Jörg Arndt
email: arndt@jjj.de

Testing polynomial irreducibility without GCDs

Keywords: finite fields, irreducible polynomial, GCD corps fini, polynôme irréductible, PGCD

We determine classes of degrees where testing irreducibility for univariate polynomials over finite fields can be done without any GCD computation.

Recall Rabin's test for the irreducibility of a polynomial over GF(p) where p is prime [1, p.7]: A polynomial C ∈ GF(p)[x] of degree d is irreducible if and only if

x p d ≡ x mod C (1)
and, for all prime divisors l i of d,

gcd x p d/l i -x mod C, C = 1 (2)
The number of GCD computations equals the number of prime divisors of d.

We call a polynomial C a pseudo irreducible (P I) if it has no linear factor and relation [START_REF] Michael | Probabilistic algorithms in finite fields[END_REF] holds. We denote by I d the set of all irreducible polynomials of degree d, and by P I d the set of all pseudo irreducibles of degree d.

A composite polynomial C of degree d can be a P I only if it has no square factor: relation (1) tells us that C must divide the polynomial x p d -x which has no square factors. Let the factorization of a composite polynomial C that is a P I into irreducible polynomials C j be

C = f j=1 C j (3)
then the degrees of all factors d j := deg(C j ) must be divisors of d (the polynomial x p d -x is the product of all irreducibles whose degrees divide d).

We define

L := lcm j=1...f (d j ) (4)
For a composite C that is a P I we have x p L = x mod C. This motivates our next definition.

We call a polynomial C of degree d that is a P I and for which the relation

x p k = x mod C for 1 ≤ k ≤ d -1 (5)
holds, a strong pseudo irreducible (SP I). We write SP I d for the set of all polynomials of degree d that are SP I.

A composite C that is a SP I must factor into mutually distinct irreducibles C j where all degrees d j must divide d (so L divides d) and L = d.

The following algorithm (SP I-test) determines whether a polynomial C ∈ GF(p)[x] of degree d is a SP I:

1. If C has a linear factor then return false.

2. Set t := x.

3. Do the following d -1 times: set t := t p mod C, if t = x then return false.

4. Set t := t p mod C, if t = x return false.

5. Return true.

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The detection of linear factors is cheap if the characteristic is small. For example, a binary polynomial has no linear factor if its constant term is one and its weight is odd. The costs of the test by evaluation at the nonzero elements of GF(p) is

∼ (p -1) d.
Next we will identify classes of degrees where I d = SP I d , that is, testing for strong pseudo irreducibility is sufficient to determine irreducibility. Note that the test does not involve any GCD computations.

For a composite C that is SP I we have a factorization as in relation (3) and We have d = a r + b s ≡ a r mod s and, since d has s as a factor, a must be a multiple of s: a = u s. The minimal value is u = 1, since a = 0, hence the minimal value of R is R min := s r.

d = f j=1 d j (
Therefore the factorization of C must contain a product of irreducible polynomials of degrees d j = r kj where 1 ≤ k j ≤ e such that their product has degree

INRIA ≥ R min = s r.
If the product of all irreducible polynomials of degrees r k (where 1 ≤ k ≤ e) has degree < R min then there is no partitioning of d that makes a composite polynomial C a SP I.

Write D(n) for the degree of the product of all irreducible polynomials whose degrees are greater than one and divide n. The polynomial x p n -x gives the product including linear irreducibles. There are p linear irreducibles, so we have D(r e ) = p r e -p. Depending on the ratio of the costs of powering and computing the GCD this test may be faster than Rabin's test. In the following we look at characteristic two. The speed of multiplication (and thereby powering) is very dependent on the weight of the polynomial modulus C. One of the most favorable cases for powering is when C is a trinomial. For fixed degree let S, M , and G be the time required for squaring, multiplication, and computing a GCD, respectively. For d = 859, 433 Zimmermann [pers. comm.] gives the ratios G/M ≈ 30 and G/S ≈ 8, 800 for a general purpose CPU (AMD Opteron 2.4GHz). When a hardware multiplier is available the ratio is much more in favor of the powering.

An average-case analysis of Rabin's algorithm is given in [START_REF] Panario | Analysis of Rabin's polynomial irreducibility test[END_REF]. The fact that reducibility of a composite polynomial is almost always detected with the computation of the first GCD suggests that it may be a bad idea to postpone all GCDs. Instead one may proceed as in the test above, but compute gcd(s 1 , C) as soon as s 1 is computed and postpone only the remaining GCDs. Even for degrees d where SP I d = I d one might want to either compute the first GCD or exclude factors of small degrees by other means.

Theorem 2

 2 6) That is, the degrees of the irreducible factors are a partition of d into numbers d j that are divisors of d where 2 ≤ d j ≤ d and L = d. Theorem 1 If d is a prime power then I d = SP I d . Proof: Let d = r e where r is prime (and e ≥ 1). The divisors of d that can appear in the partition given in relation (6) are r, r 2 , . . . , r e-1 . We have L ≤ r e-1 < d so the polynomial cannot be a SP I. If d is the product of two primes then I d = SP I d . Proof: Let d = r s where r and s are distinct primes. The divisors of d allowed in the partition are r and s. Let the partition be d = a r + b s. Now d = r s ≡ a r mod s, and as r and s are coprime, a must be a multiple of s. The choices for a are the following: firstly, a = 0 corresponding to the partition d = 0 r + r s (i.e., r factors of degree s) but then we have L = s < d; secondly, a = s corresponding to the partition d = s r + 0 s (i.e., s factors of degree r) but then we have L = r < d again. Now we give classes of degrees d where I d = SP I d for polynomials over GF(p).Here the conditions for the degree depend on the characteristic p.Let d = r e s where r and s are distinct primes. In the partitioning d = d j we have d j = r u s v where u and v are nonnegative and u + v < e + 1.We split the partitioning into two sets according to whether the divisors are pure powers of r: d = R + R where R = j:gcd(s,dj )=1 d j and R = d -R = j:gcd(s,dj ) =1 d j . For L = d the divisors r e must occur in the partitioning so R = 0. Also for L = d some divisor d j must have a factor s so R = 0. Now we determine the minimal value of R. As R is a sum of powers of r, R must be divisible by r. As R is a sum of multiples of s, R must be divisible by s. Thereby a partitioning with L = d corresponds to a partition d = R + R = a r + b s where a = 0 and b = 0.

For a composite C

  of degree d = r e s that is a SP I we must have R min ≤ D(r e ), use R min = r s and divide by r to obtain s ≤ (p r e -p)/r. Thus we have proved: Theorem 3 If d = r e s where r and s are different primes and s > (p r e -p)/r we have I d = SP I d . For example, over GF(2) we have I d = SP I d for the following cases where the bound is < 10 9 : d = 4 s and s > (2 4 -2)/2 = 7, d = 8 s and s > (2 8 -2)/2 = 127, d = 9 s and s > (2 9 -2)/3 = 170, d = 16 s and s > (2 16 -2)/2 = 32, 767, d = 25 s and s > (2 25 -2)/5 = 6, 710, 886, and d = 27 s and s > (2 27 -2)/3 = 44, 739, 242. The rapid growth of the bound renders the Theorem impractical for large characteristic. We give the cases where the bound is < 10 9 for characteristic 3: d = 4 s and s > 39, d = 8 s and s > 3, 279, d = 9 s and s > 6, 560, and d = 16 s and s > 21, 523, 359. For degrees d where SP I d = I d the following test that delays the computation of the GCDs can be used: 1. Compute the factorization d = f i=1 p ei i . 2. Set t 0 := x.3. Do the following fork = 1, 2, . . . , d -1: set t k := t p k-1 mod C, if k = d/p i then save s i := t k , if t k = x then return false. 4. Set t d := t p d-1 mod C, if t d =x return false. 5. Do the following for i = 1, 2, . . . , f : set g := gcd(s i , C), if g = 1 then return false.6. Return true.

