ordonnancement des tâches sur des machines de type NUMA, pour ainsi réduire la latence et gagner en performances. D'autres tests vont être menés notamment sur la gestion de placement de tâches avec X-Kaapi sur les architectures NUMA.

Remerciements :

Ce travail a été partiellement supporté par le projet IRSES2011-295217 HPC-GA. Les auteurs remercient les personnes du BSC pour leurs commentaires et leur aide sur ces travaux.

FIGURE 4 -

 4 FIGURE 4 -Maille : 2000x2000 permettant de toujours décroitre le temps de calculs. L'exécution d'un graphe de tâches moins équilibré comme pour le BiCGStab aurait pu montrer plus de disparités.

  Dans ce papier, nous expérimentons une heuristique basée sur l'affinité au sein d'un banc NUMA, qui utilise des piles de tâches par banc NUMA pour placer les tâches là où sont allouées les données.

	Compas'2016 : Parallélisme / Architecture / Système
	Lorient, France, du 5 au 8 juillet 2016
	2.3. X-Kaapi
	X-Kaapi [8] est une bibliothèque C permettant la création non bloquante de tâches. Le parallé-
	lisme est explicite alors que la détection de synchronisations est implicite. Les machines visées
	sont aussi bien homogènes que hétérogènes [9, 12]. X-Kaapi propose plusieurs APIs (C, C++,
	Fortran) et fournit un moteur exécutif OpenMP avec la compatibilité des binaires sur le moteur
	exécutif libGOMP de GCC. X-Kaapi est structuré autour de la notion d'ordonnancement par
	vol de tâches et des ouvriers qui exécutent le code des tâches et prennent des décisions locales
	d'ordonnancement.

2.4. HARTS

Le moteur exécutif HARTS

[START_REF] Gratien | An abstract object oriented runtime system for heterogeneous parallel architecture[END_REF]

repose sur des concepts abstraits. La bibliothèque est basé sur plusieurs modèles. Le modèle d'architecture est décrit via Hwloc

[START_REF] Broquedis | hwloc : a Generic Framework for Managing Hardware Affinities in HPC Applications[END_REF]

pour fournir des informations sur la topologie de l'architecture. Le modèle de tâche permet de décrire une tâche avec plusieurs représentations selon les architectures. Ces tâches sont ensuite gérées via un pool centralisé de tâches. Un algorithme est décrit par un Graphe Acyclique Direct (DAG) avec les dépendances entre tâches. Les DAGs et les tâches sont persistants entre différentes itérations d'un programme. Le modèle de données permet d'encapsuler les données associées aux tâches. Le modèle d'exécution maintient la cohérence entre ces modèles, en distribuant les tâches prêtes sur les ressources disponibles via l'ordonnanceur. Une racine du DAG est donnée à chaque ressource lors de l'exécution.

  est décrite dans l'algorithme 1, puis ensuite traduite avec l'API dans le listing 1. Nous avons implémenté cette API au dessus de différents moteurs exécutifs tels que OpenMP, OmpSs, X-Kaapi et HARTS. La plupart des moteurs exécutifs détruisent les tâches après leur exécution et on les ré-instancie à chaque itération. Parmi ceux présentés, HARTS est le seul à permettre la persistance des tâches : la structure du DAG est construite une seule fois à travers

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		Listing 1 -BiCGStab sequence
	Algorithm 1: BiCGStab Algorithm	AlgebraKernelType alg;
	Matrix A; Vector b, p, pp, r, v; Scalar a; do pp = inv(P).p; v = A.p ; r += v; a = dot(p,r); if(a==0) break; ...; while (|r|<tol*|b|);	Matrix A; Vector p,pp,r,v; double alpha; SequenceType seq = alg.newSequence(); alg.exec(precond,p,pp,seq) ; alg.mult(A,pp,v,seq); alg.axpy(1.,r,v,seq); alg.dot(p,r,alpha,seq); alg.assertNull(alpha,seq); while(!iter.stop()) { alg.process(seq); }

l'API et les tâches sont instanciées à cette construction. Pour les autres, on stocke des structures de tâches propres à l'API (avec les pointeurs sur les données associées au calcul de la tâche) dans un vecteur, i.e. une séquence. A chaque exécution de la séquence, les tâches sont instanciées dans le moteur exécutif visé. Lors de la création de la tâche, certaines annotations peuvent être fournies pour le placement de celles-ci selon une stratégie définie. Certains mo-

