Jean Dolbeault

An Zhang

Flows and functional inequalities for fractional operators

Keywords: fractional Gagliardo-Nirenberg-Sobolev inequality, fractional Sobolev inequality, fractional fast diffusion equation, self-similar solutions, asymptotic behavior, intermediate asymptotics, rate of convergence, entropy methods, carré du champ, Rényi entropy powers, entropy -entropy production inequality, self-similar variables, linearization AMS Subject Classifications: Primary: 26A33, 35R11, Secondary: 35A23, 35B33, 35B40, 35K65

This paper collects results concerning global rates and large time asymptotics of a fractional fast diffusion on the Euclidean space, which is deeply related with a family of fractional Gagliardo-Nirenberg-Sobolev inequalities. Generically, self-similar solutions are not optimal for the Gagliardo-Nirenberg-Sobolev inequalities, in strong contrast with usual standard fast diffusion equations based on non-fractional operators. Various aspects of the stability of the self-similar solutions and of the entropy methods like carré du champ and Rényi entropy powers methods are investigated and raise a number of open problems.

Introduction

Recently many papers have been devoted to the extension of results involving second order partial differential operators to fractional operators. More specifically, fractional diffusion equations have been studied from the point of view of existence, comparison and regularity of the solutions, not only in the linear case but also in the framework of nonlinear models of porous medium or fast diffusion type. Concerning modeling issues, fractional diffusions are usually motivated by microscopic jump processes. One can refer to [START_REF] Bouchaud | Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications[END_REF][START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractional dynamics approach[END_REF] for an extended review of models arising from various areas of physics. We will not go in this direction. Relying on known theoretical results like the ones of [START_REF] Caffarelli | Regularity of solutions of the fractional porous medium flow[END_REF][START_REF] Caffarelli | Nonlinear porous medium flow with fractional potential pressure[END_REF][START_REF] Caffarelli | Asymptotic behaviour of a porous medium equation with fractional diffusion[END_REF][START_REF] De Pablo | A general fractional porous medium equation[END_REF][START_REF] De Pablo | Classical solutions for a logarithmic fractional diffusion equation[END_REF], our approach aims at the description of fundamental qualitative properties of such equations in relation with closely associated, basic functional inequalities.

Standard fast diffusion or porous medium equations have simple features which arise from the homogeneity of the nonlinear term or from the fact that the diffusion operator is of second-order. These features explain the special role of selfsimilar solutions, known as Barenblatt-Pattle, in the large time regimes: see [START_REF] Vázquez | Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type[END_REF] for an historical presentation. Also remarkable is the fact discovered in [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF] that the Barenblatt-Pattle solutions are optimal for some Gagliardo-Nirenberg-Sobolev inequalities, which are essential to measure the asymptotic stability of the selfa Jean Dolbeault, Email: dolbeaul@ceremade.dauphine.fr b Corresponding author: An Zhang, Email: zhang@ceremade.dauphine.fr similar solutions using relative entropy methods. Entropy functionals are indeed deeply connected with fast diffusion equations, as these equations can be seen as gradient flows of the entropies with respect to the Wasserstein's distance (see [START_REF] Otto | The geometry of dissipative evolution equations: the porous medium equation[END_REF]).

In the context of fractional fast diffusion or porous medium equations, it is therefore very natural to question the role of self-similar solutions in terms of large time asymptotics and related functional inequalities. However, fractional order derivatives and nonlinearities do not combine well and this raises a number of difficulties. Self-similar solutions which generalize the Barenblatt-Pattle solutions to fractional diffusion have been studied in connection with large time asymptotics in [START_REF] Caffarelli | Nonlinear porous medium flow with fractional potential pressure[END_REF][START_REF] Caffarelli | Asymptotic behaviour of a porous medium equation with fractional diffusion[END_REF] and [START_REF] Biler | Barenblatt profiles for a nonlocal porous medium equation[END_REF][START_REF] Biler | The nonlocal porous medium equation: Barenblatt profiles and other weak solutions[END_REF][START_REF] Carrillo | Exponential convergence towards stationary states for the 1D porous medium equation with fractional pressure[END_REF]. There is a certain flexibility in the generalization of the standard nonlinear diffusion equations: this has been investigated in [START_REF] Huang | Explicit Barenblatt profiles for fractional porous medium equations[END_REF] and [START_REF] Stan | Transformations of self-similar solutions for porous medium equations of fractional type[END_REF].

In the present paper, we shall specifically rely on the fractional fast diffusion equation

∂u ∂t = ∇ • √ u ∇(-∆) -s u m-1 2
with m < 1, which seems particularly well adapted to entropy -entropy production inequalities as we shall see below. Such an equation has already been considered in [START_REF] Stan | Transformations of self-similar solutions for porous medium equations of fractional type[END_REF]Equation (MG)] from the point of view of self-similarity. Our purpose is to clarify the interplay of this equation with fractional Gagliardo-Nirenberg-Sobolev inequalities (see Section 2.3), and observe that, generically, optimal functions for these inequalities differ from self-similar solutions that are supposed to govern the large time behavior of the solutions of the evolution equation (Proposition 2.3). Beyond some results about, e.g., the existence of optimal solutions for the interpolation inequalities (Proposition 2.1), we raise a number of open questions concerning the large time asymptotics in Section 3 and the applicability of the Bakry-Emery, or carré du champ, method in Section 4.

Preliminaries: a fractional interpolation inequality and a fractional fast diffusion flow

The fractional Sobolev inequality

According to [START_REF] Chen | Remainder terms in the fractional Sobolev inequality[END_REF], for any α ∈ (0, d), with q = 2 d d-α , the fractional Sobolev inequality in R d can be written as

w 2 H α 2 (R d ) ≥ S d,α R d |w| q dx 2 q ∀ w ∈ H α 2 (R d ) (1)
where

H α 2 (R d ) is the space of all tempered distributions w such that ŵ ∈ L 1 loc (R d ) and w 2 H α 2 (R d ) := R d |ξ| α | ŵ| 2 dξ < ∞ .
Here ŵ denotes the Fourier transform of w and the optimal constant is given by

S d,α = 2 α π α 2 Γ( d+α 2 ) Γ( d-α 2 ) Γ( d 2 ) Γ(d) α/d .
Up to translations, dilations and multiplications by a nonzero constant, the optimal function is

w ⋆ (x) = 1 + |x| 2 -d-α 2 ∀ x ∈ R d .
It is easy to check that w ⋆ solves (-∆)

α 2 w ⋆ = C d,α w d+α d-α ⋆ where C d,α = 2 α Γ( d+α 2 ) Γ( d-α 2 ) . (2)
With the notation D α := ∇ (-∆) α-1 2 , we get that (-∆)

α 2 = D * α/2 D α/2
and (1) can then be written as

w 2 H α 2 (R d ) = D α/2 w 2 L 2 (R d ) ≥ S d,α w 2 L q (R d ) ∀ w ∈ H α 2 (R d ) .
The case α = 2 was established in [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF][START_REF] Talenti | Best constant in Sobolev inequality[END_REF], but one may refer to [START_REF] Rosen | Minimum value for c in the Sobolev inequality φ 3 ≤ c ∇φ 3[END_REF] in case d = 3 and even to [START_REF] Bliss | An integral inequality[END_REF] for some early considerations on optimal functions. In the fractional case α = 2, the dual form of the optimal inequality, i.e., the Hardy-Littlewood-Sobolev inequality, was established in [START_REF] Lieb | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF], while considerations on the case α = 1 in connection with trace issues can be found in [START_REF] Escobar | Sharp constant in a Sobolev trace inequality[END_REF]. Later work include [START_REF] Swanson | The best Sobolev constant[END_REF] and [START_REF] Chang | Fractional Laplacian in conformal geometry[END_REF] among many other related papers. The issue of the symmetry of the optimal functions, up to translations, was considered in [START_REF] Lieb | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF] and established in [START_REF] Chen | Classification of solutions for an integral equation[END_REF] using the moving planes method.

A fractional Gagliardo-Nirenberg-Sobolev inequality

With q = 2 d d-α , a simple Hölder interpolation shows that

w L 2p (R d ) ≤ w 1-ϑ L p+1 (R d ) w ϑ L q (R d ) with ϑ = d p p -1 d + α -p (d -α) .
Together with Sobolev's inequality, this provides an interpolation inequality of Gagliardo-Nirenberg-Sobolev type. More precisely, we have the following statement.

Proposition 2.1 Assume that α ∈ (0, d) and p ∈ (1, q] with q = 2 d d-α . With the above notations, the following Gagliardo-Nirenberg-Sobolev inequality

w L 2p (R d ) ≤ C GNS w ϑ H α 2 (R d ) w 1-ϑ L p+1 (R d ) ∀ w ∈ D(R d ) (3)
holds with an optimal constant C GNS ≤ S ϑ/2 d,α . Moreover, there exists a positive optimal function in

H α 2 (R d ) ∩ L p+1 (R d ) which solves the Euler-Lagrange equation (-∆) α 2 w + w p -w 2 p-1 = 0 , x ∈ R d . (4)
In the above statement D(R d ) denotes the space of smooth functions with compact support, but it is simple to extend it by density to the space w ∈ H α 2 (R d ) :

R d w p+1 dx < ∞ . See [26, Theorem 2.1] for details. When p = q 2 = d d-α , then ϑ = 1 and C GNS = S d,α
. The symmetry of the optimal functions is not considered here and the interested reader is invited to refer for instance to [27] for a related problem.