RR n°0123456789

http://www.inria.fr ISSN 0249-6399

This work was partly funded by the INRIA Associate Team "Algorithms, Numbers, Computers" (http://www.loria.fr/~zimmerma/anc.html). Ce travail a été réalisé en


CESAER Ghib Marie-Luce

Larkham Krystyna
email: krystynalarkham@googlemail.com

CESAER Wavresky Pierre
email: pierre.wavresky@enesad.inra.fr

IEA Luca Lucian

Small farm in Romania: evolution under localization constraint?

Keywords: small farms, farm evolution, Romania, restructuration, historical and geographical determinants JEL: Q15, Q18, P32, J11

. In contrast, the role of localization, linked to historical background or physical constraints, as a factor is more limited in the literature. Other institutional factors such as inheritance and succession have not really been approached until now. We choose to investigate those factors using multiple regression on the 3 recent data collections in Romania.

Introduction

A focus on small farms requires a brief tour of work on the definition of farms [START_REF] Ghib | Mise en oeuvre de la politique des structures roumaines : une nécessaire définition de l'activité agricole pour un ciblage des mesures[END_REF]. This question has long been addressed in terms of economies of scale, but since the late 90s has been discussed under the umbrella of multifunctionality, along with new meanings of agriculture including the integration of environmental and social aspects of this activity. Finally, at the European level, enlargement to include the countries of Central and Eastern Europe, with the diversity of distribution of agricultural structures raises the question of small farms, particularly through their inclusion in the Common Agricultural Policy. The polarised structure of Romanian farms is strongly marked by many small farms but also by the social role of these structures, making it a particularly interesting case. This paper is the beginning of work on the determinants of change in Romanian agricultural structures. We present here the first results concerning the relationship between distribution and evolution of structures based on structural factors. Indeed, we find in the literature and through surveys of agricultural Romanian actors, assumptions related to the localization of small farms and their evolution. The location factors most commonly related to structural change were defined: mountain zones, the history of land in the region (prior to communism), and also to the relative wealth of a Judet 1.

Small farm definition and hypothesis on their localization

Definitions

The issue of small farms leads us first to the definition of agricultural activity. It is noted that the definition of small farms is still not yet determined, despite many attempts at both national and international levels [START_REF] Lund | Aspects of the Definition and Classification of Farms[END_REF][START_REF] Remy | Les petites exploitations dans la politique agricole, Pour (194)[END_REF]. A recent OECD publication (OECD, 2009) made a comparison of the definition of agricultural activity, a prerequisite to the definition of the thresholds of classes. The 17 countries studied were shown to define agricultural activity on various criteria that can be linked to income from business (Canada, Finland, Poland), sales (the United States, Australia, Japan, Korea) , potential production (Austria, Germany, Netherlands), factors of production such as area (Denmark, Finland, France, Switzerland, Japan, Korea) or the size of herd (Switzerland, Denmark), but also the number of work units engaged in the business (France and Germany).

Similarly, the small holding, the farm or the farm household are entities that provide the basis for the development of this activity. Again, definitions vary. In the Romanian case that we treat here, this definition has been difficult to elucidate for the establishment of the Common Agricultural Policy in Romania [START_REF] Ghib | Mise en oeuvre de la politique des structures roumaines : une nécessaire définition de l'activité agricole pour un ciblage des mesures[END_REF]. The latest document, the National Rural Development Plan (PNDR, 2008), tells of a typology of three categories: subsistence farms (0 to 2 ESU2 ), the semi-subsistence farms (2 to 8 ESU ) and commercial farms (more than 8 ESU). Within this article, we will use as a definition of "small farms" subsistence or semi-subsistence farms, measured either by ESU or by default surface area in hectares3 .

Factors influencing the evolution of farms

Firm size as a factor in their growth is an old subject of research, as far back as [START_REF] Gibrat | L'élevage en Roumanie, une tradition laitière et un avenir pour les viandes ovine et bovine[END_REF]. Gibrat proposed a theory, called the law of proportional effect, that the rate of growth of a firm is 1 Judete are the Romanian Administrative unit equivalent to NUTS 3 : 'Nomenclature d'Unités Territoriales Statistiques'. independent of its original size. Growth would be a purely stochastic phenomenon resulting from the operations of many random independent factors. His theory was applied to farms for the first time by [START_REF] Upton | The Growth of Farms[END_REF] and [START_REF] Shapiro | Farm size and growth in Canada[END_REF], who arrive at conflicting conclusions about its validity for agriculture. Other more recent work, in turn, invalidates the theory on empirical data, in particular in the work of [START_REF] Weiss | Farm growth and survival: econometric evidence for individual farms in Upper Austria[END_REF].

We do not develop in this short article a test of Gibrat on Romanian data. Instead we propose a synthesis of the data aggregated by Judet examining changes according to the categories of physical size.

Indeed, recent statistics collected on farms in Romania suggest that a restructuring is underway. The general trend is a decrease in the number of farms and cultivated areas:

Table 1: Number of Farms and area farmed categorised by area

Sources: RGA 2002, ASA 2005, ASA 2007 This decrease at the national level was higher in the first period 2002 -2005 between 2005 and 2007 for Very Small Farms (VSF, <1 ha). However, for Small Farms (SF), 1 to 5 ha, the trend is completely different-a slight increase on the first period and decrease the second, more than 6% in number and more than 5% surface.

The medium sized farm (MF) of 5 to 100 ha experienced a sharp increase between 2002 and 2005 with nearly 40% increase in number, which remained positive between 2005 and 2007 but capped at less than 5%. In contrast, the holdings of more than 100 ha significantly decreased between 2002 and 2005 ( -12.36%), before experiencing an increase between 2005 and 2007 by 8%.

These developments confirm the gradual establishment of a restructuring process, probably linked to the agricultural policies implemented during these two periods. In particular, we can highlight the changes by the Minister G. Flutur from 2005. But we could also attribute these changes to anticipation of the arrival of subsidies and the first effects of the SAPARD program started in 2002. These areas were therefore previously subjected to different political regimes but also different land regimes, [START_REF] Axenciuc | Evolutia economica al Romaniei. Cercetari statistico-istorice 1859-1947[END_REF] which have helped shape the present picture of distribution of land structures.

Here we call upon the theories of historical institutionalism, according to which historical institutions affect present phenomena.

According to [START_REF] Thelen | Comment les institutions évoluent : perspectives de l'analyse comparative historique, L'année de la régulation[END_REF], the institutions may be resilient in the face of major historical ruptures, ie shocks that are a priori expected to break previous models (...).

Here we test the reappearance or reformation of land structures after three major shocks. The first of these shocks is the land reform of 1945 with the seizure of the lands of Saxons 4 in Romania but also any property over 50 hectares and their redistribution to veterans and war widows. The second was the communist period and the major impact of collectivization, and the third is the land reform of 1991 to the fall of communism with the redistribution and restitution of land to more than 4 million people.