  En observant les courbes, on voit qu'elles ont toutes la même tendance en fonction du nombre de threads. Nous avons préalablement vérifié que les performances des différents moteurs exécutifs ne varient pas beaucoup selon la taille du système. OpenMP et X-Kaapi présentent des performances similaires, bien que le second a démontré lors d'une précédente expérience ses faibles taux d'insertion de tâches. HARTS présente des performances plus élevées que les autres car il n'a pas à instancier les tâches à chaque itération contrairement aux autres moteurs exécutifs. Sur un nombre inférieur d'itérations, les coûts d'insertion de tâches dans HARTS sont légèrement supérieurs à X-Kaapi. Facteur NUMA et BiCGStab tations OpenMP et HARTS sur une machine composée de 24 bancs NUMA, chacun ayant une mémoire de 32Gb et un processeur 8 coeurs cadencé à 2,40GHz. L'initialisation parallèle est effective pour que les données soient allouées sur des pages mémoires proches du processeur concerné. Les résultats sont illustrés dans la figure3. Nous pouvons premièrement remarquer une perte d'efficacité de notre application sur un grand nombre de threads, comme déjà vu dans la section 4.1, qui peut s'expliquer par de trop lourdes communications et un manque de parallélisme. HARTS perd lentement en efficacité, alors que les performances d'OpenMP stagnent jusqu'à 12 bancs NUMA avant de se dégrader brusquement. HARTS distribue les tâches cycliquement entre les processeurs. Celles avec le même identifiant de partition sont donc toujours exécutées par le même processeur tout en évitant de rapatrier des données distantes. Après vérification, OpenMP ne place pas les tâches selon la localité des données, donc la latence augmente et les performances se dégradent. Dans le cas de X-Kaapi et OmpSs, chacun permet d'annoter les tâches pour les placer selon la topologie NUMA. X-Kaapi utilise un mot clé à la définition d'une tâche pour gérer le placement d'une tâche selon la localité d'une donnée. OmpSs fait appel à une fonction interne du moteur exécutif pour placer explicitement la tâche sur un banc NUMA. En pratique, nous avons vérifié que ces politiques sont bien respectées, mais nos expérimentations sont toujours en cours.Dans un deuxième temps, nous avons regardé plus en détails la multiplication d'une matrice creuse par un vecteur (SpMV), opération essentielle lors de la résolution de tels systèmes linéaires. Contrairement à l'expérience précédente où seules les matrices sont distribuées, les vecteurs le sont également sur l'ensemble des bancs NUMA disponibles afin de réduire les coûts de latence et la contention mémoire. Chaque vecteur est ainsi découpé selon le partitionnement de graphe de la matrice, et à chaque domaine est associé une portion de vecteur. Nous avons fait tourner nos tests avec le moteur exécutif XKaapi (avec interface OpenMP), mais également OpenMP 4.0 provenant de Gcc 5.2.0 et aussi du compilateur Clang version 3.8.0. Les résultats obtenus sur la même machine que l'expérience précédente sont illustrés Figure4, sur une matrice provenant de M Lp , en prenant la mesure de 500 itérations de SpMV. Grâce à sa politique d'ordonnancement des tâches prenant en compte la localité des données, X-Kaapi obtient des performances supérieures aux autres moteurs exécutifs. En revanche pour les

