Mostafa Adimy
email: mostafa.adimy@inria.fr

Abdennasser Chekroun
email: chekroun@math.univ-lyon1.fr

Tarik-Mohamed Touaoula
email: tarik.touaoula@mail.univ-tlemcen.dz

Global asymptotic stability for an age-structured model of hematopoietic stem cell dynamics

Keywords: Age-structured PDE, delay differential-difference system, Lyapunov-Krasovskii functional, cell dynamics, hematopoietic stem cells AMS Subject Classifications: 34D20, 34D23, 34K06, 92C37

We investigate a system of two nonlinear age-structured partial differential equations describing the dynamics of proliferating and quiescent hematopoietic stem cell populations. The method of characteristics reduces the age-structured model to a system of coupled delay differential and renewal difference equations with continuous time and distributed delay. By constructing a Lyapunov-Krasovskii functional, we give a necessary and sufficient condition for the global asymptotic stability of the trivial steady state, which describes the population dying out. We also give sufficient conditions for the existence of unbounded solutions, which describes the uncontrolled proliferation of hematopoietic stem cell population. This study may be helpful in understanding the behavior of hematopoietic cells in some hematological disorders.

Introduction

Many biological systems are self-regulating and hence, they involve nonlocal terms (called delays) in feedback loops. These delays are due to the inherent time lag between the sensing and control signals. The retardation usually originates from maturing processes or finite signaling velocities. Sometimes the delays can be of considerable length and cannot be neglected. In general, the delays can represent incubation periods [START_REF] Cooke | Stability analysis for a vector disease model[END_REF], gestation times [START_REF] Aiello | Analysis of a model representing stage-structured population growth with state-dependent time delay[END_REF], cell division processes [START_REF] Mackey | Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis[END_REF], or can simply be used to model unknown intermediate steps in reaction chains, accounting only for the time required for these processes to occur.

A classical approach to introduce delays in differential systems starts from certain agestructured models (transport PDE). The age-structured systems can be transformed in a natural way using the method of characteristics to a system of threshold-type delay differential-difference equations (see [START_REF] Adimy | A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia[END_REF][START_REF] Kuang | Delay Differential Equations: With Applications in Population Dynamics[END_REF][START_REF] Smith | Reduction of structured population models to threshold-type delay equations and functional differential equations: a case study[END_REF]). This approach is used in this paper to reduce an age-structured model to a system composed with a delay differential equation and a renewal difference equation with continuous time and distributed delay.

Mathematical modeling of hematopoietic stem cell (HSC) dynamics has been extensively studied in the past 40 years. In 1978, Mackey [START_REF] Mackey | Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis[END_REF] proposed a mathematical model of HSC dynamics formed by a system of two delay differential equations. This system describes the evolution of proliferating and quiescent HSCs. The delay describes the average cell cycle duration. The model of Mackey stressed the influence of some factors, such as the apoptotic rate, the introduction rate from the quiescent phase into the proliferating phase, and the cell cycle duration, playing an important role in the behavior of the system. Since then, Mackey's model has been improved by many authors. It has been analyzed by Pujo-Menjouet and Mackey [START_REF] Pujo-Menjouet | Contribution to the study of periodic chronic myelogenous leukemia[END_REF], and Pujo-Menjouet et al. [START_REF] Pujo-Menjouet | Long period oscillations in a G 0 model of hematopoietic stem cells[END_REF], in order to prove the existence of long period oscillations, characterizing situations observed in chronic myelogenous leukemia, a cancer of HSCs. Many authors, including [START_REF] Adimy | Global stability of a partial differential equation with distributed delay due to cellular replication[END_REF][START_REF] Adimy | Modelling and asymptotic stability of a growth factor-dependent stem cell dynamics model with distributed delay[END_REF][START_REF] Adimy | Discrete-maturity structured model of cell differentiation with applications to acute myelogenous leukemia[END_REF][START_REF] Adimy | Stability and hopf bifurcation for a cell population model with state-dependent delay[END_REF][START_REF] Adimy | A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia[END_REF][START_REF] Adimy | Stability and hopf bifurcation in a mathematical model of pluripotent stem cell dynamics[END_REF][START_REF] Adimy | Modelling hematopoiesis mediated by growth factors with applications to periodic hematological diseases[END_REF] and the references therein, analyzed various versions of Mackey's model and investigated the effect of perturbations of the parameters of the system on the behavior of the cell population. In all these studies, the authors assumed that just after division all cells enter immediately the quiescent phase. This restrictive assumption allows to reduce the model to a delay differential system.

In the general situation in a cellular population and in particular, in HSC population (see [START_REF] Bradford | Quiescence, cycling, and turnover in the primitive hematopoietic stem cell compartment[END_REF][START_REF] Takizawa | Dynamic variation in cycling of hematopoietic stem cells in steady state and inflammation[END_REF]), it is believed that just after cell division the daughter cells follow one of the two pathways: they enter the quiescent phase or they return to the cell cycle to divide again. Contrary to the above works, we take into account this situation. We reconsider the model of Mackey and we suppose that just after division only a part of daughter cells enter the quiescent phase (longterm proliferation) and the other part of cells return immediately to the proliferating phase to divide again (short-term proliferation). This assumption leads to a modification of Mackey's model, which cannot be reduced to a classical delay differential system as in the previous works. We obtain a system of coupled delay differential and continuous time difference equations with distributed delay. We focus our mathematical analysis on this delay differential-difference system. We show that the behavior of the system is related to both short and long proliferating terms. We focus on the influence of coefficients of the system on the global asymptotic stability of the trivial steady state and the existence of unbounded solutions. This model may help to explain the behavior of HSCs in some hematological diseases.

In the next section (Section 2), we present the age-structured model that will be considered in this paper. In Section 3, we use the characteristics method to reduce the model to a coupled delay differential and renewal difference equations with continuous time and distributed delay. In Section 4, we give a necessary and sufficient condition for the global asymptotic stability of the trivial steady state. In Section 5, we establish the existence of unbounded solutions of the system.

Age-structured model of HSC dynamics

Let consider a population of HSCs, located in the bone marrow. The period of life of each HSC is divided into two phases: quiescence and proliferation (see [START_REF] Mackey | Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis[END_REF]). Denote by q(t, a) (respectively, p(t, a)) the density of quiescent (respectively, proliferating) HSCs with time t ≥ 0 and age a. The age represents the time spent by a cell in one of the two phases, thus it ranges, in quiescent phase, from 0 to infinity (which means that, cells can stay their entire life in the quiescent phase), and in the proliferating phase from 0 to τ > 0. Quiescent cells cannot divide, and provide they not died (with a rate δ ≥ 0 witch take into account the differentiation), they can be introduced in the proliferating phase with a rate β ≥ 0. The nature of the trigger signal for introduction in the proliferating phase is not clear. However, the work of Sachs [START_REF] Sachs | The molecular control of hemopoiesis and leukemia[END_REF] (see also [START_REF] Mackey | Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis[END_REF]) shows that we can reasonably think that it strongly depends on the entire quiescent cell population, that is,

β := β(Q(t)) with Q(t) = +∞ 0 q(t, a)da, t ≥ 0.
The function β is assumed to be differentiable and strictly decreasing with lim Q→+∞ β(Q) = 0. Typically, β is a Hill function (see [START_REF] Mackey | Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis[END_REF][START_REF] Pujo-Menjouet | Long period oscillations in a G 0 model of hematopoietic stem cells[END_REF][START_REF] Pujo-Menjouet | Contribution to the study of periodic chronic myelogenous leukemia[END_REF]). The proliferating phase is the active part of the cellular development. As soon as cells enter the proliferating phase, they perform a series of process leading to division at mitosis before reaching the maximum age a = τ . If they do not die by apoptosis (programmed cell death) with a rate γ ≥ 0, they are committed to divide with a rate d τ (a) depending on their age 0 < a < τ . The function d τ is assumed to be L 1 loc ([0, τ ), R + ) such that τ 0 d τ (a)da = +∞.