Despite these three upheavals in agricultural structures, political and land regimes anterior at 1945 will also contribute to the history of regions, and the slow rate of present restructuring is differentiated by region. These differences are explained by the persistence and transfer of agricultural know-how maintained during communism on the plots in areas where individual farming prevailed in contrast to regions where the population was predominantly manual or subject to large landowners and which may have lost this element of agricultural knowledge. Juvancic (2006) shows that the phenomenon is reversed in Slovenia: farms disappear fastest in the most urbanized areas, which by default have the most favourable conditions, as the non-agricultural labour market is more developed therefore alternative income strategies become more important. Furthermore, he believes that the relative persistence of smaller farms in remote areas may be due to the rigidity of the land market in these areas.

• factors related to the richness of the Judet

The relationship between regional wealth and maintenance of exploitations is often perceived as unfavorable. The richer a region the greater the importance of alternative employment and exits from agriculture become easier. Conversely, poor regions would see a continuation in agricultural activity, reinforcing its role as a social buffer [START_REF] Pouliquen | L'agriculture néo-paysanne roumaine : le tampon social contre la relance globale Revue d'études comparatives[END_REF]. GDP is here a proxy for regional wealth.

In contrast, in countries where agricultural activity is farming is essentially part-time, this agriculture practised in this region is peri-urban, highly intensive in small areas. The reasons for the development of these farms are also heavily influenced by land markets in conjunction with ongoing with urban sprawl.

The results concerning the contribution of different economic size classes to the total economic size of Judet are presented in Table 2. The variable HISTO, which measure the part of holding below 3 ha in 1941, plays the strongest role in the explanation of the presence of small farm in a Judet: the regions which held many small farms before the collectivization of land are still characterized by this feature at present. Influence of the GDP is lower and less significant: nevertheless it appears that in rich regions holdings tend to be large. Finally, large farms are somewhat less represented in the mountain areas.

For regressions on the evolution, we chose the change over the period 2002-2007 as more significant than for other periods. The explanation of the evolution by class of area is lower than the contribution in terms of economic size at a given time. Unlike in Slovenia (Juvancic, 2006), mountain areas appear to favour a relative preservation of large farms at the expense of the very small. Knowing that we take into account the total agricultural area, including also the pastures, we can explain this correlation by the sheep sector, which has a strong presence in mountain regions and known for its dynamism (Institut de l'élevage, 2007).

We note also the representation of SF (but not VSF) in the Judet which are "traditionally" refuges of small farms (using the HISTO variable, the percentage of holdings of less than 3 ha in 1941). The importance of this variable on structure of holdings is therefore accentuated. Moreover, it is evident that the evolution of the number of MF does not depend on the percentage of deprived areas, history or wealth of the region: these variables are expected to play on the survival, growth or decrease of holdings at the extremes of farm size.

When looking at trends for the period on all farms, only GDP and 'less favoured areas' are involved. The Judet with a low GDP are those for which farm structures are most 'resistant' to change. This confirms the role of agriculture as a social buffer described by [START_REF] Pouliquen | L'agriculture néo-paysanne roumaine : le tampon social contre la relance globale Revue d'études comparatives[END_REF].

Conclusions

Within this study, we have been able to show correlations between contribution by economic size class or developments, and the explanatory variables commonly proposed to explain regional differences in terms of agricultural structure and its evolution. Thus the historical variable is particularly important, indicating that earlier agricultural structures have persisted despite 40 years of collectivization. Equally, three land reforms have failed to erase the effects of the land distribution of 1941.

Mountain areas are not particularly characterised by the presence of smaller farms, contrary to what one might expect. Large farms are also absent from mountain regions, but where their numbers change there is an increase. It would be interesting to analyse the role of sheep farming in terms of specialisation and farm size in mountain farms.

Finally, the lack of correlations within the analysis corresponding to medium sized exploitations leads us to question the chosen thresholds.

  variable which represents the percentage of farms below 3ha by Judet in 1941 Sources: Agricultural Census 1941, map made by the authors with Philcarto http://philcarto.free.fr Romania in its current borders is a young country. The joining in the region of Transylvania with former princely provinces in the south and east occurred in 1918 and was formalised by the Treaty of Trianon (1920) and the defeat of Hungary alongside Germany.

4•

  People of German origin living in Transylvania since 1200-1300 4 Fa it a ve c P hilca rto * 23 /0 4 /2 0 0 9 22 :3 8 :5 5 * h ttp ://p h ilcarto .fre e .fr rectangles de l'histogramme sont proportionnelles au nombre d'unites spatiales dans chaque classe définie sur la variable : '% of farm <3 ha in 1941' maximum= 24 pour la classe n° 2 Factors relating to less favoured areas Sources: PDRN, 2008, map made by the authors using Philcarto http://philcarto.free.fr Map 2: Distribution by percentage of the total agricultural areas classified as ZMD and ZDS/ ZSD Mountain areas and those classed as 'disadvantaged' (ZMD, ZSD and ZDS 5 ) are assumed to have handicaps affecting agricultural activity, and farms are expected to disappear faster in these areas. This assumption is the basis of the measures the EU has put in place since 1975 to aid these disadvantaged areas based on zoning according to altitude, slope, or the degradation of agricultural land (EC, 2009).

Fa

  it a ve c P h ilca rto * 2 2 /0 4 /2 0 0 9 1 4 :5 9 :4 9 * h ttp ://p h ilca rto .fre e .fr rectangles de l'histogramme sont proportionnelles au nombre d'unites spatiales dans chaque classe définie sur la variable : '% ZMD' maximum= 14 pour la classe n° 1 Fa it a ve c P h ilca rto * 2 2 /0 4 /2 0 0 9 1 4 :5 9 :4 9 * h ttp ://p h ilca rto .fre e .fr

Table 2 : Factors explaining contributions in ESU

 2

		ctb0_2	ctb_sup8
	constant	18.8	56.5
	ZMD	-	-0.14 **
	HISTO	0.72 ***	-0.66 ***
	GDP	-0.002 **	0.003 **
	R²	0.43	0.47
	-: insignificant variable
	** : significantly different from 0 at 5%
	*** : significantly different from 0 at 1%

Table 3 : Explanation of evolution in percentage of farm number given by area.

 3

		evol02_07 VSF	evol02_07 SF	evol02_07EM 5 -100 ha	evol02_07GE > 100 ha	total
		0_1 ha	1 -5 ha
	Constant	-15.2	-16.2	-	11.8	-5.39
	ZMD	-0.16 **	-	-	0.40 ***	-
	ZSD_ZDS	-	-	-	-	0.12 **
	HISTO	-	0.21 *	-	-0.83 ***	-
	GDP	-	-	-	0.006 ***	-0.00148 ***
	R²	0.12	0.07	0.04	0.37	0.23
	-: insignificant variable
	* : significantly different from 0 at 0%
	** : significantly different from 0 at 5%
	*** : significantly different from 0 at 1%

European Size Unit

Subsistence and semi subsistence exploitations are defined in the first version of the National Strategic Plan as those with a surface area of less than 5 ha[START_REF] Ghib | Mise en oeuvre de la politique des structures roumaines : une nécessaire définition de l'activité agricole pour un ciblage des mesures[END_REF]

ZMD: Zone Montana Defavorizate, altitudes up. to

600m or 400-600m with average slope > 15%. ZDS: Zone Semnificativ Defavorizata, ie low quality of soil, unfavorable climate or humidity and relief conditions. ZSD: Zonele de Conditii Naturale Specifice, areas with natural handicap.

relationship is reversed for survival, but growth of farms remains lower in the more affluent (Juvancic, 2006).

Data and variables

We conducted multiple regressions to test the hypotheses presented earlier with the following elements.