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						Clang
	TIME (SECONDS)	20 40				GCC XKaapi
		0
		2	4	8	10 12	16 32 48 64 92 128 160 192
		FIGURE 1 -Accélération du BiCGStab sur 2 bancs NUMA CORES
	4.1. Comparatif d'efficacité parallèle Dans le but de comparer les différentes implémentations d'un solveur BiCGStab avec un pré-conditionneur polynomial [3], nous exécutons successivement cette application sur p threads, FIGURE 2 -Prec. Diagonal -N = 2 188 842 FIGURE 3 -Vecteurs et matrices distribués
	1. Version en développement

parmi p = {1, 2, 4, 8, 10, 12, 16}, sur la collection de matrices M Lp avec différentes tailles N. Nous calculons ensuite pour chaque exécution l'accélération, S(p) = T seq /T p . Les résultats sont présentés dans la Figure

1

. En analysant les courbes, on peut voir qu'elles ont toutes la même tendance, ce qui est sans doute dû au fait que nous parallélisons cet algorithme sur le même modèle. Lorsqu'on passe sur le second banc NUMA (i.e. p > 8), on voit que les écarts augmentent entre les courbes. Sur un seul banc NUMA, l'écart entre les courbes s'explique par les surcoûts liés aux mécanismes de gestion des tâches. En revanche, au delà d'un socket, les mécanismes mis en places par les moteurs exécutifs pour prendre en charge la localité des données à l'ordonnancement sont également impliqués et expliquent des écarts de plus en plus grands : ces écarts sont d'autant plus importants lorsque l'initialisation parallèle du système est inactive.

4.2. Surcoût des moteurs exécutifs

Nous cherchons maintenant à évaluer de manière plus générale les écarts de surcoûts entre les différents moteurs exécutifs. Nous exécutons un solveur BiCGStab avec préconditionneur diagonal sur des matrices issues de M R de taille N différentes, sur plusieurs threads d'un seul banc NUMA, p = {1, 2, 4, 8}, avec un nombre fixe d'itérations, N iter = 1000. Nous analysons ici l'efficacité parallèle, Eff(p) = ((T seq /T p )/p), de chaque implémentation. Les résultats sont présentés dans la figure

2

, pour une taille de matrice donnée. Le surcoût d'un runtime par rapport à un autre est représenté par la différence entre les deux courbes d'efficacité.

4.3. Placement de tâches sur bancs NUMA

L'optimisation des temps de latence passe par l'ordonnanceur qui se doit de prendre en compte la localité des données. Pour évaluer ce facteur, nous avons exécuté un solveur parallèle avec un préconditionneur polynomial sur p threads, équitablement répartis sur N Numa bancs NUMA, où p = {8, 24, 48, 96, 144, 192} et N Numa = {1, 4, 6, 12, 18, 24}. Le système utilisé de taille N = 62 480 000 est issu de M Lp . Nous comparons les comportements observés pour les implémen-autres, malgré que les réductions de temps de calculs en fonction du nombre de coeurs utilisés, les performances sont moindres car les tâches ne sont pas bien placées. Cela se vérifie dès que l'on passe sur plus d'un banc NUMA (i.e. p > 8) où là le temps de calcul augmente par rapport au runtime X-Kaapi. En revanche, les tâches s'occupant des mêmes partitions sont toujours exécutées sur les même coeurs. De ce fait, l'exécution de celles-ci bénéficient d'un effet de cache

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Guillaume R Coudevylle

Kathleen Martin Ginis

Jean-Pierre Famose

Kathleen Martin

Kathleen A Martin Ginis

DETERMINANTS OF SELF-HANDICAPPING STRATEGIES IN SPORT AND THEIR EFFECTS ON ATHLETIC PERFORMANCE

Keywords: self-handicapping, self-esteem, self-confidence, performance, sport

HAL is

Self-handicapping in Sport 3 Determinants of Self-handicapping Strategies in Sport and their Effects on Athletic Performance Self-handicapping is a self-presentational strategy whereby people make claims or engage in behaviors prior to an upcoming performance, in order to have a reasonable excuse for a possible failure [START_REF] Berglas | Drug choice as a self-handicapping strategy in response to non contingent success[END_REF]. From an attributional perspective, self-handicaps blur the relationship between ability and performance. Thus, a self-handicapping athlete who performs poorly can attribute failure to the performance impediment rather than ability or competence whereas the athlete who performs well creates the impression of being especially competent and talented, because success was achieved despite impediments [START_REF] Tice | Esteem protection or enhancement? Self-handicapping motives and attributions differ by trait self-esteem[END_REF].