(

) 1
This later assumption describes the fact that cells which did not die have to divide before they reach the maximal age τ . We suppose that just after division, a fraction (1 -K) with K ∈ [0, 1], of daughter cells enter the quiescent phase (long-term proliferation) and the other fraction K of daughter cells return immediately to the proliferating phase to divide again (shortterm proliferation). Then, the part of dividing cells with age a ∈ (0, τ ) that enter the quiescent phase is 2(1 -K)d τ (a) and the part that return to the proliferating phase is 2Kd τ (a).

Then the densities q(t, a) and p(t, a) satisfy, for t > 0, the system of age-structured partial differential equations

                                   ∂q(t, a) ∂t + ∂q(t, a) ∂a = -(δ + β (Q(t))) q(t, a), a > 0, ∂p(t, a) ∂t + ∂p(t, a) ∂a = -(γ + d τ (a))p(t, a), 0 < a < τ, q(t, 0) = 2(1 -K) τ 0 d τ (a)p(t, a)da, p(t, 0) = +∞ 0 β(Q(t))q(t, a)da + 2K τ 0 d τ (a)p(t, a)da, lim a→+∞ q(t, a) = 0. (2)
The initial conditions are nonnegative functions given by q(0, a) = q 0 (a), a > 0,

p(0, a) = p 0 (a), 0 < a < τ. (3)
We assume that

q 0 ∈ W 1,1 (0, +∞), p 0 ∈ W 1,1 (0, τ ) and d τ (•)p 0 (•) ∈ L 1 (0, τ ), (4)
where W 1,1 (a, b) is the space of L 1 -functions on the interval (a, b) such that there derivatives are also L 1 -functions on (a, b).

The two boundary conditions in [START_REF] Adimy | Global stability of a partial differential equation with distributed delay due to cellular replication[END_REF], for a = 0, describe the flux of cells entering each phase: new proliferating cells are quiescent cells introduced with a rate β and cells coming from the division of proliferating cells that return to the cell cycle, with a rate Kd τ , to divide again, and new resting cells are coming from the other part of dividing cells (with a rate (1 -K)d τ ). It is important to note that the boundary conditions in [START_REF] Adimy | Global stability of a partial differential equation with distributed delay due to cellular replication[END_REF], for a = 0 and a → +∞, are not required to hold at t = 0 (see [START_REF] Iannelli | Mathematical theory of age-structured population dynamics[END_REF][START_REF] Gurtin | Non-linear age-dependent population dynamics[END_REF]). Indeed, these relations are satisfied at t = 0 if and only if the initial functions q 0 and p 0 satisfy the following compatibility condition

               q 0 (0) = 2(1 -K) τ 0 d τ (a)p 0 (a)da, p 0 (0) = β(Q 0 )Q 0 + 2K τ 0 d τ (a)p 0 (a)da, lim a→+∞ q 0 (a) = 0, (5)
where

Q 0 = +∞ 0 q 0 (a)da.
The assumptions (4) and ( 5) provide us the existence of a solution of (2) (see [START_REF] Iannelli | Mathematical theory of age-structured population dynamics[END_REF]); actually, the assumption ( 4) is enough to provide a solution of (2) in the sense stated in Theorem 4.3 on page 16 of [START_REF] Iannelli | Mathematical theory of age-structured population dynamics[END_REF].

The treatment of the system (2) will be developed in the next sections; in fact, instead of studying (2) directly, it will be transformed into a delay differential-difference system which is derived in the next section.

Reduction to a coupled retarded functional differential equations

By using the method of characteristics (see [START_REF] Iannelli | Mathematical theory of age-structured population dynamics[END_REF][START_REF] Webb | Theory of Nonlinear Age-Dependent Population Dynamics[END_REF]), the solutions p(t, a) of ( 2) are given, for 0 < a < τ , by One can check that a → D τ (a) represents the density of division of proliferating cells, i.e. D τ (a) is the probability density for a cell to divide at age a. In particular, thanks to [START_REF] Cruz | Stability of functional differential equations of neutral type[END_REF] we have

p(t, a) =      e -γt Π τ (a) Π τ (a -t) p(0, a -t), 0 < t ≤ a, e -γa Π τ (a)p(t -a, 0), t > a, (6) with
τ 0 D τ (a)da = 1.
Integrating system (2) over the age, remembering that a > 0 in the quiescent phase and 0 < a < τ is the proliferating phase, and by using the boundary and initial conditions, we obtain the following system We put, for t ∈ [-τ, 0],

                                                                     Q (t) = -(δ + β (Q(t))) Q(t) +2(1 -K) ×          t 0 e -γa D τ (a)u(t -a)da + τ t e -γt D τ (a) Π τ (a -t) p 0 (a -t)da, t < τ, τ 0 e -γa D τ (a)u(t -a)da, t > τ, P (t) = -γP (t) + β (Q(t)) Q(t) -(1 -2K) ×          t 0 e -γa D τ (a)u(t -a)da + τ t e -γt D τ (a) Π τ (a -t) p 0 (a -t)da, t < τ, τ 0 e -γa D τ (a)u(t -a)da, t > τ, u(t) = β(Q(t))Q(t) +2K ×          t 0 e -γa D τ (a)u(t -a)da + τ t e -γt D τ (a) Π τ (a -t) p 0 (a -t)da, t < τ, τ 0 e -γa D τ (a)u(t -a)da, t > τ, (7)
φ(t) := e -γt Π τ (-t) p 0 (-t). (8)
Then, system (7) can be written, for t > 0,

                 Q (t) = -(δ + β (Q(t))) Q(t) + 2(1 -K) τ 0 e -γa D τ (a)u(t -a)da, P (t) = -γP (t) + β (Q(t)) Q(t) -(1 -2K) τ 0 e -γa D τ (a)u(t -a)da, u(t) = β(Q(t))Q(t) + 2K τ 0 e -γa D τ (a)u(t -a)da, (9)
with initial conditions

Q(0) = Q 0 , P (0) = P 0 and u(t) = φ(t) for -τ ≤ t ≤ 0.
Thanks to ( 6) and ( 8), we obtain

P (t) = τ 0 e -γa Π τ (a)u(t -a)da, t > 0.
Since Q and u do not depend on P , we will focus on the study of the system

       Q (t) = -(δ + β (Q(t))) Q(t) + 2(1 -K) τ 0 e -γa D τ (a)u(t -a)da, u(t) = β(Q(t))Q(t) + 2K τ 0 e -γa D τ (a)u(t -a)da, (10)
with initial conditions given by

Q(0) = Q 0 and u(t) = φ(t) for -τ ≤ t ≤ 0. ( 11
)
System ( 10) is a coupled of differential and difference equations with distributed delay. The case when d τ is a Dirac distribution concentrated at τ ,

d τ (a) = δ τ (a) := +∞, a = τ, 0, a = τ,
was considered in the previous work [START_REF] Adimy | A delay differential-difference system of hematopoietic stem cell dynamics[END_REF], in the special case where γ = 0. In that case, we have

τ 0 d τ (a)da = 1, (12)
and the integral in the system (10) takes the form

τ 0 e -γa D τ (a)u(t -a)da = e -γτ u(t -τ ).
Then, the system (10) can be reduced to a delay differential-difference model with a discrete delay (see [START_REF] Adimy | A delay differential-difference system of hematopoietic stem cell dynamics[END_REF]). In all this paper, we assume that Q 0 ≥ 0, φ is a nonnegative and continuous function on

[-τ, 0], Q → β(Q)Q is a Lipschitz-continuous
Then, the existence and uniqueness of nonnegative solutions of ( 10)-( 11) can be obtained in the same manner as in [START_REF] Hale | Existence, uniqueness and continuous dependence for hereditary systems[END_REF]. From a solution (Q, u) of ( 10)- [START_REF] Cooke | Stability analysis for a vector disease model[END_REF], we can obtain a solution of (2). Indeed, under the conditions (4) and ( 5), we can proceed as in Theorem 4.2 on page 15 and Theorem 2.3 on page 51 of [START_REF] Iannelli | Mathematical theory of age-structured population dynamics[END_REF], to prove that the couple of functions (q, p) defined by

q(t, a) =          q 0 (a -t) exp - a a-t (δ + β(Q(s))) ds , 0 < t ≤ a, 2(1 -K) τ 0 e -γs D τ (s)u(t -s)ds exp - a 0 (δ + β(Q(s))) ds , t > a, and
p(t, a) =      e -γt Π τ (a) Π τ (a -t) p 0 (a -t), 0 < t ≤ a, e -γa Π τ (a)u(t -a), t > a,
satisfies the system (2).