Data

The data used are those of the last data collection at the National Agricultural Census 2002 and the last two surveys of agricultural structures in 2005 and 2007 in Romania. In this article, we use the aggregated data at NUTS 3 level, ie within the 42 Judet.

Dependant variables

The dependent variables chosen are twofold. We first tested a ratio of economic size (ESU), ie the contribution of different economic classes in size to the total economic size Judets put together: ctb0_2 for the contribution of agricultural units from 0 to 2 ESU, ctb2_8 for the contribution of agricultural units from 2 to 8 ESU and ctb_sup8 for the contribution of agricultural units of more than 8 ESU. These data were available from the structure survey in 2005. In a second step, we tested the trend in the number of operations by Judet on 3 periods 2002-2005, 2005-2007 and 2002 -2007, translated as a percentage of the starting year. These changes have been made for the 4 categories of surface for the holdings. We selected the period 2002-2007 which will be noted by the following 'evol02_07*'.

Explanatory variables

The explanatory variables were selected to test the hypotheses presented earlier. Disadvantaged areas are captured by two variables: the mountain 'ZMD' and other disadvantaged areas 'ZDS_ZSD'. These two variables are constructed from information in Annex 4 of NRDP for the period 2007-2013.

To take into account the historical factor, we used an indicator noted 'HISTO' based on the percentage of holdings by Judet less than 3 ha in 1941 [START_REF] Cresin | Recensamintul agricol al Romaniei din 1941[END_REF]. We chose to work on data from 1941 because it is a date prior to the agrarian reform of 1945 but above all before the communist period that has significantly changed the distribution of sizes of structures by collectivization and the establishment of state farms. Finally we also tested the role of 'GDP' in the explanation of dependent variables chosen. The equations used are as follows: ctb* = f (ZMD, ZDS_ZSD, HISTO, GDP) evol02_07*= f (ZMD, ZDS_ZSD, HISTO, GDP)

Results

A first analysis on the whole data set showed a strong role of the GDP variable, but this was strongly influenced by the Judet of Bucharest. We have therefore chosen to remove it from our sample because its features are very important. The


Stefano Bianchini
email: bianchin@sissa.it

Bernard Hanouzet
email: bernard.hanouzet@math.u-bordeaux1.fr

Roberto Natalini
email: r.natalini@iac.cnr.it

Dissipative Hyperbolic Systems: the Asymptotic Behavior of Solutions

Introduction

In this talk we shall review some recent results about qualitative beahavior of smooth solutions to the Cauchy problem for general hyperbolic symmetrizable m-dimensional systems of balance laws of the form

u t + m α=1 (f α (u)) xα = g(u), (1)
with the initial condition u(x, 0) = u 0 (x),

where u = (u 1 , u 2 ) ∈ Ω ⊆ R n1 × R n2 , with n 1 + n 2 = n. We assume that there are n 1 conservation laws in the system, namely that we can take g(u) = 0 q(u)

, with q(u) ∈ R n2 .

(3)

As well known, even for nice initial data, smooth solutions may break down in finite time, due to the appearance of singularities. On the other hand, sometimes dissipative mechanisms induced by the source term can prevent the formation of singularities, at least for small initial data, as for instance for the compressible Euler equations with damping [START_REF] Nishida | Nonlinear hyperbolic equations and related topics in fluid dynamics[END_REF][START_REF] Sideris | Decay and singularities of solutions of the three-dimensional Euler equations with damping[END_REF].

Recently, it was proposed in [START_REF] Hanouzet | Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF] a quite general framework of sufficient conditions to have the global existence in time of smooth solutions. Actually, for systems which are endowed with a strictly convex entropy function E = E(u), a first natural assumption is the entropy dissipation condition, see [START_REF] Chen | Hyperbolic conservation laws with stiff relaxation terms and entropy[END_REF]. Unfortunately, it is easy to see that this condition is too weak to prevent the formation of singularities. A quite natural supplementary condition can be imposed to entropy dissipative systems, following the classical approach by Shizuta and Kawashima [START_REF] Kawashima | Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications[END_REF][START_REF] Shizuta | Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation[END_REF]. It is possible to prove that this condition, which is satisfied in many interesting examples, is also sufficient to establish a general result of global existence for small perturbations of equilibrium constant states, see [START_REF] Hanouzet | Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF][START_REF] Yong | Entropy and global existence for hyperbolic balance laws[END_REF]. These results will be shortly reviewed in Section 2.

Here, we intend to present a short survey of the new results of [START_REF] Bianchini | Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF], about the asymptotic behavior in time of the global solutions, then always assuming the existence of a strictly convex entropy and the Shizuta and Kawashima condition. However, let us recall that, starting from the seminal paper by T.P. Liu [START_REF] Liu | Hyperbolic conservation laws with relaxation[END_REF], there were some previous studies on 2 × 2 systems, see [START_REF] Natalini | Recent results on hyperbolic relaxation problems, Analysis of systems of conservation laws[END_REF] for some references. For more general models, we recall [START_REF] Zeng | Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation[END_REF], for a model of gas dynamics in thermal nonequilibrium, and [START_REF] Ruggeri | Stability of constant equilibrium state for dissipative balance laws system with a convex entropy[END_REF], about stability of constant equilibrium states for general hyperbolic systems in one space dimension, under zero-mass perturbations.

The paper is organized as follows: Section 2 is devoted to recall some basic results about hyperbolic systems with entropy dissipation and the Shizuta-Kawashima condition. In this section we also introduce the decomposition of the linearized system, which will be called the Conservative-Dissipative form, which will be necessary to cast the decay results in the optimal way. Section 3 contains some precise results about the asymptotic behavior of the Green kernel for the linearized problem. Finally, Section 4 is devoted to the study of the decay properties of the nonlinear system. Not only we shall present the decay results for both the conservative and the dissipative part of the solution, but we shall show also that the conservative variable approaches the conservative part of the solution of the corresponding linearized problem, faster that the decay of the heat kernel for m ≥ 2. Also, it is possible to see that the solution of the parabolic problem given by the Chapman-Enskog expansion, is a good approximation of the conservative part of the solution of the nonlinear hyperbolic system. For m ≥ 2 the Chapman-Enskog operator is linear, while, in one space dimension, the decay of the nonlinear part has a stronger influence, and so we can only show the faster convergence towards the solution of a parabolic equation with quadratic nonlinearity. Let us point out that the present results can be useful to design more accurate numerical approximations of these problems, which are increasingly accurate for large times, see [START_REF] Aregba-Driollet | Asymptotic high-order schemes for dissipative hyperbolic systems[END_REF] for some preliminary results in this direction.

Finally, we notice again that these results were obtained by assuming all the time the condition (SK). Unfortunately, this condition is not satisfied by many models, as for instance for the model studied in [START_REF] Zeng | Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation[END_REF], where however global existence of solutions has been established. For the Kerr-Debye system, studied in [HH00, CH106, CH206], condition (SK) is not always verified for m = 1, see [START_REF] Hanouzet | Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF], and never verified for m ≥ 2.

2 A short review of some basic structures for the entropy dissipative hyperbolic systems

In the following we shall consider a general m-dimensional system of balance laws given by equation (1), with the source term g = g(u) given by (3). According to the general theory of hyperbolic systems of conservation laws [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF], we shall assume that the system satisfies an entropy principle: there exists a strictly convex function E = E(u), the entropy density, and some related entropy-flux functions F α = F α (u), such that for every smooth solution u ∈ Ω to system (1), there holds

E(u) t + m α=1 (F α (u)) xα = G(u) , (4)
where

F α ′ = (f ′ α ) T E ′ and G = E ′ • g. Let us introduce the set γ of the equilib- rium points: γ = {u ∈ Ω; g(u) = 0}.
We shall assume that system (1) is non-degenerate in the sense that around a given ū ∈ γ, it holds q u2 (ū) is non singular.