Theorists have highlighted the important conceptual distinction between behavioral and claimed self-handicapping (e.g., [START_REF] Hirt | Self-reported versus behavioral selfhandicapping: Empirical evidence for a theoretical distinction[END_REF][START_REF] Leary | Behavioral self-handicapping vs. self-reported handicaps: A conceptual note[END_REF].

However, effort has not been directed toward determining whether each has different determinants. The majority of the published sport studies have focused on determinants of claimed self-handicapping (e.g., [START_REF] Kuczka | Self-handicapping in competitive sport: Influence of the motivational climate, self-efficacy, and perceived importance[END_REF][START_REF] Martin | Self-handicapping in physical achievement settings: The contributions of self-esteem and self-efficacy[END_REF] or on determinants behavioral self-handicapping (e.g., [START_REF] Berglas | Drug choice as a self-handicapping strategy in response to non contingent success[END_REF][START_REF] Pyszczynski | Determinants of reduction of intended effort as a strategy for coping with anticipated failure[END_REF]. The few studies that have examined both types have assumed common determinants (e.g., [START_REF] Elliot | Achievement goals, self-handicapping, and performance attainment: A mediational analysis[END_REF][START_REF] Thompson | Self-handicapping status, claimed self-handicaps and reduced practice effort following success and failure feedback[END_REF].

Behavioral self-handicaps are a more costly, riskier strategy than claimed self-handicaps.

For example, not practicing can provide an excuse for poor performance, but also decreases the likelihood of a successful performance. In contrast, simply claiming to be too anxious or tired also serves as an excuse for poor performance, but does not actually reduce the likelihood of success (see [START_REF] Hirt | Self-reported versus behavioral selfhandicapping: Empirical evidence for a theoretical distinction[END_REF]. Given these differences, it is possible that different factors underlie the use of these two strategies (see Coudevylle, Famose, Martin Ginis, & Gernigon, 2007).

Knowledge of these factors would contribute to a better understanding of predictors of selfhandicapping and the development of interventions to alter the use of self-handicaps.

Self-handicapping in Sport 4

Athletes who use behavioral self-handicaps may take such a risk because they are convinced of an impending performance failure. In line with this reasoning, [START_REF] Pyszczynski | Determinants of reduction of intended effort as a strategy for coping with anticipated failure[END_REF] showed that participants in a low-probability of success condition intended to exert less effort on an upcoming highly ego-relevant task, than did participants in a highprobability of success condition. As social cognitive theory proposes that maladaptive behaviors--such as reduced effort--result primarily from low expectations of mastery in an uncertain situation [START_REF] Bandura | Social foundations of thought and action: A social cognitive theory[END_REF][START_REF] Bandura | Self-efficacy: The exercise of control[END_REF], it is plausible that the use of behavioral self-handicaps may be linked to situation-specific low levels of performance self-efficacy.

In contrast, athletes who use claimed self-handicaps may suffer from more general selfdoubts; they may not necessarily believe that failure is imminent, but they may have weak beliefs about their general levels of capability, success and self-worth. Within the self-handicapping literature, feeble self-beliefs have been conceptualised as low self-esteem (see [START_REF] Tice | Esteem protection or enhancement? Self-handicapping motives and attributions differ by trait self-esteem[END_REF].

Athletes with low self-esteem may not want to risk sabotaging an upcoming performance with a costly behavioral self-handicap, but just in case they do fail, they may desire the protection of a relatively low-risk, claimed self-handicap. Indeed, Martin and Brawley (2002, Study 2) found that low self-esteem was associated with greater claimed self-handicapping prior to a fitness test.