Global asymptotic stability of the trivial steady state

Consider the auxiliary linear homogeneous difference equation

u(t) = D(u t ), t ≥ 0, (14)
where the history function u t ∈ C([-τ, 0], R) is defined, for t ≥ 0 and u ∈ C([-τ, +∞), R), by u t (θ) = u(t + θ) for θ ∈ [-τ, 0], and the operator D : C([-τ, 0], R) → R is given by

D(ψ) = 2K τ 0 e -γa D τ (a)ψ(-a)da.
Remark that the condition ( 13) is equivalent to

D := sup ψ ≤1 |D(ψ)| < 1, (15)
with ψ = sup θ∈[-τ,0] |ψ(θ)|. The condition [START_REF] Hale | Existence, uniqueness and continuous dependence for hereditary systems[END_REF] says that the zero solution of the linear difference equation ( 14) is globally asymptotically stable (see Theorem 3.5 on page 275 of [START_REF] Hale | Introduction to Functional Differential Equations[END_REF]).

The second equation of ( 10) can be written

u(t) -D(u t ) = β(Q(t))Q(t), t > 0. ( 16
)
We put

ν γ (τ ) = τ 0 e -γa D τ (a)da. (17)
Then, the inequality (13) (or ( 15)) becomes

2Kν γ (τ ) < 1. ( 18
)
As mentioned before, we investigate the global asymptotic stability of the trivial steady state using a Lyapunov-Krasovskii functional approach and a notion called input-to-state stability property of system [START_REF] Bradford | Quiescence, cycling, and turnover in the primitive hematopoietic stem cell compartment[END_REF] (see [START_REF] Gu | Lyapunov-krasovskii functional for uniform stability of coupled differential-functional equations[END_REF][START_REF] Pepe | A new Lyapunov-Krasovskii methodology for coupled delay differential and difference equations[END_REF]).

The following lemma gives a fundamental inequality that leads to the input-to-state stability property of system [START_REF] Bradford | Quiescence, cycling, and turnover in the primitive hematopoietic stem cell compartment[END_REF]. Lemma 1. Let (Q, u) be the solution of system (10) associated to the initial condition (Q 0 , φ) ∈ R × C([-τ, 0], R). Then there exist α, C > 0 such that, for t > 0,

|u(t)| ≤ C φ e -αt + sup 0≤s≤t |β(Q(s))Q(s)| , ≤ C φ e -αt + β 0 sup 0≤s≤t |Q(s)| , ( 19
)
with φ = sup θ∈[-τ,0] |φ(θ)| and β 0 := β(0) = max x≥0 β(x).
Proof. This result follows from Theorem 3.5 on page 275 of [START_REF] Hale | Introduction to Functional Differential Equations[END_REF].

We state the following lemma.

Lemma 2. Let (Q(t), u(t)) be a solution of [START_REF] Bradford | Quiescence, cycling, and turnover in the primitive hematopoietic stem cell compartment[END_REF]. Then, lim t→+∞ u(t) = 0 if and only if

lim t→+∞ β(Q(t))Q(t) = 0.
Proof. The proof of this result is based on the equation ( 16), the inequalities [START_REF] Mackey | Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis[END_REF] and ( [START_REF] Cruz | Stability of functional differential equations of neutral type[END_REF], Lemma 3.5).

Remark that when β : R + → R + is a Hill function (see [START_REF] Mackey | Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis[END_REF][START_REF] Pujo-Menjouet | Long period oscillations in a G 0 model of hematopoietic stem cells[END_REF][START_REF] Pujo-Menjouet | Contribution to the study of periodic chronic myelogenous leukemia[END_REF]) given by

β(x) = β 0 θ n x n + θ n , with β 0 > 0, θ > 0 and n > 1, ( 20
)
the only possible way to have lim

t→+∞ (β(Q(t))Q(t)) = 0 is lim t→+∞ Q(t) = 0 or lim t→+∞ Q(t) = +∞.
We now prove the result dealing with the global asymptotic stability of the trivial steady state of ( 10) and its instability.

Theorem 1. (i) Assume that δ > 0 and δ β 0 > 2ν γ (τ ) -1 1 -2Kν γ (τ ) . ( 21
)
Then, the trivial steady state of (10) is globally asymptotically stable.

(ii) Assume that δ β 0 < 2ν γ (τ ) -1 1 -2Kν γ (τ ) . ( 22
)
Then, no nontrivial solution of the system (10) tends to the trivial steady state.

Proof. (i) Consider the continuous functional

V : R + × C([-τ, 0], R + ) → R + , (Q 0 , φ) → V (Q 0 , φ),
defined by

V (Q 0 , φ) = Q 0 + 2(1 -K) 1 -2Kν γ (τ ) τ 0 e -γa D τ (a) 0 -a φ(θ)dθda.

It is not difficult to check that

Q 0 ≤ V (Q 0 , φ) ≤ ϑ ( (Q 0 , φ) ) , with (Q 0 , φ) := max{|Q 0 |, φ } and ϑ(x) = 1 + 2(1 -K)τ ν γ (τ ) 1 -2Kν γ (τ ) x, x ≥ 0.
Let (Q, u) be a solution of [START_REF] Bradford | Quiescence, cycling, and turnover in the primitive hematopoietic stem cell compartment[END_REF]. We have

V (Q(t), u t ) = Q(t) + 2(1 -K) 1 -2Kν γ (τ ) τ 0 e -γa D τ (a) t t-a u(θ)dθda.
The function t → V (Q(t), u t ) is differentiable on (0, +∞) and

d dt V (Q(t), u t ) = Q (t) + 2(1 -K) 1 -2Kν γ (τ ) τ 0 e -γa D τ (a)[u(t) -u(t -a)]da.
Then, using [START_REF] Bradford | Quiescence, cycling, and turnover in the primitive hematopoietic stem cell compartment[END_REF] one can check that

d dt V (Q(t), u t ) = -(δ + β(Q(t)))Q(t) + 2(1 -K) τ 0 e -γa D τ (a)u(t -a)da + 2(1 -K) 1 -2Kν γ (τ ) τ 0 e -γa D τ (a) [β(Q(t))Q(t) + D(u t )] da - 2(1 -K) 1 -2Kν γ (τ ) τ 0 e -γa D τ (a)u(t -a)da.