Sketch of the proof. This result is out of the scope of the present paper and the proof is relatively standard, so let us give only a simple sketch. A reader interested in further details is invited to refer to [START_REF] Trabelsi | Optimal functions in fractional Gagliardo-Nirenberg-Sobolev inequalities[END_REF] and [START_REF] Hajaiej | Existence of minimizers of functionals involving the fractional gradient in the absence of compactness, symmetry and monotonicity[END_REF] for further details in similar cases. The above result can actually be seen as a special case of those dealt with in [START_REF] Hajaiej | Fractional Gagliardo-Nirenberg and Hardy inequalities under Lorentz norms[END_REF]. The convergence of sequences in the critical case has been analyzed in details in [START_REF] Gérard | Description du défaut de compacité de l'injection de Sobolev[END_REF]. We can therefore assume that p ∈ (1, q).

An optimal function for (3) can be obtained by considering

I M := inf w 2 H α 2 (R d ) + w p+1 L p+1 (R d ) : R d w 2p dx = M . An optimisation on λ > 0 of w λ = λ d 2p w(λ•)
shows that the minimization of I M is equivalent to the characterization of the optimality case in (3), and also that

I M = M γ I 1 with γ = d + α -p (d -α) d -p (d -2 α) < 1 .
With these preliminary observations, a concentration-compactness analysis can be done as follows. Let us consider a sequence (w n ) n∈N of nonnegative function such that R d w 2p

n dx = M for any n ∈ N and

lim n→∞ w n 2 H α 2 (R d ) + w n p+1 L p+1 (R d ) = I M .
1) The following result has been stated in [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case. II[END_REF]

, Lemma I.1, p. 231]. Lemma 2.2 Let R > 0 and p ∈ [1, 2 * /2). If (w n ) n is bounded in H 1 (R d ) and if lim sup n→∞ BR(x) |w n | 2p dy = 0 for any x ∈ R d , then lim n→∞ w n L r (R d ) = 0 for any r ∈ [2, 2 * ).
Here 2 * = ∞ if d = 1 or 2, and 2

* = 2 d/(d -2) if d ≥ 3
. By an analogue of this result in H α 2 (R d ), with 2 * /2 replaced by q, vanishing cannot occur, which means that, after eventually replacing w n by w n (• + y n ) for some unbounded sequence (y n ) n∈N of points in R d , we have lim

R→∞ lim n→∞ BR(0) |w n | 2p dy > 0 .
2) Since γ < 1, it is straightforward to check that

I M1+M2 ≤ I M1 + I M2 ,
which prevents dichotomy. As a consequence, for any ε > 0, there exists some R > 0, large enough, such that lim n→∞ BR(0)

|w n | 2p dy ≥ M -ε .
In other words, this means that the sequence (w 2p n ) n∈N is tightly relatively compact in the sense of measures.

3) Concentration is forbidden by Sobolev's inequality [START_REF] Bouchaud | Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications[END_REF]. Altogether, this proves that (w n ) n∈N is weakly relatively compact in L 2p (R d ) (see for instance [START_REF] Autuori | Elliptic problems involving the fractional laplacian in n[END_REF]Lemma 2.2]) and the conclusion holds by lower semi-continuity.

Let us take the logarithm of both sides in (3). An optimal function is a minimizer of the functional

w → ϑ log w H α 2 (R d ) + (1 -ϑ) log w L p+1 (R d ) -log w L 2p (R d ) .
A variation shows that

ϑ (-∆) α 2 w w H α 2 (R d ) + (1 -ϑ) w p w L p+1 (R d ) - w 2 p-1 w L 2p (R d ) = 0 .
Using the homogeneity in (3) and a multiplication by a constant, the coefficients can be adjusted so that w solves (4).

A fractional fast diffusion equation

Let us consider the case

α = 2 (1 -s) .
Any nonnegative solution u of

∂u ∂t = ∇ • √ u ∇(-∆) -s w with w = u m-1 2 (5)
which is smooth enough and has good decay properties as |x| → ∞ is such that

E ′ = 2 m (1-m) 2 m-1 I , (6)
where the entropy E and the Fisher information I are defined respectively by

E[u] := R d u m dx and I[u] := R d ∇ (1-s) u m-1 2 2 dx ,
and ∇ (1-s) is defined by

∇ (1-s) w := ∇(-∆) -s/2 w .
Above we consider E[u(t, •)] as a function of t if u solves ( 5) and

E ′ denotes the t-derivative of E[u(t, •)].
Because the right hand side in ( 5) is in divergence form, we know that

M := R d u dx
is independent of t ≥ 0 for any solution u which is smooth enough, and with enough decay as |x| → +∞. Let us assume that

m ≥ m 1 := 1 2 + 1 q = 1 - α 2 d . If m = m 1 and v ⋆ := w q ⋆ , then v ⋆ (x) = 1 + |x| 2 -d satisfies v m1-1 2 ⋆ = w ⋆ and (-∆) 1-s v m1-1 2 ⋆ = C d,α v d+α 2 d ⋆ = C d,α 1 + |x| 2 s-1 √ v ⋆
according to [START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractional dynamics approach[END_REF]. Assume now that m ∈ [m 1 , 1). If we introduce a generalized Rényi entropy power functional F as in [START_REF] Savaré | The concavity of Rényi entropy power[END_REF][START_REF] Dolbeault | Nonlinear diffusions: Extremal properties of Barenblatt profiles, best matching and delays[END_REF], defined by

F := E σ with σ := 2 ϑ 1 -ϑ p + 1 + 1 = m -m c 1 -m , p = 1 2 m -1 and m c := 1 - α d ,
then by applying Proposition 2.1 to w = u m-1 2 we obtain that

F ′ ≥ κ using (6)
, where the constant κ depends only on d, s (or α), m and M, and can be computed as

κ := 2 m (1 -m) 2 m -1 σ M C 2p GNS 1 p ϑ = 2 m 2 m -1 (m -m c ) M C 2p GNS d+α-p (d-α) d (p-1)
.

Self-similar solutions

Let us introduce the time-dependent change of variables

u(t, x) = 1 R d v log R, x R (7)
where R = R(t) is given by the ordinary differential equation

R µ-1 dR dt = 1 , R(0) = 1 , (8)
and the parameter µ is defined by

µ := d (m -m c ) with m c = 1 - α d .
If m is in the range [m 1 , 1), it is straightforward to check that µ is positive and the scale R has the explicit expression

R(t) = (1 + µ t) 1/µ ∀ t ≥ 0 .
If u solves [START_REF] Caffarelli | Asymptotic behaviour of a porous medium equation with fractional diffusion[END_REF], it is then easy to check that v is a solution of

∂v ∂t = ∇ • √ v ∇(-∆) -s v m-1 2 + x √ v . (9)
This change of variable is classical in case of non-fractional operators but has also been considered in the case of the fractional Laplacian, for instance in [START_REF] Biler | Barenblatt profiles for a nonlocal porous medium equation[END_REF][START_REF] Biler | The nonlocal porous medium equation: Barenblatt profiles and other weak solutions[END_REF]. A detailed analysis of self-similar profiles is available in [START_REF] Stan | Transformations of self-similar solutions for porous medium equations of fractional type[END_REF]. In the fast diffusion case, for an equation related with [START_REF] Caffarelli | Asymptotic behaviour of a porous medium equation with fractional diffusion[END_REF], we refer to [START_REF] Vázquez | Optimal estimates for fractional fast diffusion equations[END_REF], where the role of Barenblatt type solutions for regularization effects has been clarified.

Proposition 2.3 Let p = 1 2 m-1 .
Under the assumptions of Proposition 2.1, if w is an optimal function for (3), then v = w 2p is not a bounded, positive, stationary solution of [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF]

such that I[v] is finite, unless s = 0, α = 2 or m = d+1 d+2 (1-s) .
In other words, we claim that (4) does not provide stationary solutions of (9) as it is the case in the non-fractional case, except in the special case corresponding to m = d+1 d+2 (1-s) , which was observed in [START_REF] Huang | Explicit Barenblatt profiles for fractional porous medium equations[END_REF][START_REF] Stan | Transformations of self-similar solutions for porous medium equations of fractional type[END_REF].

Proof. We argue by contradiction and consider a solution of

∇ • √ v ∇(-∆) -s v m-1 2 + x √ v = 0 . We deduce that w = v m-1 2 solves 0 = ∇(-∆) -s v m-1 2 + x √ v = ∇(-∆) -s w + x w p
where p = 1/(2 m -1) and, as a consequence,

-(-∆) 1-s w + ∇ • (x w p ) = 0 .
With α = 2 (1 -s), if w simultaneously solves (4) up to a multiplication by a constant, a scaling and a translation, this means that w p = v is such that

∇ • (x w p ) = a w 2 p-1 -b w p
for some positive constants a and b that can be adjusted. With

a = p p -1 and b = p p -1 -d ,
we find that

w(x) = 1 + |x| 2 1 1-p ∀ x ∈ R d .
The result follows from [14, Theorem 3.1].