(5)

We shall also assume that system (1) is entropy dissipative around the given equilibrium point ū ∈ γ and u ∈ Ω, in the sense that we have

(E ′ (u) -E ′ (ū)) • g(u) ≤ 0. (6)
Following [START_REF] Godunov | An interesting class of quasi-linear systems[END_REF][START_REF] Friedrichs | Systems of conservation equations with a convex extension[END_REF], it is now possible to symmetrize our system by introducing a new variable, the entropy variable, which is just given by W = E ′ (u). This change of variable is very useful in the statement of the global existence results. Actually, since E is a strictly convex function, we can inverse E ′ to recover the original variable u by the inverse map Φ .

= (E ′ ) -1 . Let us set now

A 0 (W ) . = Φ ′ (W ), C α (W ) . = Df α (Φ(W ))A 0 (W ), and
G(W ) = g(Φ(W )) = 0 Q(W )
. It is easy to see that the matrix A 0 (W ) is symmetric positive definite and, for every α = 1. . . . , m, C α (W ) is symmetric. Then, selecting W as the new variable, our system reads

A 0 (W )W t + m α=1 C α (W )W xα = G(W ). (7)
In the following, without loss of generality, we can always suppose ū = 0 ∈ γ and consider system (1) with g(0) = 0, f α (0) = 0, and endowed with a quadratic entropy function E. Next, we focus our investigation on a slightly restricted class of entropy dissipative non-degenerate systems, namely the systems such that

Q(W ) = D(W )W 2 , with D(0) negative definite. (8)
To continue our analysis of smooth solutions for dissipative hyperbolic systems, we need some supplementary coupling conditions to avoid shock formation. A very natural condition was first introduced by Shizuta and Kawashima in [START_REF] Shizuta | Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation[END_REF], for hyperbolic-parabolic systems. Let us introduce this condition for the original unknown, i.e. for system (1).

Definition 1. The system (1) verifies condition (SK), if every eigenvector of m α=1 Df α (0)ξ α is not in the null space of Dg(0), for every ξ ∈ R m \ {0}. About the existence of a solution, we recall the following result [START_REF] Hanouzet | Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF][START_REF] Yong | Entropy and global existence for hyperbolic balance laws[END_REF].

Theorem 1. Assume that system (1) is strictly entropy dissipative and condition (SK) is satisfied. Then there exists δ > 0 such that, if u 0 s ≤ δ, with s ≥ [m/2] + 2, there is a unique global solution u of (1)-( 2), which verifies

u ∈ C 0 ([0, ∞); H s (R m )) ∩ C 1 ([0, ∞); H s-1 (R m )),
and such that, in terms of the entropy variable

W = (W 1 , W 2 ), sup 0≤t<+∞ W (t) 2 s + +∞ 0 ∇W 1 (τ ) 2 s-1 + W 2 (τ ) 2 s dτ ≤ C(δ) W 0 2 s , (9)
where C(δ) is a positive constant.

Let us consider now the linearized version of system (1), namely, setting

A α = Df α (0) and B = Dg(0), u t + m α=1 A α u xα = Bu, u ∈ R n , x ∈ R m , t ∈ R + , (10)
with B of the form

B = 0 0 D 1 D 2 , D 1 ∈ R n1×n2 , D 2 ∈ R n2×n2 , (11)
with n = n 1 + n 2 . Set A(ξ) = m α=1 A α ξ α .
According to the previous discussion, we can assume that (H1) there is a symmetric positive definite matrix A 0 such that A α A 0 is symmetric, for every α = 1, . . . , m, and

BA 0 = 0 0 0 D , where D ∈ R n2×n2 is negative definite; (H2) any eigenvector of A(ξ) is not in the null space of B, for every ξ ∈ R m \{0}.
To give a precise picture of the asymptotic behavior of the solutions for large times, we have to transform the original system in a suitable form, using a special projection of the original variables.

Definition 2. The partially dissipative system (10) is in Conservative-Dissipative form (C-D form) if the matrices A α are symmetric and there exists a negative definite matrix D ∈ R n2×n2 , such that

B = 0 0 0 D . ( 12
)
In the following w 1 . = w c is called the conservative variable, while w 2 . = w d is the dissipative one.

We shall prove in the following that there exists a linear change of variable such that (10) takes the C-D form under assumptions (H1). We shall construct the C-D variables using the projection Q 0 on the null space and the complementing projection Q -= I -Q 0 . We can compute Q 0 by using the classical representation formula, which yields

Q 0 = A 1/2 0 (A 0,11 ) -1 0 0 0 A 1/2 0 .
Note that, due to the assumptions on A 0 , this projector is symmetric. In particular we can choose left and right projectors

L 0 ∈ R n×n1 , R 0 ∈ R n1×n , so that Q 0 = R 0 L 0 , L 0 R 0 = I ∈ R n1×n1 , L 0 = R T 0 ,
and they are uniquely given by

R 0 = A 1/2 0 (A 0,11 ) -1/2 0 , L 0 = (A 0,11 ) -1/2 0 A 1/2 0 . (13)
We define the complementary projection

Q -such that Q - . = I -Q 0 = R -L -, L -R -= I ∈ R n2×n2 , L -= R T -.
The matrices R -, L -are uniquely given by

R -= A -1/2 0 0 ((A -1 0 ) 22 ) -1/2 , L -= 0 ((A -1 0 ) 22 ) -1/2 A -1/2 0 . (14)
If we set w 1 = L 0 A -1/2 0 u and w 2 = L -A -1/2 0 u, we obtain the following result.

Proposition 1. If u is a solution to system (10), then, under assumption (H1),

w = M u =   (A 0,11 ) -1/2 0 ((A -1 0 ) 22 ) -1/2 (A -1 0 ) 21 ((A -1 0 ) 22 ) 1/2   u (15)
is a solution of a system in C-D form.

In the following, we are always going to assume that the unknown u is chosen in such a way that (10) is in conservative-dissipative form. In this case, we say that also system (1) is in conservative-dissipative form and we shall set

u = (u c , u d ) ∈ R n1 × R n2 .
Example 1. The p-system with relaxation. Let us consider system

   ∂ t u + ∂ x v = 0 , ∂ t v + ∂ x σ(u) = h(u) -v, (16)
with σ ′ (u) > 0. One symmetrizer for the linearized version of ( 16) is given by

A 0 =   1 a a λ 2
  , with λ = σ ′ (0) and a = h ′ (0). The matrix A 0 is positive definite if it holds the subcharacteristic condition λ > |a|. It is easy to verify that assumption (H1) is verified and the proper C-D form of the system is found for the new unknowns

u c u d = M u v =   1 0 -a(λ 2 -a 2 ) -1 2 (λ 2 -a 2 ) -1 2   u v , see formula (2.29) in [BRS06].
3 The Green kernel for linear dissipative systems

The one dimensional case

We want to characterize the Green kernel Γ (t) for a linear dissipative hyperbolic system. The fact that we are in dimension one will help us in inverting the Fourier transforms, hence giving explicit form to the principal parts of Γ (t). We can consider directly a system in C-D form, according to the results of Section 2. So we write our system as

w t + Aw x = Bw, (17)
where w = (w c , w d ) ∈ R n1 × R n2 . We assume that the matrix A is symmetric and the matrix B verifies (12). We assume also that (17) verifies condition (SK). We want to present a precise description of the Green kernel Γ (t, x) of (17), which is the solution to problem

   Γ t + AΓ x = BΓ Γ (0, x) = δ(x)I, . (18)
by extending the approach used in [START_REF] Zeng | Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation[END_REF].