It follows that

d dt V (Q(t), u t ) = -δ - 2ν γ (τ ) -1 1 -2Kν γ (τ ) β(Q(t)) Q(t) +2(1 -K) τ 0 e -γa D τ (a)u(t -a)da + 4K(1 -K)ν γ (τ ) 1 -2Kν γ (τ ) τ 0 e -γa D τ (a)u(t -a)da - 2(1 -K) 1 -2Kν γ (τ ) τ 0 e -γa D τ (a)u(t -a)da, = -δ - 2ν γ (τ ) -1 1 -2Kν γ (τ ) β(Q(t)) Q(t).
Consequently,

d dt V (Q(t), u t ) = -w(Q(t))Q(t), t > 0, with w(x) = δ - 2ν γ (τ ) -1 1 -2Kν γ (τ ) β(x) , for x ≥ 0.
If 2ν γ (τ ) > 1, then from [START_REF] Kuang | Delay Differential Equations: With Applications in Population Dynamics[END_REF] and since the function β is decreasing, it follows that w is an increasing function on [0, +∞). Furthermore, the condition [START_REF] Pujo-Menjouet | Long period oscillations in a G 0 model of hematopoietic stem cells[END_REF] implies that w(0) > 0. If 2ν γ (τ ) ≤ 1, then ( 21) is always satisfied and w(x) ≥ δ > 0, for all x ≥ 0. We conclude that V is a Lyapunov-Krasovskii functional. This, together with the input-to-state stability estimation [START_REF] Mackey | Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis[END_REF], implies the global asymptotic stability of the trivial steady state (see [START_REF] Gu | Lyapunov-krasovskii functional for uniform stability of coupled differential-functional equations[END_REF], Theorem 3).

(ii) We now suppose that the condition ( 22) is satisfied. Let (Q, u) be a solution of [START_REF] Bradford | Quiescence, cycling, and turnover in the primitive hematopoietic stem cell compartment[END_REF]. Assume by contradiction that Q(t) → 0 as t → ∞. By the continuity of the function β, we have β(Q(t)) → β 0 := β(0) as t → ∞. Consequently, for all > 0 there exists A( ) such that, for all t ≥ A( ),

β 0 -< β(Q(t)) ≤ β 0 . (23)
We fix 0 < < β 0 . By integration the system (10) over t ∈ (A( ), +∞), we get

         Q(A( )) = +∞ A( ) [δ + β(Q(t))] Q(t)dt -2(1 -K) +∞ A( ) τ 0 e -γa D τ (a)u(t -a)dadt, +∞ A( ) u(t)dt = +∞ A( ) β(Q(t))Q(t)dt + 2K +∞ A( ) τ 0 e -γa D τ (a)u(t -a)dadt. (24)
By applying Fubini's theorem we obtain

+∞ A( ) τ 0 e -γa D τ (a)u(t -a)dadt = τ 0 e -γa D τ (a) +∞ A( )-a u(t)dtda, ≥ ν γ (τ ) +∞ A( ) u(t)dt. (25)
From ( 23), the second equation of ( 24) and ( 25), it follows

+∞ A( ) u(t)dt ≥ (β 0 -) +∞ A( ) Q(t)dt + 2Kν γ (τ ) +∞ A( ) u(t)dt.
Then, thanks to [START_REF] Kuang | Delay Differential Equations: With Applications in Population Dynamics[END_REF], we obtain

+∞ A( ) u(t)dt ≥ β 0 - 1 -2Kν γ (τ ) +∞ A( ) Q(t)dt. (26)
We deduce from the first equation of ( 24) and ( 25), that

Q(A( )) ≤ (δ + β 0 ) +∞ A( ) Q(t)dt -2(1 -K)ν γ (τ ) +∞ A( ) u(t)dt, ≤ δ -β 0 2ν γ (τ ) -1 1 -2Kν γ (τ ) + 2(1 -K)ν γ (τ ) 1 -2Kν γ (τ ) +∞ A( ) Q(t)dt.
As a consequence of ( 22), one can choose ∈ (0, β 0 ) such that Q(A( )) < 0. This leads to a contradiction with the positivity of the solutions. Then, Q(t) does not tend to zero as t → +∞.

Remark 1.

1. One can apply the same reasoning as in the proof of Theorem 1-(i), to obtain the global stability of the trivial steady state (but not the global asymptotic stability) under the condition

δ β 0 = 2ν γ (τ ) -1 1 -2Kν γ (τ ) .
For the definition of the global stability, see for instance [START_REF] Hale | Introduction to Functional Differential Equations[END_REF]. 2. If δ = 0 and ν γ (τ ) < 1/2, then (21) is satisfied and the trivial steady state is globally asymptotically stable. 3. If δ = 0 and ν γ (τ ) > 1/2, then [START_REF] Pujo-Menjouet | Contribution to the study of periodic chronic myelogenous leukemia[END_REF] is satisfied and the nontrivial solutions of the system [START_REF] Bradford | Quiescence, cycling, and turnover in the primitive hematopoietic stem cell compartment[END_REF] do not converge to zero. 4. Although the solution Q(t) does not tend to zero, u(t) may tend to zero (for instance, when Q(t) tends to infinity and β is a Hill function (20)).

Existence of unbounded solutions

We now study the special case δ = 0. We show in the next proposition, that the solutions of (10) may be unbounded on [0, +∞). Proposition 1. Assume that δ = 0 and

2(1 -K)ν γ (τ ) > 1. ( 27
)
Let (Q, u) be a solution of [START_REF] Bradford | Quiescence, cycling, and turnover in the primitive hematopoietic stem cell compartment[END_REF]. Suppose that there exists x > 0 such that the function x → β(x)x is decreasing for x ≥ x. Assume that there exists t ≥ 0 such that Q(t) > x and Q is increasing on [t, t + τ ]. Then, Q is increasing on [t, +∞) and lim t→+∞

Q(t) = +∞. Proof. Let a ∈ [0, τ ]. We have x < Q(t) ≤ Q(t + τ -a) ≤ Q(t + τ ). Then, it follows that β(Q(t + τ ))Q(t + τ ) ≤ β(Q(t + τ -a))Q(t + τ -a). Consequently, Q (t + τ ) = -β(Q(t + τ ))Q(t + τ ) + 2(1 -K) τ 0 e -γa D τ (a)u(t + τ -a)da, = -β(Q(t + τ ))Q(t + τ ) +2(1 -K) τ 0 e -γa D τ (a)β(Q(t + τ -a))Q(t + τ -a)da +4K(1 -K) τ 0 e -γa D τ (a) τ 0 e -γs D τ (s)u(t + τ -a -s)dsda, ≥ (2(1 -K)ν γ (τ ) -1) β(Q(t + τ ))Q(t + τ ), > 0.
Hence, Q is increasing on [t, T ] for some T > t + τ . Suppose by contradiction that there exists T > t + τ such that Q (T ) = 0 and Q (t) > 0 for all t ∈ [t, T ). Then, x < Q(t + τ ) < Q(T ). We repeat the same calculation as above and we get

0 = Q (T ) ≥ (2(1 -K)ν γ (τ ) -1) β(Q(T ))Q(T ) > 0.
This gives a contradiction. Then, Q is increasing on [t, +∞). We now suppose by contradiction that Q is bounded on [t, +∞). Consequently, the limit L := lim t→+∞ Q(t) exists. As Q is an increasing function on [t, +∞), we have lim t→+∞ Q (t) = 0. Letting t tend to infinity in both sides of the system (10), we obtain

β(L)L 2ν γ (τ ) -1 1 -2Kν γ (τ ) = 0.
Hence L = 0. This leads to a contradiction. We conclude that lim t→+∞ Q(t) = +∞.

The term 2(1 -K)ν γ (τ ) in ( 27) is the portion of dividing cells that enter the quiescent phase and it is assumed to be greater than 1, with δ = 0. So, it is no surprise that lim t→+∞ Q(t) = +∞.