The primary purpose of the present experiment was to examine self-efficacy and selfesteem as predictors of claimed and behavioral self-handicapping. It was predicted that selfefficacy would be negatively correlated with the use of a behavioral self-handicap whereas selfesteem would be negatively correlated with the use of claimed self-handicaps. A secondary purpose was to compare the relationship between behavioral and claimed self-handicaps, and athletic performance. Presumably, behavioral self-handicaps would have a negative effect on performance because they lower one's chance of success [START_REF] Hirt | Self-reported versus behavioral selfhandicapping: Empirical evidence for a theoretical distinction[END_REF][START_REF] Leary | Behavioral self-handicapping vs. self-reported handicaps: A conceptual note[END_REF]. Indeed, reduced preparatory effort (i.e., a behavioral self-handicap) has been linked with Self-handicapping in Sport 5 poorer performance (e.g., [START_REF] Elliot | Achievement goals, self-handicapping, and performance attainment: A mediational analysis[END_REF]. In contrast, claimed self-handicaps may improve performance by reducing pressure to perform well [START_REF] Leary | Behavioral self-handicapping vs. self-reported handicaps: A conceptual note[END_REF][START_REF] Ryska | Effects of trait and situational self-handicapping on competitive anxiety among athletes[END_REF]. Two studies involving children produced conflicting results regarding the effects of claimed handicaps on performance [START_REF] Elliot | Achievement goals, self-handicapping, and performance attainment: A mediational analysis[END_REF][START_REF] Ryska | Effects of situational self-handicapping and state self-confidence on the physical performance of young participants[END_REF]. Thus, further research is warranted to address this issue. We hypothesized that performance on a basketball task would be negatively related to behavioral self-handicapping and positively related to claimed selfhandicapping.

Method

Participants

Participants were 31 competitive basketball players, 16 men (M age = 20.8 years; SD = 4.8) and 15 women (M age = 19.6 years; SD = 3.9). All players competed at the French regional level-a sufficiently competitive level to ensure that participants would be personally invested in the experimental task and its outcome.

Measures

General Self-esteem was measured by the French version [START_REF] Vallières | Traduction et validation canadienne-française de l'échelle de l'estime de soi de Rosenberg[END_REF]) of Rosenberg's Self-Esteem Scale (Rosenberg, 1965). Participants rated the 10 scale items on a four-point Likert scale (1 = strongly disagree; 4 = strongly agree).

Self-efficacy was measured using a procedure adapted from [START_REF] Bandura | Analysis of self-efficacy theory of behavioral change[END_REF].

Participants indicated if they believed that they could make 1, 2, 3, 4, and 5 successful passes through the obstacle course (yes/no response) and if so, their degree of confidence (0% = not sure; 100% = totally sure). A mean score was calculated across the items to which participants responded 'yes'.

Claimed Self-handicapping was measured using a state measure similar to that used and validated by Martin and Brawley (2002, Study 2). The scale consisted of 14 impediments that athletes may Self-handicapping in Sport 6 use as self-handicaps such as: "I am feeling tired," "I have personal concerns in this moment," and "I am ill." The impediments were those most frequently cited by athletes in studies of selfhandicapping in sport (e.g., [START_REF] Carron | Group effects and self-handicapping[END_REF]. Participants indicated whether each impediment was present (yes/no), and the extent to which it would interfere with their performance (i.e., perceived impact), using a scale ranging from 0 (not at all) to 6 (extremely). A total score was calculated by summing the impact scores for the impediments that were present.

Two items were reversed scored before being added to the total.

Behavioral Self-handicapping. Within the context of sports participation, a lack of preparatory effort constitutes an index of behavioral self-handicapping [START_REF] Prapavessis | Self-presentational issues in competition and sport[END_REF].

Thus, behavioral self-handicapping was operationalized as the amount of time, in minutes, spent practicing during a warm-up period. A measurement protocol used by [START_REF] Harris | The role of uncertain self-esteem in self-handicapping[END_REF] and [START_REF] Tice | Esteem protection or enhancement? Self-handicapping motives and attributions differ by trait self-esteem[END_REF] was employed (see procedure).

Performance. Performance was scored by counting the number of successful shots and passes completed on an obstacle course (described next). Performance scores could range from 0 to 11.