The case m = m 1 is almost explicit. In order to illustrate Proposition 2.3, let us give some details. With the notations of Section 2.1, if we choose

v m1-1 2 ⋆ = w ⋆ ⇐⇒ v ⋆ = w we obtain that ∇ (-∆) -1 (-∆) 1-s v m1-1 2 ⋆ + x √ v ⋆ = ∇ (-∆) -1 C d,α w d+α d-α ⋆ + x √ v ⋆ = ∇ (-∆) -1 C d,α v d+α 2 d ⋆ -C d,2 v d+2 2 d ⋆ = 0 unless α = 2.
For later purpose, let us define the self-similar profile B as the unique radial solution of

∇(-∆) -s B m-1 2 + x √ B = 0 such that R d B dx = M.
We recall that these solutions are the so-called Barenblatt profiles when s = 0 and refer to [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF] for more details. It is straightforward to check that

u ⋆ (t, x) = 1 R d B log R, x R
is a self-similar solution of ( 5) if R = R(t) is given by (8).

Global rates of convergence and formal large time asymptotics

The generalized Rényi entropy power functional defined as in Section 2.3 by [START_REF] Caffarelli | Asymptotic behaviour of a porous medium equation with fractional diffusion[END_REF] with initial datum u 0 on the one hand, and a direct computation shows that the self-similar solution u ⋆ satisfies the relation

F[u] = R d u m dx σ is such that F[u(t, •)] ≥ F[u 0 ] + κ t ∀ t ≥ 0 if u solves
F[u ⋆ (t, •)] = F[B] (1 + µ t) ∀ t ≥ 0 .
Corollary 3.1 With the above notations and under the assumptions of Proposition 2.3, we have the estimate

κ < κ ⋆ := µ F[B] .
Proof. The inequality κ ≤ κ ⋆ is a straightforward consequence of the above computations, with u 0 = B. The strict inequality follows from Proposition 2.3. Now let us investigate the large time asymptotics at formal level. Inspired by [START_REF] Vázquez | Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type[END_REF][START_REF] Vázquez | Optimal estimates for fractional fast diffusion equations[END_REF], we expect that for any m ∈ [m 1 , 1), u(t, x) ∼ u ⋆ (t, x) as t → +∞ as in [START_REF] De Pablo | Nonlocal filtration equations with rough kernels[END_REF]Theorem 1.4], and thus

F[u(t, •)] = κ ⋆ t 1 + o(1) = F[B] (1 + µ t) 1 + o(1) .
Using the self-similar change variables ( 7)-( 8), the equivalence u ∼ u ⋆ amounts to

v(t, •) → B as t → +∞ .
So far this question is open. Next let us investigate at formal level the question of the attraction of the solutions by the self-similar solutions.

If u solves ( 5) and u ⋆ is the self-similar solution with same mass, we define the relative entropy by

E[u] = 1 m-1 R d u m -u m ⋆ -m u m-1 ⋆ (u -u ⋆ ) dx .
A straightforward computation shows that

d dt E[u(t, •)] = m m-1 R d u m-1 -u m-1 ⋆ ∂u ∂t dx -m R d u m-2 ⋆ (u -u ⋆ ) ∂u ⋆ ∂t dx = -m m-1 R d ∇ u m-1 -u m-1 ⋆ • √ u ∇(-∆) -s u m-1 2 dx + m R d ∇ u m-2 ⋆ (u -u ⋆ ) • √ u ⋆ ∇(-∆) -s u m-1 2 ⋆ dx .
At formal level, we investigate the limit as ε → 0 of

u ε = u ⋆ 1 + ε u 1 2 -m ⋆ f .
First, let us notice that

√ u ε = √ u ⋆ + 1 2 ε u 1-m ⋆ f + o(ε) , u m-1 2 ε = u m-1 2 ⋆ + m -1 2 ε f + o(ε) , u m-1 ε = u m-1 ⋆ + (m -1) ε f √ u ⋆ + 1 2 (m -1) (m -2) ε f 2 u m ⋆ + o(ε 2 ) .
As a consequence, we find that

E[u] = m 2 ε 2 R d u 1-m ⋆ f 2 dx + o(ε 2 )
and

d dt E[u(t, •)] = -m 2 ε 2 Q u⋆ [f ] + o(ε 2 )
where the quadratic form Q u⋆ is defined by

Q u⋆ [f ] := R d ∇ f √ u⋆ • (2 m -1) √ u ⋆ ∇(-∆) -s f + u 1-m ⋆ f ∇(-∆) -s u m-1 2 ⋆ dx + (m -2) R d ∇ f 2 u m ⋆ • √ u ⋆ ∇(-∆) -s u m-1 2 ⋆ dx .
If s = 0, we observe that

1 2 m-1 Q u⋆ [f ] = R d u ⋆ ∇ f √ u⋆ 2 dx -1-m 2 R d (2 -m) |∇u ⋆ | 2 u 2 ⋆ - ∆u ⋆ u ⋆ f 2 dx .
To characterize the asymptotic stability of u ⋆ , the point is to notice that

Q u⋆ [f ] controls R d u 1-m ⋆ f 2 dx.
Proposition 3.2 With the previous notations, if s = 0 and m 1 ≤ m < 1, there exists a positive constant C which depends on only on m and d such that

Q u⋆ [f ] ≥ 2 1 -m C R µ R d u 1-m ⋆ f 2 dx for any smooth function f such that R d u ⋆ dx)) such that Q u⋆ [f ] is finite for any t ≥ 0.
Proof. The result follows from the change of variables [START_REF] De Pablo | Classical solutions for a logarithmic fractional diffusion equation[END_REF] and [38, Corollary 1]: the above inequality can be rewritten in the form of a spectral gap inequality with a constant which is independent of t. Also see [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF] for earlier spectral gap computations.

In self-similar variables, Proposition3.2 shows that B is linearly stable, and this is enough to prove that u ⋆ is asymptotically linearly stable when s = 0. Actually one also knows that B is globally stable when s = 0, with explicit global rates, which is a far deeper issue. Proposition 3.2 immediately raises two open questions:

(i) Is u ⋆ asymptotically linearly stable for any s ∈ (0, n) ? (ii) Can we prove that u ⋆ is, under the appropriate mass normalization, a global attractor and deduce that the asymptotic rate of convergence of any solution towards u ⋆ is determined by the spectral gap Λ associated with the inequality

Q B [g] ≥ Λ R d B 1-m g 2 dx
for any admissible function g such that

R d B 3
2 -m g dx = 0 ?

4. On the Bakry-Emery method applied to the fractional fast diffusion equation and generalized Rényi entropy powers

In this section we do a computation based on the generalized Rényi entropy powers in case s = 0 and emphasize the points which differ when s > 0. So far such a computation has been done using the so-called pressure variable u m-1 as, for instance, in [START_REF] Savaré | The concavity of Rényi entropy power[END_REF][START_REF] Dolbeault | Nonlinear diffusions: Extremal properties of Barenblatt profiles, best matching and delays[END_REF]. Here we base our computations on

f := u m-1 2 .
With this simple change of variables, we obtain that

∂f ∂t = u m-1 m -1 2 ∆f + 1 2 |∇f | 2 f
and compute the time derivative of the Fisher information

I[u] = R d |∇f | 2 dx
as

I ′ = -2 R d (∆f ) ∂f ∂t dx = -(2 m -1) R d u m-1 (∆f ) 2 dx - R d u m-1 ∆f |∇f | 2 f dx .
The computation of the r.h.s. relies on two identities. Here we assume that integrations by parts can be taken without any special precautions. A detailed discussion of this issue can be found in [START_REF] Dolbeault | Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization[END_REF].

• First identity: we use an integration by parts to obtain

- R d u m-1 ∆f |∇f | 2 f dx = R d ∇f • ∇ u m-1 |∇f | 2 f dx = R d (∇u m-1 ) • ∇f |∇f | 2 f dx + 2 R d u m-1 Hf : ∇f ⊗ ∇f f dx - R d u m-1 |∇f | 4 f 2 dx
where Hf denotes the Hessian matrix of f . After integrating by parts again, we obtain

- R d u m-1 ∆f |∇f | 2 f dx = 2 R d u m-1 Hf : ∇f ⊗ ∇f f dx-1 2 m-1 R d u m-1 |∇f | 4 f 2 dx (10) using ∇ u m-1 = 2 (m-1) 2 m-1 u m-1 ∇f f . (11)
• Second identity: using again [START_REF] Biler | Barenblatt profiles for a nonlocal porous medium equation[END_REF], we find that

-(2 m -1) R d u m-1 (∆f ) 2 dx = (2 m -1) R d ∇f • ∇ u m-1 ∆f dx = 2 (m -1) R d u m-1 ∆f |∇f | 2 f dx + (2 m -1) R d u m-1 ∇f • ∇∆f dx .
Next we use the identity ∇f • ∇∆f = 1 2 ∆ |∇f | 2 -Hf 2 and obtain

-(2 m -1) R d u m-1 (∆f ) 2 dx = 2 (m -1) R d u m-1 ∆f |∇f | 2 f dx -(2 m -1) R d u m-1 Hf 2 dx + m -1 2 R d u m-1 ∆ |∇f | 2 dx
where, using twice [START_REF] Biler | Barenblatt profiles for a nonlocal porous medium equation[END_REF], we obtain that