In the next proposition, we construct an example of functions (3) such that the initial condition (11) of system [START_REF] Bradford | Quiescence, cycling, and turnover in the primitive hematopoietic stem cell compartment[END_REF] satisfies the assumptions of Proposition 1. First, the assumption on the function x → β(x)x is satisfied when β is given by a Hill function [START_REF] Pepe | A new Lyapunov-Krasovskii methodology for coupled delay differential and difference equations[END_REF]. We can take

x = θ (n -1) 1/n .
Proposition 2. Assume that δ = 0 and (27) holds true. Let v ≥ x be fixed, with x > 0 defined in Proposition 1. Consider the following initial condition

   q 0 (a) = exp - a v , a > 0, p 0 (a) = e -γa Π τ (a)β(v)v, 0 < a < τ.
Then, Q is an increasing function on [0, +∞) and lim t→+∞

Q(t) = +∞. Proof. First, it is clear that Q(0) = +∞ 0 exp -a v da = v > x.
Second, the initial function φ is given by

φ(t) = e -γt Π τ (-t) p 0 (-t) = β(v)v, t ∈ [-τ, 0].
Then, from [START_REF] Adimy | Stability and hopf bifurcation in a mathematical model of pluripotent stem cell dynamics[END_REF] we have

Q (0) = -β (Q(0)) Q(0) + 2(1 -K) τ 0 D τ (a) Π τ (a) p 0 (a)da, = -β (Q(0)) Q(0) + 2(1 -K)β(v)v τ 0 e -γa D τ (a)da, = -β (Q(0)) Q(0) + 2(1 -K)ν γ (τ )β(v)v. (28)
Consequently,

Q (0) = (2(1 -K)ν γ (τ ) -1) β(v)v, > 0.
So, Q is increasing on [0, η] for some η ∈ (0, τ ]. Suppose by contradiction that there exits η ∈ (0, τ ] such that Q (η) = 0 and Q (t) > 0 for t ∈ [0, η). Then, we have

0 = Q (η) ≥ -β (Q(η)) Q(η) + 2(1 -K) τ η
e -γη D τ (a) Π τ (a -η) p 0 (a -η)da,

≥ -β (Q(η)) Q(η) + 2(1 -K)ν γ (τ )β(v)v. Moreover, Q(η) > Q(0) = v ≥ x. Then, β(Q(η))Q(η) < β(v)v. Consequently, 0 = Q (η) > (2(1 -K)ν γ (τ ) -1) β(v)v > 0,
which is absurd. We conclude that Q is an increasing function on [0, τ ]. By direct application of the previous proposition, the result holds.

We note that the initial functions q 0 and p 0 in the previous proposition satisfy the assumption (4). But, they do not satisfy the compatibility condition [START_REF] Adimy | Stability and hopf bifurcation for a cell population model with state-dependent delay[END_REF]. In fact, (4) is enough to provide a solution of ( 10)- [START_REF] Cooke | Stability analysis for a vector disease model[END_REF].

Discussion

We proposed and analyzed an age-structured partial differential system describing the dynamics of hematopoietic stem cells (HSCs). We took into account the fact that a cell cycle has two phases, that is, HSCs are either in a quiescent phase or actively proliferating. We made the difference between dividing cells that enter directly to the quiescent phase (long-term proliferation) and dividing cells that return to the proliferating phase to divide again (short-term proliferation). These features were never been taken into account in previous models of HSCs dynamics. We focused our mathematical analysis on the behavior of the trivial steady state (global asymptotic stability) and the existence of unbounded solutions. We showed that the behavior of the system is related to both short and long proliferating terms. The model is relevant due to its ability to explain how an uncontrolled proliferation of HSCs can lead to either a high/low-throughput of HSCs. Thus, it can be able to characterize some hematological disorders, especially those associated to a high blood production (for instance, myeloproliferative diseases) or a low blood production (for instance, anemias). A further analysis could help in understanding the complicated dynamics of the blood cell production system.

Π

  τ (a) := exp -a 0 d τ (s)ds , 0 < a < τ. Set D τ (a) := d τ (a)Π τ (a), 0 < a < τ.

  where u(t) = p(t, 0) is the new proliferating cells and P (t) = τ 0 p(t, a)da is the total population of the proliferating cells. The initial conditions becomeQ(0) = Q 0 := +∞ 0q 0 (a)da and P (0) = P 0 := τ 0 p 0 (a)da.

0 e

 0 function, and the following inequality 2K τ -γa D τ (a)da < 1.


Marin Bougeret
email: marin.bougeret@ens-lyon.fr.

Pierre-Francois Dutot
email: pierre-francois.dutot@imag.fr.

Denis Trystram
email: ‡trystram@imag.fr.

Using oracles for the design of efficient approximation algorithms

Using oracles for the design of efficient approximation algorithms

Marin Bougeret (Speaker) * Pierre-Francois Dutot † Denis Trystram ‡

We are interested here in oracle techniques for the design of approximation algorithms. Following the classical definition, an oracle is a black box capable of answering correctly and instantaneously any question. Several classical P T AS design techniques can be expressed using oracle formalism (by allowing the algorithm to "guess" some values during the computation).

Our objective in this work is to point out the interest of oracle techniques, beyond the design of P T AS. Indeed, questions to the oracle (i.e. guessed values) leading to non polynomial algorithms must also be considered, as the complexity may be exponential, but in a parameter that is supposed to be "small". Moreover, we aim at showing how it is possible to "degenerate" questions asked to the oracle to derive fast implementations of these interactive algorithms. These ideas will be illustrated on the classical makespan minimization on uniform machines problem (Q||C max ).

Context : oracle algorithms

Given an instance I of an optimization problem, an oracle algorithm A or asks the oracle for a guess, in the form of a string r * I ∈ R I , that generally provides some information on the structure of an optimal solution. Then, the algorithm constructs a solution A or (I, r * I ) for the initial problem. From such an algorithm, it is possible to derive a "classical" algorithm A (without oracle), by either re-executing A or (I, r) for any r ∈ R I , or constructing separately r * I (using another algorithm). Taking the example of scheduling problems, a very classical question r * I is an "optimal configuration" of a wellchosen small subset of k tasks (among the n of the instance), where k is constant. Such information may allow arbitrarily good approximation ratios (like 1 + 1 k ) at the price of subset enumeration, when simulating the oracle.

Hence, it is clear that there exist deep connections between oracle algorithms and techniques for designing approximation schemes. As shown in [START_REF] Bougeret | Systèmes interactifs our la résolution de problèmes complexes[END_REF], an oracle formulation allows natural alternative definitions of several classical techniques (as those presented in [START_REF] Schuurman | Approximation schemes -a tutorial[END_REF]). Most of such techniques are based on information obtained by exhaustive enumeration or by binary search. Replacing them by oracle answers separates difficulties due to the information determination from the ones due to its utilization.

Application on the classical Q||C max problem

Let us consider the problem of minimizing the makespan when scheduling independent tasks on uniform machines as a case study. It is shortly denoted by Q||C max .

Several approximation algorithms have been proposed for this problem. The 2 ratio (achieved by the classical Longest Processing Time algorithm [START_REF] Gonzalez | Bounds for lpt schedules on uniform processors[END_REF]) has been improved to 3 2 in [START_REF] Hochbaum | A polynomial approximation scheme for scheduling on uniform processors: Using the dual approximation approach[END_REF] (using the dual approximation technique), and to 1.382-approximation in [START_REF] Chen | Tighter bound for MULTIFIT scheduling on uniform processors[END_REF]. Among all existing approximation schemes, the most relevant here are the following (we list below the time complexity to achieve a ratio of (1 + ǫ)):

• O(mn 10 ǫ 2 +3 ) in [START_REF] Hochbaum | A polynomial approximation scheme for scheduling on uniform processors: Using the dual approximation approach[END_REF] • O(( 1ǫ n 2 ) m-1 ) (also applies to R||C max ) in [START_REF] Horowitz | Exact and approximate algorithms for scheduling nonidentical processors[END_REF] • O((n + 1)

We propose an oracle algorithm based on [START_REF] Hochbaum | A polynomial approximation scheme for scheduling on uniform processors: Using the dual approximation approach[END_REF]. For any a ∈ N * , our algorithm guarantees an 1 + 1 a ratio by asking some information on the "big" tasks scheduled on each machine (i.e. whose computation requires more than a fraction 1 a of the total computation time on this machine).