The Experimental Task

The task consisted of completing three repetitions of an obstacle course laid out on a standard basketball court. The test began with the completion of two free-throws followed by running to mid-court, and then turning around backwards and performing a backward defensive shuffle to the end of the court. Then, the participant picked up a basketball and dribbled it to midcourt. At mid-court, the participant dribbled the ball between three pylons while bouncing the ball between the legs and passing it behind the back. The participant then continued to the basket and attempted a lay-up. After the lay-up, the participant repeated the entire course again and finished with a jump shot. The course was repeated a third time, finishing with a three-point shot.

Procedure

Self-handicapping in Sport 7

To maximize the likelihood of self-handicapping, an experimental situation was created in which factors that are known to elicit self-handicapping (for a review, see [START_REF] Coudevylle | Les stratégies d'autohandicap : Caractéristiques et déterminants[END_REF] were made salient. Specifically, the testing situation emphasized public performance, an emphasis on results, social comparisons of one's performance with others, and public awareness of the use of self-handicaps. Participants performed the test in the presence of spectators, coaches and other players, were told that their test results would be used to identify the best and worst players in the region, and that their scores would be compared with their teammates' scores.

After hearing and reading this information, participants completed the measures of selfesteem and self-efficacy. Then, they completed the measure of claimed self-handicapping. In order to mask the true purpose of the self-handicapping questionnaire, participants were told: "The following questionnaire is intended to know your general actual state in order to comment on your results today. It will allow those that will evaluate your competence (the experimenter, your trainer, the other players, the spectators) to consider your current personal situation when authenticating your results and comparing them with those of the other participants. Your responses to this questionnaire will be made public along with your results so that those who evaluate your performance will be better able to evaluate your competence".

After completing the claimed self-handicapping measure, the experimenter explained the warm-up procedure. In accordance with [START_REF] Harris | The role of uncertain self-esteem in self-handicapping[END_REF] and [START_REF] Tice | Esteem protection or enhancement? Self-handicapping motives and attributions differ by trait self-esteem[END_REF], participants were told that they could have as much time as they desired to warm-up. To make sure that all participants realized that a lack of preparatory effort could serve as a behavioral self-handicap, the experimenter told the participants that a lack of preparatory effort could provide an explanation for a good or bad performance on the test. He also reminded players that a good warm-up is essential to a good performance and that a bad warm-up can provoke a bad performance. Therefore, participants knew that the lack of practice before the test could be Self-handicapping in Sport 8 detrimental to their performance (see [START_REF] Harris | The role of uncertain self-esteem in self-handicapping[END_REF][START_REF] Tice | Esteem protection or enhancement? Self-handicapping motives and attributions differ by trait self-esteem[END_REF]. As no participant had previously performed the experimental task, one can assume that a warm-up would be essential to successful performance of the task.

Participants then warmed-up and the experimenter timed each participant's warm-up.

Once the participant indicated that she/he was ready, the experimental task proceeded. Upon completion of the task, the participant was debriefed regarding the true purpose of the study.

Results

Descriptive statistics

Three female participants refused to perform the warm-up and test due to injury. Thus, for analyses relating to behavioral self-handicapping and performance, n = 28. For all other analyses, n = 31. Prior to testing our hypotheses, ANOVAs were conducted to examine possible sex differences in self-handicapping (see [START_REF] Kuczka | Self-handicapping in competitive sport: Influence of the motivational climate, self-efficacy, and perceived importance[END_REF] and performance. As these ANOVAs were not significant (ps > .05), the data were collapsed across sex for all subsequent analyses.

Correlation Analyses

Table 1 shows that as predicted, claimed self-handicaps were significantly negatively correlated with self-esteem (r = -.48, p < .01), but not self-efficacy (r = -.12, p > .05), such that athletes with lower self-esteem reported a greater impact of performance impediments. Also as predicted, time spent warming up was significantly positively correlated with self-efficacy (r = .52, p < .01), but not self-esteem (r = -.35, p > .05). Thus, as self-efficacy increased, behavioral self-handicapping decreased as indexed by greater time spent practicing. In partial support of our third hypothesis, performance was correlated with time spent warming up (r = .50, p < .01), but not with the impact of claimed self-handicaps (r = .07, p > .05).