R d u m-1 ∆ |∇f | 2 dx = 2 (m-1) 2 m-1 R d |∇f | 2 ∇ • u m-1 ∇f f dx = 2 (m-1) 2 m-1 R d u m-1 ∆f |∇f | 2 f dx + 2 (m-1) 2 m-1 2 (m-1) 2 m-1 -1 R d u m-1 |∇f | 4 f 2 dx = 2 (m-1) 2 m-1 R d u m-1 ∆f |∇f | 2 f dx -2 (m-1) (2 m-1) 2 R d u m-1 |∇f | 4 f 2 dx .
Notice that the above computation will cause trouble for a generalization to the case of a function f which is defined as a non-local expression of u m-1 2 . Hence

-(2 m -1) R d u m-1 (∆f ) 2 dx = 3 (m -1) R d u m-1 ∆f |∇f | 2 f dx -(2 m -1) R d u m-1 Hf 2 dx -m-1 2 m-1 R d u m-1 |∇f | 4 f 2 dx .
We can evaluate R d u m-1 ∆f |∇f | 2 f dx using [START_REF] Otto | The geometry of dissipative evolution equations: the porous medium equation[END_REF] and get that

-(2 m -1) R d u m-1 (∆f ) 2 dx = -3 (m -1) 2 R d u m-1 Hf : ∇f ⊗ ∇f f dx -1 2 m-1 R d u m-1 |∇f | 4 f 2 dx -(2 m -1) R d u m-1 Hf 2 dx -m-1 2 m-1 R d u m-1 |∇f | 4 f 2 dx .
This provides the second identity

-(2 m -1) R d u m-1 (∆f ) 2 dx = -(2 m -1) R d u m-1 Hf 2 dx -6 (m -1) R d u m-1 Hf : ∇f ⊗ ∇f f dx + 2 (m-1) 2 m-1 R d u m-1 |∇f | 4 f 2 dx . ( 12
)
Collecting terms of our two identities [START_REF] Otto | The geometry of dissipative evolution equations: the porous medium equation[END_REF] and ( 12), we have found that

I ′ = -(2 m -1) R d u m-1 Hf 2 dx -2 (3 m -4) R d u m-1 Hf : ∇f ⊗ ∇f f dx -3-2 m 2 m-1 R d u m-1 |∇f | 4 f 2 dx ,
that we can also write as

I ′ = -(2 m -1) 1 -m1 m R d u m-1 ∆f -1 2 m-1 |∇f | 2 f 2 dx -R with R := (2 m -1) m 1 m R d u m-1 Hf 2 dx -2 3 (1 -m 1 ) + m1 m R d u m-1 Hf : ∇f ⊗ ∇f f dx + 2 m (1-m1)+m1 m (2 m-1) R d u m-1 |∇f | 4 f 2 dx .
Recalling that m 1 = (d -1)/d, the reader is invited to check that

R = 2 m-1 m R d u m-1 Lf -1 2 m-1 Mf 2 dx using the notations Lf := Hf -1 d ∆f Id and Mf := ∇f ⊗ ∇f f -1 d |∇f | 2 f Id , so that Hf : ∇f ⊗ ∇f f = Hf : Mf + 1 d ∆f |∇f | 2 f = Lf : Mf + 1 d ∆f |∇f | 2 f and Lf 2 = Hf 2 -1 d (∆f ) 2 and Mf 2 = 1 -1 d |∇f | 4 f 2 .
Finally we introduce the Rényi entropy power F defined by

F := E σ with σ := 2 d 1 1 -m -1 . If 1 -1 d = m 1 ≤ m < 1,
proving that, as a function of t, F is concave amounts to proving that

G := 2 m-1 2 m (1-m) E 2-σ F ′′ = (σ -1) 2 m (1-m) 2 m-1 I 2 + E I ′
is nonpositive, because of (6). We observe that

R d u m-1 ∆f -1 2 m-1 |∇f | 2 f + 2 m 2 m-1 I E √ u 2 dx = R d u m-1 ∆f -1 2 m-1 |∇f | 2 f 2 dx -2 m 2 m-1 2 I 2 E
and, as a consequence,

G = -(2 m -1) 1 - m 1 m E R d u m-1 ∆f -1 2 m-1 |∇f | 2 f + 2 m 2 m-1 I E √ u 2 dx -2 m-1 m E R d u m-1 Lf -1 2 m-1 Mf 2 dx .
Summarizing, when s = 0 and by assuming that there are no boundary terms in the integrations by parts, we establishes that F is a concave increasing function, and shows that

F ′ ≥ κ = κ ⋆
with the notations of Sections 2.3 and Corollary 3.1. According to [START_REF] Dolbeault | Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization[END_REF], this is equivalent to the proof of Inequality (3). The main difference with [START_REF] Dolbeault | Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization[END_REF] is that the computations are done in terms of f = u m-1 2 instead of u m . When s = 0, it would be desirable to do a similar computation with f = (-∆) -s/2 u m-1 2 but there are several obstructions. First of all, according to Proposition 2.3, we cannot expect that κ = κ ⋆ . In other words, the self-similar solution u ⋆ is not optimal for Inequality (3). We can hope that monotonicity of F ′ holds for some non optimal constant as a consequence of the Bakry-Emery method applied to the fractional fast diffusion equation and generalized Rényi entropy powers: proving it is still open. Technically another obstruction arises from the computations as we have no identity equivalent to [START_REF] Biler | Barenblatt profiles for a nonlocal porous medium equation[END_REF]. As in [START_REF] Biler | Barenblatt profiles for a nonlocal porous medium equation[END_REF][START_REF] Biler | The nonlocal porous medium equation: Barenblatt profiles and other weak solutions[END_REF] in the porous medium case, one could expect that a Stroock-Varadhan identity [START_REF] Liskevich | Some problems on Markov semigroups[END_REF][START_REF] Stroock | On degenerate elliptic-parabolic operators of second order and their associated diffusions[END_REF][START_REF] Stroock | On the support of diffusion processes with applications to the strong maximum principle[END_REF] allows to give a proof of the optimality with non-optimal constants, and also characterize the stability of the self-similar solutions as t → +∞, but this is so far also an open question.

reference: ANR-10-LABX-0098, LabEx SMP) and by the projects STAB (J.D., A.Z.) and Kibord (J.D.) of the French National Research Agency (ANR). A.Z. thanks the ERC Advanced Grant BLOWDISOL (Blow-up, dispersion and solitons; PI: Frank Merle) # 291214 for support. The authors thank Maria J. Esteban for fruitful discussions and suggestions. c 2016 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

d d-α ⋆ ⇐⇒ v ⋆ (x) = 1 + |x| 2 -d ∀ x ∈ R d ,

2 -m ⋆f dx = 0.We recall that R = R(t) is given by[START_REF] Vázquez | Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type[END_REF], and u ⋆ depends on t, so that the above inequality holds for any t ≥ 0. When s = 0, µ = d m -(d -2) and R µ (t) = 1 + µ t for any t ≥ 0. By density, it is easy to extend the above inequality to any function f ∈ L ∞ (dt, L 2 (R d , u1-m

Acknowledgements. This work is supported by a public grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" program (A.Z.,


Khaled Bayoudh

Solutions pour dissipation d'un bouchon

Dans cet article, nous proposons des nouvelles solutions pour la détection et dissipation de la congestion par la diffusion et l'exploitation des informations routières.

I. INTRODUCTION

La congestion automobile se manifeste au quotidien dans les grandes métropoles. Elle se traduit par des embouteillages, des temps de parcours qui s'allongent et une incertitude sur l'heure d'arrivée. La congestion est le résultat d'un déséquilibre, à un moment donné en un point donné, entre la demande automobile et l'offre viaire. Soit l'offre devient inférieure à la demande, comme dans le cas d'un accident qui réduit temporairement la voirie disponible. Soit la demande devient supérieure à l'offre. Ce sont, à titre d'exemple, les heures de pointe les jours de la semaine sur les voies rapides urbaines.

Comment résoudre le problème de la congestion du trafic routier ?

II. POURQUOI Y A-T-IL DES BOUCHONS SUR LES ROUTES?

Pour comprendre la congestion du trafic routier, il faut garder présent à l'esprit que c'est un phénomène qui survient lorsque la demande (le nombre de véhicules qui cherchent à utiliser une infrastructure donnée) est supérieure à la capacité de cette infrastructure. Si la demande excède la capacité, alors des véhicules seront ralentis à l'entrée de l'infrastructure, formant ainsi un bouchon. Ces véhicules excédentaires seront à chaque instant plus nombreux qu'à l'instant précédent. Comme chaque véhicule occupe une certaine longueur de voie, la longueur de la file d'attente ne fera que croître en proportion du nombre de véhicules présents dans cette file d'attente. On voit de ce qui précède que la congestion est un phénomène évolutif, à la fois dans le temps et dans l'espace.

Or, en raison du caractère maillé de l'infrastructure, cet allongement de la file d'attente peut la conduire à atteindre des points de choix d'itinéraire, et ainsi congestionner des parties du réseau où circulent des véhicules qui n'utiliseront pas l'infrastructure à l'origine de la congestion.

A. Volume de circulation supérieur à la capacité routière les voitures sont trop nombreuses et le trafic s'arrête à chaque goulot d'étranglement (rétrécissement de la route, voies qui se rejoignent, croisements de routes). La circulation est à ce titre comparable à l'écoulement d'un liquide dans un [START_REF] Lakas | Detection and dissipation of road traffic congestion using vehicular communication[END_REF] Khaled Bayoudh étudiant en Matser HiT4Med -ENISo 2015/2016 tuyau. En outre, le contexte de la mondialisation économique a largement contribué à l'augmentation des besoins en circulation des biens et des personnes. Cet accroissement en mobilité constitue une des causes principales d'apparition de plus en plus fréquente du phénomène de congestion.