Firstly, notice that the classical guess (i.e. asking the index of the big tasks scheduled on each machine) would lead to an approximation scheme with the same complexity as the one in [START_REF] Lenstra | Approximation algorithms for scheduling unrelated parallel machines[END_REF]. Thus, we show how to reduce the amount of information asked, and thus the size of R I , for small values of a (typically a = 3 or 4). We get for instance a 4 3 (resp. a 5 4 )-approximation by only asking the number of big tasks for each machine, leading to an algorithm in O(2 m poly(n, m)) (resp.O(3 m poly(n, m))). Thus, these approximation algorithms can be faster than the better approximation schemes applied for ǫ equal to 1 3 (resp. 1 4 ), as there is no constant hidden in the exponent. Secondly, we discuss efficient implementations where the algorithms avoid asking some sub-parts of the question. The key idea is to check if some additional a priori unexpected conditions become true during the execution on the particular current instance, allowing then to make a local optimal decision (without oracle query).

The approach presented in this work leads to the following natural questions for Q||C max .

• Using only one bit of information for each machine, what information should be asked to obtain a ratio better than 4 3 ?

• How to reduce the amount of information used for larger values of a ?


Weiqing Fang

Ioannis Panagiotopoulos

Frédéric Ott

François Boué

Ait-Atmane Kahina

Jean-Yves Piquemal

Guillaume Viau

Florent Dalmas

Optimization of the magnetic properties of aligned Co nanowires / polymer composites

Keywords:

We aim at combining high coercivity magnetic nanowires in a polymer matrix in a view to fabricate rare--earth free bonded magnets. In particular, our aim is to fabricate anisotropic materials by aligning the wires in the polymer matrix. We have explored the different parameters of the fabrication process in order to produce a material with the best possible magnetic properties. We show that the choice of a proper solvent allows obtaining stable nanowire suspensions. The length and the type of the polymer chains play also an important role. Smaller chains (M w <10000) provide better magnetization results. The magnetic field applied during the casting of the material plays also a role and should be of the order of a fraction of a tesla. The local order of the nanowires in the matrix has been characterized by TEM and Small Angle Neutron Scattering. The correlation between the local order of the wires and the magnetic properties is discussed. Materials with coercivity µ 0 H c up to 0.70 T at room temperature have been obtained.

Introduction

During the last decade, a significant research effort has been put into the fabrication and the investigation of the structure and the magnetic properties of magnetic metal nanowires, especially Ni, Fe and Co (Vazquez 2006) and their alloys (particularly FeNi (Brzòzka et al 1996), CoFe (Qin et al 2002), CoNi (Soumare et al 2008), FePt (Rhen et al 2004). Such wires can be produced either by electrochemical deposition of metals in uniaxial porous templates such as anodic aluminium oxide (Fert et al 1999), by organo--metallic synthesis (Wetz et al 2007), or by the polyol process (Ung et al 2007;Soumare et al 2008 andSoumare et al 2009). Such wires exhibit specific magnetic properties owing to their large shape anisotropy (Sellmyer et al 2001 and[START_REF] Mcgary | [END_REF]. The large shape anisotropy of these objects (aspects ratios >5) gives rise to significant coercive fields (µ 0 H c = 0.3--1T). The idea of using the shape anisotropy of micro--or nano--objects to produce hard magnetic materials was proposed a long time ago. This property has been used for decades in AlNiCo magnets which consist of FeCo needles in an AlNi matrix [START_REF] Mccurrie | Ferromagnetic Materials-Structure and Properties_Academic[END_REF]. Materials using FeCo elongated single domain particles (Mendelsohn 1955 andFalk 1966) were even commercialized under the name Lodex. Owing to the recent progress in the chemical synthesis techniques we are revisiting this idea.

Cobalt nanowires synthesized by reduction in liquid polyol are mono--crystalline and mono--disperse (Soumare 2009).

Their length ranges between 200--300nm and their diameter between 10--20nm. These wires exhibit remarkable hard magnetic properties for a simple 3d metal. Due to the absence of defects and to their mono--crystalline state, experimental coercivity can reach values larger than 0.5 T at room temperature. We showed in previous work that half of the observed coercivity could be accounted for by the magneto--crystalline anisotropy (Maurer 2007 andAit--Atmane 2013). The remaining part can be attributed to the shape anisotropy contribution.

There are few reports about aligned magnetic nanowires dispersed in a polymer matrix. Fragouli et al (2010) present a technique for magnetic--field--induced formation of magnetic nanowires in a polymer film. They start from a polymer/iron oxide nanoparticle casted solution that is dried under magnetic field. Nanocomposite films with aligned nanowires formed by nanoparticles are obtained. Similar results were obtained by Robbes et al (2011). Park et al (2007) have investigated the micromechanical properties of Ni nanowire/polymer composites.

In this communication we report on the use of a polymer matrix to host aligned Co nanowires in order to fabricate high coercivity anisotropic materials. Among the roles of the different parameters involved, we investigate the role of the polymer matrix (type of polymer, length of the chains), the role of the solvent, and the role of the alignment field. The magnetic bulk properties of the obtained materials were characterized by VSM magnetometry. The microstructures of the materials were investigated by means of MEB, TEM and Small Angle Neutron Scattering. The SANS measurements were performed at the Laboratoire Léon Brillouin on the PAXY spectrometer.

Sample preparation

Co nanowires synthesis

Co nanowires are synthesized by reduction in liquid polyol (1,2--butanediol) according to procedures previously reported (Soumare 2009). After the synthesis, the magnetic nanowires are separated from the butane--diol synthesis solution by centrifugation and washed 2 times with ethanol and one time with chloroform to remove the remaining organic compounds. At this stage, only a layer of laurate remains at the surface of the nanowires (Ait--Atmane 2013).

Dispersion of the wires in solvent

Several solvents were tested to obtain a dispersion of nanowires. Water has to be strictly excluded since it leads to a fast oxidation of the Co nanowires. Solvents such as toluene, butane--diol and chloroform were tested. In most of the cases, once mixing is done, decantation is complete after a time ranging from minutes to days. In practice, even if the dispersions (~0.5% wt) remain deep black, decantation of the largest magnetic clusters nevertheless takes place. In order to quantify the quality of the dispersions and the ability of a specific solvent to provide a stable suspension, the UV absorbance of different suspensions was measured as a function of time (see figure 1). The behavior is very different for the three solvents investigated, as shown in Figure 1. In the synthesis solution (butane-diol) the evolution of absorbance shows a phase--separation phenomenon. Just after shaking, the solution is dark; in the end, we obtain a supernatant of very low NW concentration butane--diol solution above a far more concentrated liquid. The sharp drop of absorbance after 150 minutes corresponds to the point where the supernatant/dense phase interface passes in front of the optical cell. In the case of toluene solvent, the decantation is very fast: the absorbance decreases from 6 to 1 after only 10 min. On the other extreme, for chloroform, we obtain our best stability results. After a first decantation of the larger particles (over the first 2--3 hours), the suspension stabilizes. In the best cases, suspensions in chloroform have been stable up to several weeks. Note that any incorporation of water leads to an instantaneous decantation.

Preparation of solid samples in a polymer matrix

We used polystyrene (PS), Poly(methyl methacrylate) (PMMA) or poly(vinyl pyrrolindone) (PVP) as a matrix for the preparation of the nanocomposites. For PVP, several molecular masses were also considered, namely M w 10 k, 40 k and 360k. A concentrated polymer chloroform solution (typically 10% v/v polymer let dissolve for several hours) is mixed with a solution of cobalt nanowires which is agitated in an ultrasonic bath during 15 min. The mixture is then mechanically agitated on a shaking table for 30 minutes. The mixture is then poured into a Teflon container and dry cast for 5 days. As the boiling point of chloroform is low (61°C), the container is cooled to about 10°C in order to reduce the evaporation speed of the solvent, which prevents the formation of bubbles on the surface of the nanocomposites film. The composites samples prepared with PS, PMMA and PVP were labeled Co--PS, Co--PMMA and Co--PVP--x, where x stands for the polymer molecular weight.