Self-handicapping in Sport 9

Discussion

The primary purpose of the present experiment was to examine self-efficacy and selfesteem as predictors of behavioral and claimed self-handicaps. The results supported the hypotheses and demonstrated that self-efficacy was negatively correlated with the use of a behavioral self-handicap (i.e., short time of preparation) whereas self-esteem was negatively correlated with the use of claimed self-handicaps. Our results support the notion that the use of behavioral self-handicaps may be linked to situation-specific low levels of self-efficacy for performance on the upcoming task. Athletes who use behavioral self-handicaps may take such a risk because they are convinced of an impending performance failure, as evidenced by their lower levels of task self-efficacy. Although behavioral self-handicappers may feel that they have nothing to lose (in terms of their performance) by using a behavioral self-handicap, they can still protect their self-and public-images by using such a strategy.

Our findings were consistent with previous research which showed that individuals who anticipated an upcoming failure intended to use a behavioral self-handicap [START_REF] Pyszczynski | Determinants of reduction of intended effort as a strategy for coping with anticipated failure[END_REF]. Importantly, our findings extended this work by showing that athletes with self-doubts will actually use a behavioral self-handicap (i.e., time spent warming up). With regard to the use of claimed self-handicaps, our results supported [START_REF] Martin | Self-handicapping in physical achievement settings: The contributions of self-esteem and self-efficacy[END_REF] findings, as claimed self-handicaps were significantly negatively correlated with self-esteem Athletes who use claimed self-handicaps seem to suffer from more general self-doubts (i.e., low self-esteem). As predicted, they may prefer to use claimed self-handicaps, and not behavioral self-handicaps, because they hope to succeed in spite of their low self-esteem.

The second purpose was to compare the relationships between behavioral and claimed self-handicaps and athletic performance. The results provided partial support for our hypotheses.

Specifically, performance on a basketball task was negatively related to behavioral self-Self-handicapping in Sport 10 handicapping, but was not related to claimed self-handicapping. Our findings are consistent with those of Elliot and colleagues (2006) who showed that reduced preparatory effort (i.e., a behavioral self-handicap) is linked with poorer performance. However, contrary to previous research [START_REF] Leary | Behavioral self-handicapping vs. self-reported handicaps: A conceptual note[END_REF][START_REF] Ryska | Effects of trait and situational self-handicapping on competitive anxiety among athletes[END_REF], our results did not show that claimed selfhandicaps improve performance. In these previous studies, it was suggested that self-handicaps improve performance by reducing performance pressure; perhaps in the present study, because social evaluative pressures were made so salient, the reduction of the anxiety from claimed selfhandicaps was not sufficient to enhance performance.

Our results make an important contribution to theory by providing the first sports-based evidence of the conceptual distinction between self-handicapping behaviors and claims. We have shown that the two strategies are indeed related to different factors and can have different consequences for performance. From an applied perspective, given that self-efficacy was related to behavioral self-handicapping which was, in turn, associated with poorer performance, our results suggest that coaches, sport psychologists, and other interventionists should endeavour to promote strong self-efficacy among their athletes. Not only is self-efficacy important for success (e.g., [START_REF] Bandura | Self-efficacy: The exercise of control[END_REF] but as the observed correlation between self-efficacy and warm-up time indicate, self-efficacy is also important for deterring the use of harmful behavioral self-handicaps. Despite these contributions, a couple of limitations warrant mention. First, given the relatively small number of participants with a similar level of ability, the generalizability of our study findings are limited. Second, the measure of behavioral self-handicapping could be confounded by factors such as over-confidence or weak motivation. Future research would benefit from testing these hypotheses with a larger, more heterogeneous sample, and controlling for potential confounds of behavioral self-handicapping.

Self-handicapping in Sport 11

Self-handicapping in Sport 13

Table 1 .

 1 Correlations, Means and Standard Deviations of the Variables of the Study

	2	3	4	5	M	SD

Note: * p < .05; ** p < .01 + Time spent warming up is the measure of behavioral self-handicapping. Thus, less warm-up time is indicative of greater self-handicapping.