B. Comportements des conducteurs

Quand les voitures sont proches les unes des autres, il suffit qu'un conducteur freine brutalement pour faire freiner en cascade tous les véhicules qui le suivent et propager ainsi un ralentissement, voire même un arrêt du trafic sur une certaine longueur. Quand le premier conducteur redémarre, de nouvelles voitures sont venues s'agglutiner et le bouchon se déplace vers l'arrière comme une onde, par un effet de dominos. En outre, plusieurs évènements peuvent provoquer ou aggraver la congestion : accidents, travaux, voitures en panne, stationnements gênants, conditions météorologiques mauvaises, etc.

III. CONS ÉQUENCES DE LA CONGESTION DU TRAFIC

ROUTIER

Elles sont nombreuses et peuvent être classées en trois catégories : économiques, sociétales et environnementales.

A. Conséquences économiques

D'un point de vue économique, la congestion du trafic routier entraîne :

-Une perte de compétitivité d'une région, d'un pays ou d'un continent, du fait des congestions qui bloquent les artères de communication routière, -Une perte de productivité, non seulement à cause des retards des personnes se rendant sur les lieux de leur travail, mais aussi en raison des livraisons tardives des approvisionnements ou des services rendus avec des retards importants voire annulés.

B. Conséquences sociétales

Les usagers de la route se plaignent amèrement des heures qu'ils perdent dans les embouteillages (plus de 1 milliard d'heures par an pour la France) parce qu'elles empiètent sur le temps qu'ils peuvent consacrer à leur travail et à leur famille. La congestion du trafic routier aura deux impacts importants sur :

-La qualité de vie des usagers, car les conducteurs soumis à des embouteillages fréquents sont sujet à des problèmes de stress, d'anxiété et d'énervement, entraînant un accroissement du risque d'accident, -Le pouvoir d'achat des usagers, car la congestion du trafic a des conséquences directes sur l'augmentation de la consommation énergétique. Dans le contexte actuel d'envolée des prix du pétrole, ceci engendre une nette diminution du pouvoir d'achat des usagers.

C. Conséquences environnementales

Enfin, d'un point de vue environnemental, la congestion du trafic routier a un impact néfaste et négatif à cause de l' augmentation de la pollution qui engendre des coûts économiques et écologiques en croissance exponentielle. Cette rubrique renvoie vers les différents produits et systèmes supports de diffusion, quel que soit le canal de diffusion de l'information (qui est traité dans la rubrique Transmission / Télécoms).

IV. CARACT ÉRISATION DE

La rubrique est structurée par type de support : -Terminaux interactifs -Diffusion passive' (broadcast) Fig. 2. Diffusion de l'information

Les supports de diffusion sont souvent liés à des technologies télécoms, et spécialisés pour certains usages (grand public / professionnel, transport public, trafic ou fret...).

Dans le coût de déploiement d'un système de transport intelligent, la diffusion est souvent avec le recueil le principal poste de coût, les traitements effectués, purement logiciels, coûtant souvent relativement moins cher que les matériels sur le terrain.

Un des objectifs de bon nombre d'applications ITS est de pouvoir diffuser l'information sur de multiples supports, ce qui suppose une certaine normalisation quant à la manière de diffuser cette information vers ces supports.

B. Systèmes coopératifs

Par systèmes coopératifs, nous entendons: Etat de coopération des acteurs du système de transport (exploitants, infrastructures, véhicules, leurs conducteurs et autres usagers de la route) dans le but d'offrir le plus sûr, le plus fiable et le plus confortable des déplacements La coopération véhicule-véhicule et véhiculeinfrastructure contribuent à ces objectifs au-delà des améliorations réalisables avec des systèmes autonomes. Autrement dit, dans les systèmes coopératifs l'intelligence est partagée entre le véhicule et l'infrastructure. Les véhicules communiquent les uns avec les autres et avec l'infrastructure, offrant ainsi le potentiel d'accroître considérablement la qualité et la fiabilité des informations disponibles sur les véhicules, leur emplacement et de l'environnement routier. Ces nouvelles informations permettent non seulement d'améliorer les services existants mais également de développer des nouveaux services pour les usagers de la route.

Dans cette vision coopérative, les véhicules circulant sur la route sont connectés à l'infrastructure via une communication continue. Ils échangent des données et des informations pertinentes relatives à la section de route traversée afin d'améliorer la sécurité et permettre une gestion coopérative du trafic. Plus précisément, au travers des protocoles de communication intégrés de type véhicule-infrastructure-véhicule, un véhicule qui détecte une situation dangereuse ou critique utilisera d'abord la communication véhicule à véhicule (V2V) pour avertir les autres véhicules se trouvant dans son entourage direct et communique cette information aux balises environnantes. Si ces informations ne peuvent pas être envoyées aux véhicules ou aux balises, le véhicule utilisera d'autres canaux de communication pour communiquer avec le centre de gestion. Le centre prend en charge l'acheminement les messages adéquats aux autres véhicules approchant la zone concernée.

1) Niveaux de coopération Sous le vocable de système coopératif on trouve trois niveaux de coopération de complexité croissance entre les véhicules et l'infrastructure : autonome, coopératif et interactif. Nous allons détailler les principales caractéristiques de ces différents types.

• Système autonome Dans ce premier niveau, le véhicule dispose simplement de capteurs embarqués capables d'apporter une connaissance sur l'environnement immédiat du véhicule. Cependant, il n'interagit pas avec les autres véhicules directement. Cependant, le véhicule dispose d'assistance à la conduite (ADAS) lui permettant de :

-mesurer les distances par rapport au véhicule précédent -repérer les obstacles, -freiner dans le cas de situation d'urgence (freinage automatique à une seconde du choc) -détecter les écarts de trajectoire par détection du marquage au sol. Pour ces fonctions, il s'appuie sur des capteurs d'images et de télémétrie laser. Il peut également embarquer un système de localisation lié à une cartographie numérique.

• Système coopératif Dans ce deuxième niveau, le conducteur dispose d'une meilleure connaissance de l'environnement, grâce à une communication avec l'infrastructure (V2I) qui permet, par exemple, de savoir les conditions de circulation en aval, le niveau d'adhérence et plus globalement d'avoir des caractéristiques sur la voie courante. Il dispose également d'une perception plus fine des véhicules environnants sur la section et sur les voies adjacentes.

Fig. 1 .

 1 Fig. 1. Avantage du transport public

  LA CONGESTION A. Par la concentration La caractérisation de la congestion qui découle le plus directement de sa définition consiste à analyser la concentration. En un point où des mesures de débit et concentration (ou débit et taux d'occupation) sont disponibles, le tracé de nuages de points débit concentration permet de déterminer une concentration critique (ou un taux d'occupation critique) au-delà de laquelle le trafic est congestionné. La valeur de ces seuils est très variable d'un lieu à l'autre, elle peut être sensible à de multiples facteurs comme le taux de poids lourds. Elle est en général comprise, sur autoroute, dans une fourchette de 15 à 25 pour cent pour le taux d'occupation, mais peut être très inférieure dans certains cas. B. Par la vitesse Toute situation où la vitesse est inférieure à la vitesse critique est congestionnée. L'observation de données réelles croisant vitesse et concentration ou vitesse et débit permet selon les cas de confirmer cette vision ou de l'infirmer selon les sites. Une caractérisation par des valeurs absolues de vitesse, intéressante par son caractère universel et la possibilité qu'elle offre de calculer simplement des indicateurs, présente évidemment l'inconvénient de regrouper, d'un site à l'autre, des situations bien différentes du point de vue de l'observation. C. Par l'observation Les calculs d'indicateurs de congestion sont souvent fondés sur une observation visuelle des conditions de trafic par les opérateurs des centres de gestion de trafic. De plus, il arrive que des opérateurs confrontent une vision issue de caméras placées en hauteur, qui donnent accès à une vision plus spatiale et une mesure de vitesse issue des boucles électromagnétiques. Ces deux visions ne sont pas toujours compatibles et peuvent donner lieu à des incompréhensions et à des indéterminations. V. TEMPS DE DISSIPATION D'UN BOUCHON Considérons maintenant le cas où le bouchon de longueur initiale L, et de débit sortant Qb, est alimenté en amont par un débit Q plus faible que Qb . La longueur de la file d'attente va maintenant décroître, puisque plus de véhicules en sortent (Qb) qu'il n'y a de véhicules qui y rentrent (Q). La concentration dans le bouchon est toujours Kb, reliée à Qb par le diagramme fondamental. Le temps qu'il faut pour que le bouchon disparaisse, , est tel que le nombre de véhicules soit nul au bout de ce temps. Or le bouchon contient au départ Nb = Kb.L véhicules et pendant Tdis il sera alimenté à l'amont par Q.Tdis véhicules, alors qu'à l'aval Qb*Tdis véhicules le quitteront. Nb + Q . Tdis -Qb . Tdis=0 Donc Kb . L + (Q -Qb) . Tdis=0 D'où : Tdis = Kb . L / Qb -Q VI. SOLUTIONS PROPOS ÉES POUR DISSIPATION D'UN BOUCHON A. Diffusion de l'information La diffusion est le pendant du Recueil de Données. Il s'agit de présenter l'information aux différents types d'utilisateur : grand public (usager, voyageur, internaute selon les cas de figure), mais aussi utilisateurs professionnels (conducteurs, agents d'exploitation, etc.).