Consider the entire function E(z) = B -zA. The Fourier transform of the solution to (18) is given by Γ (t, ξ) = e E(iξ)t . Notice that also Γ (t, z) = e E(z)t is an entire function of z. In general, if z is not an exceptional point, the function E(z), is represented by

E(z) = j λ j (z)P j (z) + j D j (z), (19)
where λ j are the eigenvalues of E(z), P j (z) the corresponding eigenprojections and D j are the nilpotent matrices, due to the fact that in general E is not diagonalizable, since we are not assuming that the matrix B is symmetric. We now study which consequences have the assumptions (H1), (H2) on E(z) and Γ (t, z) near the point z = 0 and z = ∞. Both points are in general exceptional. For z → 0, n 1 eigenvalues different from 0 converges to 0. When |z| → ∞, the matrix A is diagonalizable, but it can have common eigenvalues: then the perturbation B/z will in general remove part of this degeneracy. First we consider the case near z = 0. It has been shown in [START_REF] Bianchini | Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF], by standard representation formula, that we can expand the projector in the following way:

P (z) = I + O(z 2 ) zA 12 D -1 + O(z 2 ) zD -1 A 21 + O(z 2 ) z 2 D -1 A 21 A 12 D -1 + O(z 3 ) . ( 20
)
We introduce the right and left eigenprojectors of

P (z), R(z) ∈ R n×n1 , L(z) ∈ R n1×n , which verify P (z) = R(z)L(z), L(z)R(z) = I.
We can find the power series of L(z) and R(z

): L(z) = L 0 + zL 1 + z 2 L 1 + O(z 3 ), and R(z) = L 0 + zR 1 + z 2 R 1 + O(z 3
). We compute L 0 and R 0 by

L 0 = R T 0 = I n1 0 . (21)
The following terms can be also computed explicitely. So, we can decompose E(z) according to the right and left operators:

E(z) = R(z)F (z)L(z) + E 1 (z), (22)
where F (z) . = L(z)E(z)R(z) ∈ R n1×n1 and E 1 is a rest with a faster decay. We have

F (z) = L 0 + zL 1 + O(z 2 ) (B -zA) R 0 + zR 1 + O(z 2 ) = -zA 11 -z 2 A 12 D -1 A 21 + O(z 3 ). ( 23
)
The matrix A 11 is symmetric, from assumption (H1), so that we can write for some eigenvalues λ 1 j , with multiplicity m ′ j , and left and right eigenprojections

l j ∈ R m ′ j ×n1 , r j ∈ R n1×m ′ j , with l j = r T j , A 11 = m ′ j=1 λ 1 j r j l j . (24)
By a further decomposition, we obtain, for some projectors P jk and D jk , the projection of E(z) on the null eigenvalue as

R(z)F (z)L(z) = jk -zλ 1 j I -z 2 c jk I + O(z 3 ) P jk (z) + D jk (z). ( 25
)
Proposition 2. We have the following decomposition near z = 0

E(z) = jk Λ jk (z)P jk (z) + D jk (z) + E 1 (z), ( 26
)
where the Λ jk are diagonal n × n matrices composed by the n 1 eigenvalues λ jk , the coefficients c jk having strictly negative real part, thanks to assumption (H2).

We can perform a similar, but simpler, analysis for |z| → ∞, since we can write

E(z) = B -zA = z -A + 1 z B = z Ẽ(1/z),
where Ẽ(ζ) = -A + ζB.

Proposition 3. We have the following decomposition near z = ∞

E(z) = jk Υ jk (z)P jk (z) + D jk (z) , ( 27
)
where Υ jk is the diagonal matrix whose entries are the eigenvalues of the jk family, the coefficients b jk having strictly negative real part, thanks to assumption (H2).

Now, we are ready to estimate the global behavior for large t of the Green kernel Γ (t, x) using the local expansions contained in Propositions 2 and 3. We associate a diffusive operator with Green function K(t, x) to the expansion (26), and a dissipative transport operator with Green function K(t, x) to ( 27), and we estimate the remainder term R(t, x) = Γ (t, x) -K(t, x) -K(t, x). In the following we shall consider the Green kernel as composed of 4 parts, acting on w c , w d :

Γ (t, x) = L 0 Γ (t, x)R 0 L 0 Γ (t, x)R - L -Γ (t, x)R 0 L -Γ (t, x)R - = Γ 00 (t, x) Γ 0-(t, x) Γ -0 (t, x) Γ --(t, x) . ( 28
)
We can associate to each term of P jk in (25), the solution g jk to the parabolic equation

w t + λ 1 j w x = -c jk I + d jk w xx , w ∈ R m ′ j , ( 29
)
where c jk = -µ jk -iν jk , with µ jk > 0. The solution g jk can be computed explicitly, and we have in any case that for some c > 0

|g jk (t, x)| ≤ O(1) √ t e -(x-λ 1 j t) 2 /(ct) ∀k, (t, x) ∈ R + × R. ( 30
)
The function

K(t, x) . = X jk 2 6 6 4 rj(g jk (t, x)p jk )r T j -rj " dg jk dx p jk « r T j A12D -1 -D -1 A21rj " dg jk dx p jk « p jk r T j D -1 A21rj " d 2 g jk dx 2 p jk « r T j A12D -1 3 7 7 5 (31)
collects the principal parts of each component (28) of the Green kernel Γ (t, x).

Near z = ∞, we can associate to each projectors, the Fourier transform of the Green kernel of the transport equation

w t + λ j w x = (b jk I + djk )w, w ∈ R m jk . (32)
Since ℜ(b jk ) ≤ -c < 0, we can estimate its solution gjk by

|g jk (t, x)| ≤ Cδ(x -λ j t)e -ct ∀k, (t, x) ∈ R + × R. (33)
We associate to the kernels gjk , the hyperbolic Green function

K(t, x) = jk δ(x -λ j t)e b jk t e tD jk (∞) P jk (∞). ( 34
)
Theorem 2. Let Γ (t, x) be the Green function of system (17), under the assumptions (H1) and (H2). Let K(t, x) be the Green function of the diffusive operator given by (31) and K(t, x) the Green function of the dissipative transport operator given by (34). Then, we have the decomposition

Γ (t, x) = K(t, x)χ λt ≤ x ≤ λt, t ≥ 1 + K(t, x) + R(t, x)χ λt ≤ x ≤ λt , ( 35
)
where χ is the characteristic function, and λ := max j λ j , λ := min j λ j . The matrix R(t, x) can be written as

R(t, x) = j e -(x-λ 1 j t) 2 /Ct 1 + t O(1) O(1)(1 + t) -1/2 O(1)(1 + t) -1/2 O(1)(1 + t) -1 . (36)
where R(t, x) can for some constant c. Here λ 1 j are the eigenvalues of the symmetric block A 11 of the matrix A.