Moreover, during the preparation of the nanocomposites, the influence of the application of an external magnetic field was evaluated. Two types of experiments were devised: samples prepared by dry casting in a zero magnetic field environment and samples prepared under the application of a magnetic field H align (up to 0.8T) during the casting. Note that this field should be extremely homogeneous with low gradients. Not fulfilling this condition will result in an inhomogeneous sample since the wires will migrate to the extremities of the film during the casting. In order to minimize these gradients we used NMR grade electromagnets with very large polar pieces (200mm in diameter) (see figure 2a). We expect the casting under field to induce an alignment of the wires (see figure 2b), inducing in turn an intrinsic bulk magnetic anisotropy in the sample. This can be readily quantified by magnetometry measurements. The hysteresis cycles can be measured along the direction of the aligned wires (Ox), perpendicular to the wires (Oy) or perpendicular to the sample surface (Oz). Figure 2c shows the principle of expected hysteresis cycles measured along various directions. In the case of an isotropic sample the curve should be rather rounded reflecting the random distribution of the anisotropy directions. In the case of the aligned wires, a measurement with the field applied along the wires long axis should give a rather square hysteresis cycle with a high remanence while a measurement perpendicular to the wires should give a narrow loop with a low coercivity and remanence. We recall that the optimal shape of the hysteresis curve for a permanent magnet is a square one in which the coercivity is at least half of the remanent polarization of the material. 3 Characterization of the magnetic properties of the composites

Isotropic samples: effect of the polymer matrix on the magnetic properties

Several polymer matrices were tested: polystyrene (PS), poly(methyl methacrylate) (PMMA) and poly(vinyl pyrrolindone) (PVP). The use of PMMA led to very brittle samples. On the contrary, films with PS matrix displayed good mechanical properties. However several tests showed that before casting, the mixture of PVP, chloroform, and NWs was more stable than the equivalent mixture with PS. The mixture with PVP can be stable during 2--3 days while that with PS decant after a few hours. This led us to focus on PVP matrices. The sample is in glassy state. For high molecular weight (e.g. 360 000), the mechanical properties enable an easy handling of the sample. For low molecular weight (<40 000), the samples are quite brittle.

Table 1 shows coercivity and remanence measured on isotropic composites with three polymers of various lengths PS 192 k, PMMA 350 k, and PVP 360 k, 40 k, 10 k. No drastic effect is observed on the coercivity values and the behavior is rather close to a "pure" NW sample in which no polymer has been added (such as the samples deposited on a wafer). Thus we may infer that there has been no significant interaction between the nanowires and the polymer matrix. The normalized remanence values range between M r /M s =0.5 and M r /M s =2/π expected for a completely random--3D and in--plane--2D--random distribution of the wire axes respectively. Thus from these values the angular deviation Δθ out of the sample plane (xy--plane, see conventions in Figure 2) can be calculated assuming that the axes distribution is completely random within the sample plane but deviates by ± Δθ above and below the 1. It can be noticed that very large chains (especially of PVP360k and PMMA350k) lead to an in-plane 2D--random distribution in contrast to shorter ones --as well as to the "pure" NW sample (with no polymer addition) -- which have an intermediate distribution, partially out of plane. When the molecular weight is reduced from 360 k down to 10 k, a measurable increase of the coercivity is observed. This suggests that when small polymer chains are used, they are able to coat and separate the cobalt wires. This can be due to kinetic effects, but here we start from a solution where the different motions should be much faster that the evaporation. So that this can be better associated with "depletion effects" where long chains, due to confinement conformational constraints tend to expel out towards regions free of NW, which turns down to bring the NW together. When the wires are physically separated (by a few nm), the magnetic dipolar interactions are reduced (Maurer 2011). This minimizes the direct effects of dipolar fields between wires and thus increases the effective coercive field.

Polymer

Effect of the applied field during casting

We studied the effects of the magnitude of the magnetic field applied during the casting of the samples. Two different types of samples were studied (i) cobalt wires dispersed in a chloroform solution without polymer deposited in the form of a thin layer of Co on silicon wafers and (ii) dry cast composites (Co NW + PVP 40k), which is the aim of our study.

The layers of NWs on silicon could be readily characterized by SEM (see Figure 3). It can be observed that the alignment of the wires is invisible for low fields (0.04T). It is barely visible for µ 0 .H align = 0.1 T, but becomes obvious for µ 0 .H align =0.2 T and 0.4 T. The results of the magnetometry measurements are summarized in Figure 4 where the coercive field and the remanence have been reported as a function of H align . The alignment of the NWs in the polymer matrix can be evaluated by measuring the loop squareness S= M R /M S value. For an ideally parallel assembly, a value close to S=1 is expected while it is decreasing to 0.5 for a totally disorganized 3D assembly. The S easy =(M R /M S ) value that is derived by the measurement along the alignment direction should increase with the alignment while at the same time S=(M R /M S ) ⊥ value derived by the measurement perpendicular to the field is expected to decrease. The ratio S easy / S _hard gives a figure of merit of the anisotropy of magnetic properties obtained by the alignment. It can be noted that even if the SEM image does not show any obvious alignment for µ 0 .H align =0.04T, a measurable anisotropy is nevertheless observed (S easy / S hard =1.6). The anisotropy increases almost linearly as a function of H align reaching a ratio S easy /S hard =3 at µ 0 .H align = 0.4T. The coercive fields follow the same trend as a function of the alignment field. The evolution of the coercive field along the easy direction is however not spectacular (increasing only from 0.52 to 0.55T).

Figure 3. SEM picture of cobalt nanowires dried on a Si wafer under various magnetic fields.

The composites films Co--PVP could not be characterized through SEM due to difficulty of preparation.

Magnetometry can nevertheless be used to follow the evolution of the anisotropy. The S easy / S hard ratio evolves from 2.2 to 4.25 for an alignment field increasing from 0.2 to 0.8 T. Surprisingly, the coercive field H c,easy decreases from 0.52T down to 0.45T when the alignment field is increased from 0.2 to 0.8T (Fig 4c). In conclusion we can say that alignment fields of a fraction of a tesla should be applied to maximize the anisotropy. However, the level of anisotropy is not directly correlated with the coercive field. It is likely that the applied field magnitude modifies the clustering of the wires. Under larger drying fields, larger NW clusters are formed. This increases the dipolar interactions and consequently reduces the effective coercive field. In the case of the wires deposited on a Si substrate the drying process is faster and the formation of larger clusters is limited. 4 Characterization of the morphology and the dispersion of cobalt nanowires

Microscopic and macroscopic observations

In order to correlate more finely the local arrangement of wires with the magnetic properties, we performed systematic studies on 2 cast samples (Co--PS--192 and Co--PVP--360). TEM, Small Angle Neutron Scattering and magnetometry were performed on these samples. Figure 5 shows TEM images of ultramicrotomic slices of the (Co--PS) sample prepared without (Fig. 5a) or under (Fig. 5b) the application of an external magnetic field. In the case of the absence of an external magnetic field during drying, isotropic samples are obtained and one observes small bundles of 6--12 aligned wires (size of order 500 nm --1 µm) together with more open clusters of NW independent in orientation, with similar size. In the case of the aligned samples, one can see that the wires are well aligned with an angular dispersion of 18° FWHM (as deduced from an analysis in the imageJ software). The wires are still grouped in small bundles of 10--20 wires. The Co--PVP--360 sample preparation for the TEM could not be achieved with our The magnetometry characterization is shown in Figure 6. Looking at the values in detail shows that the Co--PS sample exhibits slightly better properties than the Co--PVP sample in terms of coercive field. For the Co--PS sample, we obtain µ 0 H c =0.70 T, while for the Co--PVP sample µ 0 H c = 0.65 T. On the other hand, the PVP sample is better aligned since for Co--PS, M r /M s = 0.9 while for the Co--PVP sample M r /M s = 0.93. This is confirmed by SANS scattering data (Fig. 7). The alignment is more marked in the PVP sample.