•

  Système interactif Ce dernier niveau est un cas de figure idéal. Tout véhicule échange des informations avec les autres véhicules environnants (V2V), tout comme avec l'infrastructure (V2I) et (I2V) et les opérateurs. Des véhicules traceurs, qui circulent en amont, font remonter des données. L'analyse de ces informations permet de diagnostiquer des risques d'accidents liés soit à l'infrastructure, soit aux comportements des conducteurs. Dans ce niveau, le conducteur bénéficie d'un accès temps réel aux caractéristiques de la route et des conditions de circulation. Sa perception de l'environnement est optimisée par une fusion des différentes sources d'information (capteurs autonomes, systèmes coopératifs, échanges d'informations). Le véhicule peut, pour sa part, détecter en temps réel les défaillances pour la trajectoire et le comportement dynamique.

Fig. 3 .

 3 Fig. 3. Systèmes coopératifs


Jean-Louis Peaucelle

Pin making in the eighteenth century

Keywords: pin making, eighteenth century, French encyclopaedists, specialisation, division of labour, productivity, Adam Smith

teaching and research institutions in France or abroad, or from public or private research centers.

Cornwall.

From the Middle Ages through until the twentieth century, women had very roomy and long dresses. Pins were used to fix these clothes and to shorten them for work. In pre-revolutionary France a lover would give one hundred or more pins to his beloved. The old expression 'pin money' attests to the presence of pins in ancient society. In the eighteenth century, French women bought an average of 100 pins each year.

Pins were mainly used in the royal court. The best pins were dedicated 'to the queen'. They were made in towns close to London or Paris and sometimes Bordeaux or Toulouse. The Paris pin makers guild was founded in 1268 6 . The guild had the monopoly to draw the wire, make and sell pins. The London pin makers were founded in 1356 and were also in a monopoly position. They had difficulty however meeting the demand in London. Many pins were imported. On two occasions a wealthy industrialist was accorded a monopoly on the trade of pins including imports from Queen Elisabeth I, King James I, and King Charles I. The monopoly always failed. 7

II

In the eighteenth century, pins were made in the country using a new technical process. The Dutch leaving around Dinan 8 had great experience in make cooper and brass objects. They invented a new pin making process. They began selling their pins to the English court at the end of the fourteenth century. The war with the Spanish King Philip II in 1566 destroyed this industry. The Dutch merchants kept their trade relationships and moved production to France, a less dangerous land with far lower wages 9 .

Pin production in France began in a little woody area near the town of Laigle, in the south of Rouen 10 . There was a tradition of smithing, with locals producing iron and ironed objects. The 6 Secousse, Ordonnances des roys de France, tome IV pp. 124-128, Boileau, Le livre des métiers pp. 124-127, Lespinasse, Les métiers et corporations pp. 255, 256, 564-575, and Franklin, Dictionnaire historique des arts, métiers et professions p. 308. invention of the blast furnace elsewhere challenged their trade. In the fifteenth century, this new technology was brought to Normandy by Walloon smiths. Pins were also made in Wallonia.

For the two hundred years prior to the 1672 war between Louis XIV and the Dutch Republic, Dutch merchants supplied the brass wire to the French pin makers and in exchange for pins. The Dutch then sold them in London.

After the breakdown in trade relations with the Dutch merchants, the eighteenth century was a golden age for French pin makers. French merchants from Normandy organized the purchasing of brass wire from Holland, Sweden, and Germany and built commercial networks across France, in Italy, and in Spain. The pin makers in Paris became resellers of the pins made in Normandy. During the eighteenth century, there were about 1300 workers in the pin making industry made up of 500 men and 800 women. They worked in their houses in the forests near Laigle in Normandy. They purchased about 250 tons of brass wire and sold 1.8 billion pins for 1 million French Livres. Half of all production was exported. 11 From 1626 onwards, English pin makers based around Gloucester used the same Dutch technology 12 .

Many tools for pin making were multi-purpose: the drawplate to make the wire of the required diameter, the gauges to verify the diameter, the solid wine acid 13 to clean the brass, the pincers to draw the wire, the shears to cut the wire and the shanks 14 , the grindstone from iron to point, the spinning wheel to turn the wire and make the head, the pots for yellowing, whitening and washing the pins, and the tray to winnow the pins.

Several tools were specific to pin making: the 'engin' to straight the wire, the half boxes to cut the wire and the shanks to the required length 15 , the heading machine which was activated by the foot, the barrel to dry and trim the pins with bran, the pad to stamp the papers, and the awl of 25 tips 11 Peaucelle, Adam Smith et la division du travail p. 105. 12 Herbert, The city of Gloucester pp. 3, 76, 108, 130, 139. Pinchbeck, Women workers pp. 273-280. The folk museum of Gloucester shows this industry. 13 In French gravelle. 14 The shanks were the pins without a head. 15 The pins had various sizes from number 1 to number 36. The more usual pins were the number 5 (2.7 cm) to the number 22 (5.4 cm).

to pierce the papers. Those tools have strange names that would appear to come from the Dutch language.

The use of each tool for a specific activity was called an 'operation'. The number of operations is however arbitrary as it is sometimes possible to distinguish two separate operations.

Some operations were also repeated, such as drawing the wire to the right diameter. The drawer used successively smaller holes until the required diameter for the pin size was reached. Twelve holes were used for the pin number three and seven for the pin number eight, for example. The wire was annealed after three holes. The shank was pointed twice to obtain a very polished point. These were operations 5 and 6. The cutting of the shanks, operations 4 and 7, also appear to have been repeated. Adam Smith didn't understand this repetitive operation. He wrote 'a fourth points it, a fifth grinds it at the top for receiving the head'. 16 Yet the explanation for repeating the cutting is simple.

It was easier to hold the parcel of wire on the grindstone if it was long. The parcels were as long as two to five pins in length. The pointer made a point at each end. After pointing, each end was cut to be a pin without a head. The eventual remaining piece in the middle came back for pointing.

Schema 1 The 18 operations with Delaire's 17 numbers in Diderot's Encyclopédie

The separation of pin making into different operations was not a division of labour.

Sometimes a worker was specialized in one or two operations such as women heading the pins and then putting pins on the papers, or drawing the pins. Sometimes, a worker performed many different operations. The principal reason was the operative time. The longest operation was the heading, which took around three seconds per pin. The women took the pins one by one. Pin making was not organized according to a division of labour with 18 workers in a single workshop working together in an assembly line. The pin industry involved around 300 workshops and only 20% of the pin makers worked in the 'big' workshops of 10 to 20 workers 18 . The majority of pins were made in the forest houses where three to five people worked. Raw materials were moved by small horses and weighed to pay wages according to quantities produced. This process is known as the 'classical' pin making process as it was described in the well-known texts of the Diderot's Encyclopédie and those of the French Academy of Sciences.

III

The published French pin making texts describe the Normandy workshops. The Paris pin makers knew of provincial production because their 1601 charter forbade reselling of these cheaper pins and also imposed taxes on them. The French Administration knew about pins making in Normandy since the 1660s when the Dutch ambassador provided statistics concerning the goods the Dutch purchased in France. Pins and combs accounted for 500 000 French Livres. 19 Exactly how the pins were made however was poorly understood. The first texts which described pin making were written in the course of a larger project to describe all arts and crafts.

In 1675, King Louis XIV asked to the French Academy of Sciences to describe the arts and crafts20 . The Academy had little motivation to undertake this work. New members were appointed for this task, in particular Billettes in 1699. Billettes most probably discovered a report written by a local trade inspector in 169221 . He ordered one illustrative plate, dated of 1702 and he wrote an unpublished report in 1700 which was contrary to the habits of the Academy. In the Academy of Sciences' archives a handwritten note gives the production rhythms (physical productivity) of some operations22 . Using the price of raw material and the selling price, Billettes computed the value added 23 . With the rhythms he calculated the daily wages. These theoretical wages were very low in comparison with the usual daily wage of the workers in Paris. The situation was an impossible one.

Billettes explored two possible solutions, either increase the productivity or decrease the wages for certain operations. Billettes thought that women should work on operations with the lowest wages24 .

It is probably for this reason the Billettes' text wasn't published.

After the death of Louis XIV in 1715, the project to describe the arts and crafts industries was repeated. Local administrative authorities made inquiries and Guéroult25 wrote a handwritten report 26 . At the Academy, Réaumur gathered this information and ordered new four plates dated 1718. However nothing was published.

In 1723, Savary published the Dictionnaire du Commerce which contained information from the central Royal Administration27 , including the Academy of Sciences. In the article 'Espingles' ('Pins') Savary described the guilds and their charters but also the relocation of pin making to Normandy. Savary saw the five existing plates and he counted how many workers were drawn.

They were 25 and he wrote 'we count 25 workers which operate successively on it'28 . This interpretation was wrong as the characters were drawn in Paris to explain the use of tools. Many operations were drawn several times. Chambers translated this text on the pins with the workers numbering 2529 . These texts contributed to create the illusion of a division of labour.

The number '18' came from Diderot's Encyclopédie where the article 'Epingle' was written by Delaire. He described in succession each of the 18 operations. Moreover he gave nouns for the workers in almost all of the operations, as if they were separate professions. Delaire didn't directly observe pin making. He rewrote several sources, in particular Guéroult's 1818 report and Perronet's 1740 report. Diderot wanted his text to appear original. The use of numbering helped structure the details. The illusion of specialized labour is the result of a writing style used to give a unique feel to this technical text.