The multi dimensional Green function

We want to state an analogous result for multi dimensional systems. We consider the Cauchy problem for the linear relaxation system in the Conservative-Dissipative form

w t + m α=1 A α w xα = Bw, w ∈ R n1+n2 , (37)
w(0, •) = w 0 . (38)
We assume that A α , α = 1, . . . , m, are symmetric matrices and we have assumptions (H1) and (H2). Set, for ξ ∈ R m ,

A(ξ) := m α=1 ξ α A α , E(iξ) = B -iA(ξ). (39)
Let us introduce the polar coordinates in R m , to say ξ = ρζ, with ρ = |ξ| and ζ ∈ S m-1 , and set E(iρ, ζ) = E(iρζ). As before, we want to study the Green kernel Γ (t, x) of (37). The solution of the Cauchy problem (37)-( 38) is given by w(t, •) = Γ (t, •) * w 0 . Let us study the expansion of E(z, ζ) = B -zA(ζ) near z = 0. We can use the result of Section 3, noting that the matrix A is simply replaced by A(ζ). We introduce the total projector P (z, ζ) corresponding to all the eigenvalues near 0, and P -(z, ζ) = I -P (z, ζ) is the projector corresponding to the whole family of the eigenvalues with strictly negative real part. As in (23), we have

F (z, ζ) . = L(z, ζ)E(z, ζ)R(z, ζ) = -zA 11 (ζ) -z 2 A 12 (ζ)D -1 A 21 (ζ) + O(z 3 ) (40) F -(z) . = L -(z, ζ)E(z, ζ)R -(z, ζ) = D -zA 22 (ζ) + O(z 2 ). ( 41
)
So we make a decomposition of the Green operator:

Γ (t) = K(t) + K(t), (42)
where, for a given constant a small enough,

K(t) . = χ(|ξ| ≤ a)R(iξ)e F (iξ)t L(iξ), K(t) . = χ(|ξ| ≤ a)R -(iξ)e F-(iξ)t L -(iξ) + χ(|ξ| > a) e E(iξ)t .
Theorem 3. Consider the linear PDE in the Conservative-Dissipative form

w t + m α=1 A α w xα = Bw, (43)
where A α , B satisfy the assumption (SK) of Definition 1, and let

Q 0 = R 0 L 0 , Q -= I -Q 0 = R -L -
be the eigenprojectors on the null space and the negative definite part of B. Then, for any function w 0 ∈ L 1 ∩ L 2 (R m , R n ), the solution w(t) = Γ (t)w 0 of (37), (38) can be decomposed as

w(t) = Γ (t)w 0 = K(t)w 0 + K(t)w 0 , ( 44
)
where the following estimates hold: for any multi index β and for every p ∈ [2, +∞],

L 0 D β K(t)w 0 L p ≤ C(|β|) min 1, t -m 2 (1-1 p )-|β|/2 L 0 w 0 L 1 + C(|β|) min 1, t -m 2 (1-1 p )-1/2-|β|/2 L -w 0 L 1 , (45) L -D β K(t)w 0 L p ≤ C(|β|) min 1, t -m 2 (1-1 p )-1/2-|β|/2 L 0 w 0 L 1 + C(|β|) min 1, t -m 2 (1-1 p )-1-|β|/2 L -w 0 L 1 , (46)
D β K(t)w 0 L 2 ≤ Ce -ct D β w 0 L 2 . ( 47
)
Example 2. Consider for example the linearized isentropic Euler equations with damping   

ρ t + divv = 0 v t + ∇ρ = -v (48)
To fix the ideas, take m = 3, n = 4 = n 1 + n 2 = 1 + 3. Clearly the system is already in the Conservative-Dissipative form and condition (SK) is satisfied. We decompose K(t, x) as

K(t, x) = G(t, x) (∇G(t, x)) T ∇G(t, x) ∇ 2 G(t, x) + R 1 (t, x), (49)
where G(t, x) is the heat kernel for u t = ∆u, and R 1 (t, x) satisfies the bound

R 1 (t, x) = e -c|x| 2 /t (1 + t) 2 O(1) O(1)(1 + t) -1/2 O(1)(1 + t) -1/2 O(1)(1 + t) -1 . ( 50
)
4 Decay estimates in the nonlinear case and more accurate asymptotic behavior

In this section we study the time decay properties of the global smooth solutions to a nonlinear entropy strictly dissipative relaxation system in conservative-dissipative form.

Nonlinear decay estimates

We now prove the decay estimates in L p (R m ; R n ), p ∈ [2, +∞], for the solution u with initial data in L 1 ∩ H s , with s sufficiently large, for the non linear equation

u t + m α=1 (f α (u)) xα = g(u), (51)
with f α (0) = g(0) = 0 and initial condition u(x, 0) = u 0 (x).

We shall assume that the system (51) is strictly entropy dissipative and condition

u d,t + m α=1 A α,21 (0)u c,xα + m α=1 A α,22 (0)u d,xα = D u d q(0)u d +L - m α=1 (A α (0)u -f α (u)) xα + (q(u) -D u d q(0)u d ) . (58)
Now, consider the linear parabolic equation

w t + m α=1 A α,11 (0)w xα + m β=1 m α=1 A α,12 (0)(D u d q(0)) -1 A β,21 (0)w xαx β = 0, (59)
and denote by u p (t) the solution of the weakly parabolic equation (59) with

u p (0) = L 0 u(0). (60)
Compute u d using (58) and decay estimates of Theorems 4 and 5. This yields the following sharper decay result.

Theorem 7. Let u p be the solution of problem (59), (60), under the assumptions of Theorem 4, for m ≥ 2 and p ∈ [2, ∞], we have the following decay estimate

D β (u c (t) -u p (t)) L p ≤ C min 1, t -m 2 (1-1 p )-|β|/2-1/2 E |β|+[m/2]+1 , (61
We can linearize the system around the constant state (ρ, v) = (1, 0), so obtaining system (48) of Example 2. In that case we can immediately apply Theorems 4, 5, 6, and 7. In particular, by eliminating v in (48), we obtain the estimate These estimates improve on previous results in [START_REF] Sideris | Decay and singularities of solutions of the three-dimensional Euler equations with damping[END_REF] and [START_REF] Coulombel | The strong relaxation limit of the multidimensional isothermal Euler equations[END_REF].

The Chapman-Enskog expansion in the case m = 1. For m = 1, the decay of u 2 convoluted with the linear kernel and integrated in time gives the same decay estimate of u, so we have to keep a nonlinear term in the diffusive expansion. Set à = 1 2 L 0 D 2 uc f (0) -A 12 (0)(D u d q(0)) -1 D 2 uc q(0) and B = A 12 (0)(D u d q(0)) -1 A 21 (0). We rewrite (57) as u c,t + A 11 (0)u c + Ã(u c , u c )

x + Bu c,xx = S x , (63)
where S is some given faster decaying term. We replace (59) by the nonlinear diffusive equation w t + A 11 (0)w + Ã(w, w)

x

+ Bw xx = 0. ( 64
)
Theorem 8. For m = 1, let u p be the solution of problem (64), (60), under the assumptions of Theorem 4. For p ∈ [1, ∞], and a fixed µ ∈ [0, 1/2), if E 1 sufficiently small with respect to (1/2 -µ), then we have

D β (u c (t) -u p (t)) L p ≤ C min 1, t -1 2 (1-1 p )-µ-β/2 F β+4 , (65)
where C = C(µ, F β+σ) , for σ large enough, where F 1 = E 1 and, if β ≥ 1,

F β+1 =    E β+1 , if p ∈ [2, ∞], E β+1 + D β u(0) L 1 , otherwise.
Example 4. The p-system with relaxation. We can apply Theorem 8 to the Example 1. In this case, as already shown in [START_REF] Chern | Long-time effect of relaxation for hyperbolic conservation laws[END_REF], the Chapman-Enskog expansion is given by the semilinear parabolic equation u p,t + h ′ (0)u p,x + 1 2 h ′′ (0)(u 2 p ) x -(λ 2 -a 2 )u p,xx = 0. (66)

  )with C = C(E |β|+σ ), for σ large enough. Example 3. Consider the isentropic dissipative Euler equations    ρ t + div(ρv) = 0, (ρv) t + div(ρv ⊗ v) + 1 γ ∇ρ γ = -v.