In order to get a more quantitative interpretation of the magnetization data we have fitted the loops using the Stoner--Wohlfarth (SW) model for an assembly of particles with the angular distributions suggested by the remanence values. More specifically we have assumed the following distribution of angular orientations: (i) distribution of the azimuthal angle φ between the wire axis and the alignment field direction within the sample plane and (ii) distribution of the angle θ between the wire axis and sample plane. Note that θ as defined here is the complementary of the polar angle typically used in spherical coordinates. Gaussian distributions with zero average values and standard deviations σ φ , σ θ respectively have been assumed. The results are shown as the continuous lines in Fig. 6. The agreement is fair considering that the SW model is based on the hypothesis of reversal by homogeneous magnetization rotation of non--interacting magnetic entities, which may be violated in our case mainly by dipolar interactions (Maurer, 2011). Consequently the deviations of the fitted curves are maximal in the demagnetization quadrant of the loops where interaction assisted nucleation events are more likely to occur. The PS sample has identical σ φ , σ θ values of 20°. Note that these values are consistent with the direct space observations.

The PVP is better aligned within the plane (σ φ =14°) but the out of plane value is similar. A further indication of departure from the ideal SW behavior is that the best value of anisotropy field required to describe the easy axis curves (H K (par)) is about 2/3 of that of the hard axis curve H K (per). In principle these two values should coincide. Theoretically for a wire of hexagonal Co with an aspect ratio of 20 and its crystallographic c--axis along its length, a value of H K =2.3 T is expected, of which 1.56 T due to shape anisotropy and the rest 0.74 T due to the magneto-crystalline anisotropy. This theoretical value is closer to the ones obtained by the perpendicular measurement. This is due to the fact that in the field range close to saturation, homogeneous reversible rotation mechanisms are dominant whereas in the range near the magnetization switching point more complex mechanisms and irreversible jumps may occur. We have used the value that best describes the former range for the hard axis loop fit.

Furthermore in the curves measured with the applied field perpendicular to the easy direction (hard axis), there is an apparent contradiction between the anisotropy value which must be used to account for the high field range (approach to saturation) as opposed to the value that predicts the correct loop shape. In conclusion the real anisotropy field is best represented by H K (per) while the deviation of the ratio H K (par)/ H K (per) from unity gives a measure of the dominance of non--homogeneous reversal mechanisms. In that sense the PVP sample which is better aligned shows a larger deviation from the ideal SW--behavior and despite its more marked anisotropy has a lower coercivity. An explanation can be that alignment allows the wires to become more closely packed, which increases dipolar interactions. Progress in orientation and separation can thus be contradictory in some situations.

Discussion

Macroscopic observations show that the NW dispersed in (PVP + chloroform) decant slower than dispersion in (PS + chloroform). A natural conclusion is that the nanowires are better dispersed in the (PVP--chloroform) solution, and are present in the solution as smaller clusters. During casting, when starting from a more dispersed suspension it is expected that a better local alignment is obtained. This is confirmed by SANS data (Fig. 7) which show a better degree of orientation of the nanowires at the nanometric scale. It is confirmed by bulk magnetic measurements in which one observes that the ratio S easy / S hard is higher in PVP compared to PS composites. However, the better orientation of the wires does not lead to overall better magnetic properties since the coercive field of the PVP sample is smaller than in the PS sample. This can be interpreted by the fact that the wires in the PVP sample are better aligned but are also in closer contact. This leads to an overall loss of effective coercivity due to dipolar interactions. On the other hand, when the system is more disordered, the wires are physically more separated and thus the dipolar interactions are reduced. Thus in order to optimize the properties of these composites, a subtle balance between order and magnetic properties must be found.

Conclusions

We have quantified the effects of various parameters on the fabrication of Co nanowires/polymer composites for the fabrication of permanent magnets. In order to optimize the magnetic properties, magnetic fields as large as possible should be applied. The field gradients must however be minimized to obtain an homogeneous material.

Small molecular weight of the polymer chains enable to obtain better macroscopic magnetic properties by leading to a better separation of the nanowires (less aggregation, better dispersion). This minimizes the local magnetic dipolar interactions between individual wires and provides better bulk magnetic properties. In the case of anisotropic materials, being able to produce highly aligned materials does not lead to optimal magnetic properties especially in terms of coercive field. This can also be accounted for by the increased magnetic dipolar interactions between wires which are in close contact when perfectly aligned. One has thus to find an optimum between the quality of the dispersion, the degree of alignment and the bulk magnetic properties. Our experimental observations are presently being modeled using micromagnetic simulations to figure out the exact role of the local ordering between wires on the magnetic properties. The best achieved coercivity was 0.70T, with a remanence M r /M s of 0.9.

Assuming a Co volume fraction of 60%, the maximal energy product (BH) max of such a material would be on the order of 260 kJ/m 3 which is in the range of the best SmCo magnets ((BH) max ~ 120--200 kJ/m 3 ). This is not yet competitive with the best NdFeB magnets ((BH) max ~ 440kJ/m 3 ). However such compounds might present interest in high temperature applications (above 250°C) (Ait--Atmane, 2013) where NdFeB magnets cannot be used unless very expensive Dy is added as for example in automotive applications.

This work was partially supported by the EU--NMP funded project "Rare--Earth Free Permanent Magnets" REFREEPERMAG.

Ait

Figure 1 :

 1 Figure 1: UV absorbance of cobalt nanowires dispersed in different solvents (1,2--butanediol, toluene, chloroform) (~0.5% wt) as a function of time.

Figure 2 :

 2 Figure 2: (a) The wires are aligned between the polar pieces of an electromagnet. (b) Axis convention in an aligned sample; (c) Calculated in--plane hysteresis loops of an isotropic (black) and of an aligned sample (blue, easy axis; red, hard axis).

  sample plane. The calculated standard deviation values σ θ of the out of plane orientation distributions are given in the last row of table

Figure 4 :

 4 Figure 4: Coercivity and remanence of the cobalt nanowires dispersed on a Si substrate (a,b) and in PVP 40k casting under magnetic field (c,d) .

Figure 5 .

 5 Figure 5. TEM pictures of cobalt nanowires dispersed in PS 192k and dried without (a) and (b) with the application of an external magnetic field (µ 0 H = 0.6 T).

Figure 6 .Figure 7 :

 67 Figure 6. Hysteresis loops of cobalt nanowires dispersed in PS 192k and PVP 360k, aligned at room temperature.

  --Atmane K et al (2013) Journal of Solid State Chemistry Vol. 197, p. 297-303 Brzòzka K, Gawroński M, Jezuita K (1996) Journal of Magnetism and Magnetic Materials Vol. 157-158,p. 167-168 Falk R--B (1966), J. Appl. Phys. Vol.37, p.1108. Fert A, Piraux L (1999) Journal of Magnetism and Magnetic Materials Vol.200 p.338--358 Fragouli D, Buonsanti R, Bertoni G, Sangregorio C, Innocenti C, Falqui A, Gatteschi D, Cozzoli P D, Athanassiou A and Cingolani R (2010) ACS Nano Vol 4 NO. 4. p1873--1878 Maurer T, Ott F, Chaboussant G, Soumare Y, Piquemal J--Y, Viau G (2007) Appl. Phys. Lett. Vol. 91 172501.

Table 1 .

 1 Coercivity and remanence of isotropic composites with three different polymers.

		NW powder	PS 192k PMMA 350k PVP 360k	PVP 40k	PVP 10k
		on Si
	µ 0 H c (T)	0.6	0.58	0.57	0.56	0.61	0.68
	M r /M s	0.57	0.56	0.63	0.65	0.59	0.57
	σ θ (deg)	27	31	8	0	22	27