Perronet 30 was a civil engineer in Normandy. He observed pin making in 1739 and took accurate measures 31 . In his 1740 report he computed the production cost for all sizes of pins. In 1755, he gave his report to Diderot for Delaire's article 'Epingle' but he was disappointed by the result. He later gave his report to Duhamel, a member of the Academy of Sciences. In 1761 Duhamel published the Description of the arts and crafts, and pin making was amongst the first published with one plate in 1702 and four plates in 1718 32 . He mixed several sources, including Réaumur's handwritten manuscript copied from Guéroult, Perronet's document, his own notes, and the observations of Chalouzière, a local judge. Perronet looked to publish an entire original report and was helped by Diderot who published a second article called the 'Epinglier' in the 1765volume of the plates 33 .

Perronet wrote manuscript about wiredrawing in 1739 34 . His report describes the first operations. Drawing was repeated up to 12 times to obtain the required diameter of the pins.

Perronet gave the rhythms. He computed the wages of the drawers and the production cost of each wire size. He transmitted a copy to his friend Diderot and to Duhamel, his colleague in the Academy of Sciences 35 . The accompanying drawings were the source of the first plate in the Diderot's Encyclopédie. However the entire text wasn't printed. The production rates for the first operations were omitted 36 .

These texts have been referenced, copied and abstracted in many publications, periodicals and dictionaries. Schema 3 shows the French texts about pin making that existed in the eighteen century.

Almost all were copies. The texts can all be traced back to their sources. When the texts were rewritten the main ideas were transcribed yet many details were omitted, the fact that women 37 were specialized in heading and sticking, for example. The idea of specialized workers on each the 18 operations was added for stylistic reasons. 31 Perronet, 'Description de la façon dont on fait les épingles'. 32 Duhamel, L'Art de l'épinglier. 33 Perronet, Encyclopédie, tome 1 of the plates, article 'Epinglier'. 34 Perronet, 'Explication de la façon dont on tire le fil de leton'. 35 Perronet was member of the Academy or Architecture. He became an associate member of the Academy of Sciences in 1765.

36

Billettes counted the drawing operation but only for one hole.

37

During the seventeenth and the eighteenth century in Paris women worked in a small number of female guilds. The men worked in the other exclusively male guilds. There were only three original sources -Guéroult, Perronet, and Chalouzière -who gave information about productivity, although often as a range of values. There was no well-known standard at the time.

Delaire's article (1755) was well considered and the new edition of Savary's Dictionnaire du Commerce in 1759 used the full text. After its publication by the Academy of Sciences many periodical reviews referred to it. 38 The commercial success of the great Diderot's Encyclopédie opened a new market for short dictionaries 39 . They often had an article entitled 'Epingle'. From 1782 to 1791, the great publisher Pankoucke reprinted all the texts describing the arts and crafts trades including that of pin making in 1783 40 , and those of the Academy and Diderot's Encyclopédie. 38 different workmen, between the drawing of the brass wire, and the sticking of the pin in the paper' 41 .

Twelve years later another technical dictionary distinguished the articles "Pin" and "Pinmaker". This second article described the production process in London. 'This is an art that requiring very little previous learning. The pin-makers of London keep shop, and some of them carry on a very great trade. The work, for the greater dispatch, is carried on by different hands; one hand being employed in cutting the brass wire into proper lengths for the size of the pins, a second in making the heads, a third in putting them on, and a fourth in pointing them, every one of these work with surprising swiftness. Those masters, who work privately and chiefly for the shops, take five or ten pounds with an apprentice who must work from six to nine. When out of this time, he may earn in those hours twelve or fourteen shillings a week; or with fifty pound may set up master.

Those who keep shop take twenty ponds with an apprentice, and employ women in sticking the pins on paper, a sheet of which holds five hundred; these masters employ five hundred pounds and some of them above thousand pounds in trade.' 42 This text contradicts itself. On the one hand labour was divided between five workers (cutting, making the heads, putting the heads, pointing, and sticking on paper 43 ). On the other hand the work was performed by a master and an apprentice. Perhaps, women weren't counted. They stuck the pins on paper and perhaps headed the pins too. There were four or five labourers in a workshop and may have specialized.

Workshop production was 500 pounds of pins per year. According to Adam Smith, 'There are in a pound upwards of four thousand pins of a middling size.' 44 The annual workshop production was two millions pins or 6,667 pins a day 45 that is 1,667 pins a day for the each four workers. If production doubled, the productive power of labour in this workshop of five workers was 2,667 pins a day per person. These figures are very low in comparison to Adam Smith's figure of 4,800.

41 Barrow, A new and complete Dictionary of arts and sciences. 42 Mortimer, A new and complete dictionary of trade. 43 The process was shorter than the described by the French process. There were only the following operations: 4, 8-9-10 gathered, 11, 4, and 18. The order of the heading (11) and the pointing (4) were inversed.

44

Size X with the Perronet's sizes. 45 Based on 300 days of labour in a year.

The author didn't say at which time these observations were made. In the eighteenth century English pin production was based around Gloucester using the same production process as that used in France. The production described by the author was most probably that of the old process used by the guild of London.

In 1767, Arthur Young visited Mr Champion's copper works and pin making workshop in Warmley, near of Bristol. He saw 'the whole process, from the melting of ore, to making it into pins, pans etc. The liquid ore pouring out the furnace into clay moulds. [...] After being several times melted, it poured into a flat mould of stone, to make it into thin plates, about four feet long and three broad. Those plates are then cut into 17stripes and these again, by particular machine into many more very thin ones, and drawn out to the length of 17 feet, which are again drawn into wire, and done up in bunches of 40 s value each; about 100 of which are made here every week, and each makes 100,000 pins. The wires are cut into them and completed here, employed a great number of girls, who with little machines, worked by their feet, point and head them with great expedition; and each will do a pound and half in a day. The heads are spun by a woman with a wheel, much like a common spinning wheel, and then separated from one another by a man, with another little machine like a pair of sheers [...] All the machines and wheels are set in motion by water; for raising which, there is a prodigious fire engine, which raises, as it is said, 3000 hogshead every minute' 46 .

The women made 'a pound and half in a day' of pins, i.e. 6000 common pins. The process was shorter than the one used in Normandy with the only operations being those of cutting, pointing, making the head, and heading (operation 4 to operation 11).

In Gloucester, the pin workshops 'employ near 400 hands of whom a great number are women and children: good hands at pointing and sticking earn from 10 to 12 s and 15 s a week, children of 8, 10 and 11 years old, earn 2d and 3 d a day, but some journeymen do not get more than 7s, 8 and 9s a week; the wages in general are good.' 47 The last text on pin making prior to that of Adam Smith was an article in the first edition of 46 Young, A six weeks tour through the Southern pp. 184-185. 47 Young, A six weeks tour through the Southern p. 141.

the Encyclopaedia Britannica in 1771. 'Pin: in commerce, a little necessary implement made of brass wire, used chiefly by the women in adjusting their dress. The perfection of pins consists in the stiffness of the wire ad its whiteness, in the heads being well turned, and in the fineness of the points. The London pointing and whitening are in most repute, because our pin-makers, in pointing, use two steel-mills, the first of which form the point, and the latter take of all irregularities, and renders it smooth, and as it were polished; and in whitening, they use block-tin granulated: whereas in other countries they are said to use a mixture of tin, lead and quick silver.' 48 After Adam Smith's book ( 1776), all the English texts on pin making evoke the division of labour.

IV Today pin making in the eighteen century is well understood 49 . How does it match with the theory of the division of labour? Adam Smith read four of texts 50 and he developed them to formulate the example of his theory. He was misled by the way operations were enumerated. He believed that labour was divided and the productive power of the pin makers was 4800 pins per day.

Is this a good illustration of his theory? Was labour divided in pin making? Why was physical productivity so high? Would the productive power of labour in pin making be increased by the division of labour?

There are two sides to any answer to these questions. First, considering the information available at the time how well could Adam Smith have reasoned? Secondly, with what we know today, how true is the theory of the division of labour? Four arguments are presented for the first question: the daily production of each operation, a three way division of labour, the physiological limit to labour speed, and production without the division of labour.

First, the operative times for the different operations were very badly matched. The eleventh operation (heading) lasted more than the all of the 17 other operations. If the theory is true, it is possible to accelerate the process by dividing this operation. The French pin makers trialled this change. To head the pins a child was employed to thread the head in the shank. But this was useless 51 . This attempt to divide labour did not increase labour productivity.

Secondly, Adam Smith extracted the following information from Delaire's article (1755): 'So that in the making of a pin there are about 18 persons employed. These in a day will make about 36000 pins, and this comes to the same thing as if each one made about 2000.' 52 From Macquer's Dictionnaire portatif (1766) Adam Smith computed that 10 workers would produce 48000 pins per day. And from the Chambers' Cyclopaedia (1728) 25 workers cooperated to make pins. Adam

Smith then had three cases, at three dates 1728, 1755, and 1766, with three divisions of labour, 25, 18, and 10 workers. The oldest text involved a greater division of labour but it was not the more productive. The daily production with 18 workers was 2000 pins per person, and 4800 pins with 10 workers. These texts are completely contrarily to the theory by which the division of labour would increase the productive power of labour 53 .

Thirdly Adam Smith saw the market as the only upper limit to the division of labour. 'The Division of Labour is Limited by the Extent of the Market' 54 . He neglected physiology as another limit. He knew horse racing. An ordinary horse can't run faster than the winner. Animal and human muscles have a maximal speed. Dividing labour will not lead to production increases when these limits have been met. In French pin making, productivity was increased by working on batches of pins.

Fourthly Adam Smith invented a fictitious case to enhance the value of the division of labour.

It was not a real case. It evokes the beginning of an apprenticeship. 'A workman not educated to this business (which the division of labour has rendered a distinct trade), nor acquainted with the use of the machinery employed in it (to the invention of which the same division of labour has probably given occasion), could scarce, perhaps, with his utmost industry, make one pin in a day, and 51 Duhamel, L'Art de l'épinglier, p. 34. 'Pour enfiler la tête à moindre frais, il y avoit des fabriques où l'on faisoit enfiler par des enfants ; mais il falloit ajuster les têtes au bout des hanses. On a maintenant trouvé qu'il étoit plus expéditif de faire enfiler & frapper par le même ouvrier' 52 Adam Smith, Lectures on Jurisprudence, Monday, March 28, 1763 (pp. 341-342) 53 As matter of fact, that information is false. We can't conclude that a lesser division of labour is the more productive. 54 Adam Smith, The Wealth of Nations, Book 1 chapter 3. same technology. Some were small with only 3 workers, and others were bigger with 20. The organisation of labour varied according to the size of the workshop. No workshop has a significant advantage over another. The productivity of labour and wages were the same. More divided labour wasn't more productive. The theory of the division of labour does not hold true in Smith's first example, that of pin making.

Adam Smith forgot to describe pin making technology. This technology was illustrated at length in Diderot's Encyclopédie. Smith believed that the division of labour led to the invention of new tools. For pin making, technology did change in the nineteenth century. An integrated machine was invented for the operations 3 to 11. This machine increased the physical productivity and decreased the division of labour.

Adam Smith had enough information about pin making to refute the idea of production increases through the division of labour. We also today know that the bigger pin making workshops did not hold a competitive advantage. The division of labour did not increase the pin making productivity. From a Popperian perspective, the theory is falsified.

  relationships between the actors in the French pin making industry

V

  The aim of this text was to confront how pins were really made in the eighteenth century with the Adam Smith's theory. No fit is observed. The facts contradict the theory. This result poses a number of problems.Does the theory hold true in others cases?Under what conditions would be the theory hold true? What parts of Adam Smith's reasoning about the division of labour are incorrect? What facts which Adam Smith evoked still remain true while the theory is incorrect? Why did Adam Smith reason incorrectly? Does the theory true if we give another meaning to the division of labour?What have been the consequences of following a theory which has later been proven incorrect?What are the theoretical consequences of rejecting this theory? Why has almost every reader believed the veracity of this theory?These results lead to many questions. Their answers and the debates they provoke should provide rich opportunities for future research.Footnote referencesAnonymous, Articles et status accordez, dès l'an 1336, par les roys de France aux maistres épingliers-éguilliers-alleiniers, faiseurs de burins et autres outils, servant aux orphèvres (Paris, 1729) Aymard, M., (ed), Dutch capitalism and world capitalism, Capitalisme hollandais et capitalisme mondial (Cambridge, Paris, 1982). Babbage, C., On the economy of machinery and manufactures (1832). Barrow, A new and complete Dictionary of arts and sciences… by a society of gentlemen (1754). Boileau, E., Le livre des métiers : XIIIe siècle, Lespinasse (de) R. Bonnardot F. ed. (Paris, 1879). Chambers, E., Cyclopaedia or an Universal dictionnary of Arts and Sciences (1728). Delaire, A., article 'Épingle', Encyclopédie, tome V (Paris, 1755). Diderot, D., Le Rond d'Alembert, J., Encyclopédie ou dictionnaire raisonné des sciences, des arts et des métiers (Paris, 1751-1780). Dronne, R., L'Aigle, son histoire, ses monuments, ses industries (Paris, 1953). Duhamel du Monceau, H.-L., Ferchault de Réaumur, R.-A., Perronet, J.-R., Legrand de Boislandry de Chalouzière, B.-F., L'Art de l'épinglier (Paris, 1761). Duval, L., Les épingles et les aiguilles de Laigle (Alençon, 1900). Fauduet, I., Fibules préromaines, romaines et mérovingiennes du musée du Louvre (Paris, 1999). Franklin, A., Dictionnaire historique des arts, métiers et professions exercés dans Paris depuis le

48

  Encyclopaedia britannica 1771, Reprint 1997, ed. Frank Kafker. London: Routedge &Thoemmes Press. Vol. III p. 479 Article 'Pin'.49 Peaucelle, Adam Smith et la division du travail.50 Peaucelle, 'Adam Smith's use of multiple references', Journal des sçavans, 1761, Duhamel, L'Art de l'épinglier, Delaire article 'Épingle', Encyclopédie, and Macquer, Dictionnaire portatif. Ross, The Life of Adam Smith, did not mention that Adam Smith visited a French pin making workshop (chapter 13).

Adam Smith, Wealth of Nations, Book 1 Chapter 1. These were the fifth and the sixth operations, because Adam Smith omitted the first.

Marchand, 'La fabrication des épingles', p. 35.

Huet, Le grand trésor historique, pp. 108-109. The combs were made in Ezy sur Eure at 70 km far of Laigle.

Pinault, L'Encyclopédie. Salomon-Bayet, 'Un préambule historique'. Moureau, Le Roman vrai de l'Encyclopédie. The goal of the Royal Administration is unknown but in France the Protestants had got many industrial activities. Perhaps, the King prepared the Revocation of the Edit de Nantes (1685).

Savary, Dictionnaire Universel de Commerce, Préface p. XIX 'Dès l'an 1692, les inspecteurs du Commerce avaient eu l'ordre d'envoyer à la Cour des états de leurs départements'.

Peaucelle, Manin, 'Billettes and the economic viability of the pin making'.

Naturally he didn't use this expression.

Billettes was right; see Babbage, 1832, chapter XIX.

He was a civil engineer in Alençon prior toPerronet.

Guéroult, 'Description générale de la fabrique d'épingles'.

Savary quoted the judgement of the general lieutenant of the Police in Paris dated of July 16 th 1695 who sentenced several thousand of iron pins to burning. They were made in Normandy. Chambers translated 'By a sentence of the lieutenant de Police, July 1695, the seizure of some millions of those pins [in iron] are confirmed.'

Savary, Dictionnaire Universel de Commerce, article 'Espingles' 'Il n'y a guère de marchandise qui se vendent moins cher que les épingles ; & cependant il n'y en point qui passent par plus de mains, avant que de pouvoir être mises sur le marché. L'on compte jusqu'à plus de vingt cinq Ouvriers qui y travaillent successivement, depuis que le fil de léton a été tiré de la filière, jusqu'à ce que l'Espingle soit attachée au papier.'

Chambers, Cyclopaedia, Article 'Pin' 'Notwithstanding that there in scarce any commodity cheaper than pins, there is none that passes through more hands ere they come to be sold. They reckon twenty-five workmen successively in each pin, between the drawing of the brasswire and the sticking of the pin in the paper'.

Perronet succeeded Guéroult. He created in 1747 the first school of engineers in France, l'Ecole des Travaux Publics (see[START_REF] Vacant | premier ingénieur du roi et directeur de l'École des ponts et chaussées[END_REF].

Another dictionary was printed in London during the same period. The article entitled 'Pin' brought new information. The organisation of labour had only six workers in London. 'The number of hands employed in this manufacture is very great, each pin passing though the hand of six

certainly could not make twenty' 55 . At the end of the apprenticeship, how many pins did an apprentice make? Adam Smith didn't consider this situation. However, this situation was a standard test at the end of apprenticeship, necessary to become a master ('chef-d'oeuvre') in the ancient pinmaker guild. In France the 'chef-d'oeuvre' 'consists in one thousand of pins which after have been finished must be seen and opened out in the attendance of all the masters to speak their opinions' 56 .

How long did it take to produce these thousand pins? Probably a half a day and the test of all the pins lasted the remaining half-day. The productive power of labour would be some 2000 pins per day and per pin-maker working without the division of labour. The difference in production levels between a process where there was a division of labour and a process where labour was not divided was not 240 times but only 2.4 times. This gap can be explained by the length of the process and by the older technology used. In the guilds, the pin-makers drew out the wire several times. They headed the pins with a hammer and not with the heading machine like in Normandy.

Despite the structure of the Delaire's 'pin' article, Adam Smith had sufficient information to reject the theory of the division of labour. Today we know that the theory was false in the case of pin making as we have others arguments that Adam Smith could not have know: the pin makers were not specialized, many different sized workshops existed, and new pin making technology in the nineteenth century decreased the division of labour.

The pin makers weren't specialized in 18 operations although a certain specialisation did exist.

First, the merchants were not the workers. Secondly, women worked on two operations only, the heading and the putting onto papers 57 . Thirdly the cutter worked so fast 58 that he worked for several workshops. Fourthly the drawer always repeated the second operation (Adam Smith's first operation) using several holes on the drawplate. For the 13 others operations, no details exist in the original sources to suggest any sort of specialization.