D

  β (ρ(t)-ρ w (t)) L p + D β (ρ(t)-ρ p (t)) L p ≤ C min 1, t -m 2 (1-1 p )-|β|/2-1/2 ,where ρ w and ρ p are respectively the solutions of the m-dimensional dissipative wave equation equation and the m-dimensional heat equation ρ w,t + ρ w,tt -∆ρ w = 0, ρ p,t -∆ρ p = 0.

  Theorem 4. Let u(t) be a smooth global solution to problem (51), (52). LetE s = max u(0) L 1 , u(0) H s ,and assume E [m/2]+2 small enough. Let p ∈ [2, +∞]. The following decay estimate holds

SK) is satisfied. Under the assumptions of Theorem 1, we consider the global solution u of (51)-(52).

with C = C(E |β|+σ ), for σ large enough. For m = 1, this estimate holds for p ∈ [1, +∞].

We now study the faster decay of the dissipative variables u d (t).

Theorem 5. Under the assumptions of Theorem 4, we have the following decay estimates for the dissipative part of u:

Decay to the linearized profile

We consider here the difference among the solution of the nonlinear equation ( 51) and the linearized one

We can show that, if the dimension m ≥ 2, then the solution to the nonlinear equation converges to the linearized solution. Using the linear estimates, and with a special argument for the case m = 2, we have the following result. The following result does not hold for m = 1.

Theorem 6. Let u l be the solution of problem (55), ( 52), under the assumptions of Theorem 4, for m ≥ 2 and p ∈ [2, ∞], we have the following decay estimate

with C = C(E |β|+σ ), for σ large enough.

The Chapman-Enskog expansion

Next we show how the solutions to the parabolic Chapman-Enskog expansion approximate the conservative part of the solutions to the nonlinear hyperbolic problem (1). We use the conservative-dissipative decomposition of u to yields (57)


domain: math info

text:
Mapping spatial processes in environmental management

Pairwise Markov random field (2)

• Multiple interacting variables

• Independence given neighborhood ⇒ Pairwise Markov random field

• Interaction graph G = (V, E)
• ψ i : "weights" on states of vertex i

• ψ ij : correlations "strength" between neighbor vertices

• Z: normalizing constant / partition function

P (x) = 1 Z i∈V ψ i (x i ) (i,j)∈E ψ ij (x i , x j )
CompSust'09 -Cornell University -june 2009

Hidden Markov random field (1)

Hidden variables

Observations

• a ∈ {0, 1} |V | : subset of V selected for sampling

• Independent observations:

P (o|x, a) = i∈V P i (o i |x i , a i )
Question: How to reconstruct hidden map x using sampling actions?

1. Hidden map model • Independent observations:

P (o|x, a) = i∈V P i (o i |x i , a i )
Updated Markov random field (Bayes' theorem)

P (x|o, a) = 1 Z i∈V ψ ′ i (x i , o i , a i ) (i,j)∈E ψ ij (x i , x j )
where

ψ ′ i (x i , o i , a i ) = ψ i (x i )P i (o i |x i , a i )

Hidden map reconstruction (1) 00 11

Hidden variables

Observations

Reconstruction

Local (MPM):

x * i = arg max x i P i (x i |o,

Approximate spatial sampling (1)

Approximate the computation of

a * = arg max a -c(a) + o P (o|a)V M P M (o, a)
• Explore cells where initial knowledge is the most uncertain:

marginal P i (x i |o, a) closest to 1 2 ã = arg max a -c(a) + f   i,a i =1 min P i (X i = 1), P i (X i = 0)  
• Marginals computation is itself NP-hard ⇒ approximation using belief propagation (sum prod) algorithm

Adaptive spatial sampling (1) Adaptive spatial sampling (2)

• a sampling strategy δ is a tree • Distribution on maps = Potts model

• a trajectory in δ: τ = (a 1 , o 1 , . . . , a K , o K ) Value of a leaf U (τ ) = - K k=1 c(a k ) + V M P M (o 0 , o 1 , . . . , o K , a 0 ,
P e (x | α, β) = 1 Z exp i∈V α e i eq(x i , 1) + β (i,j)∈E eq(x i , x j )
• Distribution of observation given map,

P a i (o i | x i , θ) o i \ x i 0 1 0 1 1 -θ a i 1 0 θ a i with θ 0 < θ 1 -p. 34

•

  Building pest occurrence map in order to eradicate • Observations costly • Errors in mapping also costly Mapping spatial processes in environmental management Different problems depending on observations nature • Data visualization -Complete observations (everywhere) -Perfect observations (No errors/missing data) ⇒ How to visualize data? • Map reconstruction -Complete observations -Noisy observations ⇒ How to reconstruct the "true" map? • Sampling and map construction -Incomplete observations (not everywhere) -Noisy observations ⇒ Where to observe? / How to reconstruct? Random Field model is used for: • Representing current uncertain knowledge about map to reconstruct • Updating knowledge after observations • Defining a unique criterion for -map reconstruction from observed data -spatial sampling actions selection

  {0, 1} |V | : subset of V selected for sampling

  a), ∀i ∈ V Question: How to reconstruct hidden map x using sampling actions? 1. Hidden map model 2. Updated model after sampling result 3. Hidden map reconstruction 4. Sampling action optimizationHidden map reconstruction (2) ):x * i = arg max x i P i (x i |o, a)Value of reconstructed map Expected number of well classified sites in x * V M P M (o, a) = f i∈V max x i P i (x i |o, a) CompSust'09 -Cornell University -june 2009 Sampling action optimization (2) Hidden variables Observations • a ⊆ V selected for sampling • Independent observations o result ⇒ How to optimize the choice of a? U (a) = -c(a) + o P (o|a)V (o, a) a * = arg max a U (a) • The computation of a * is hard! (NP-hard) • Only feasible for small problems or needs approximation! CompSust'09 -Cornell University -june 2009

•

  a 1 , . . . , a K ) Value of a strategy V (δ) = τ U (τ )P (τ | δ) Heuristic adaptive spatial sampling • Exact computation is PSPACE-hard ! ⇒ Heuristic algorithm -on line computation -approximate method for static sampling at each step -p. 20 CompSust'09 -Cornell University -june 2009 Concluding remarks • A framework for spatial sampling optimization: -based on Hidden Markov random fields -different map quality criteria -extended to "adaptive" sampling • Problems too complex for exact resolution ⇒ Heuristic solution based on approximate marginals computation • Empirical validation on simulated problems: -Comparison of SSS, ASS and classical sampling methods (random sampling, ACS) -Markov random fields parameters learned from real data -ASS > SSS > classical methods Questions? o, a) = arg max x P (x|o, a) V (o, a) = f (P (x * |o, a)) Question: How to reconstruct hidden variable X using sampling actions? • x * (o, a) is the best reconstruction given sampling result (o, a) • V (o, a) is the value of reconstructed variable after sampling result (o, a) eradication (at previous year): e i ∈ {0, 1}, i = 1, . . . n • search actions: passive search or active search, a i ∈ {0, 1}, i = 1, . . . n • observations: no nest detected / at least one nest detected, o i ∈ {0, 1}, i = 1, . . . n CompSust'09 -Cornell University -june 2009 HMRF model for fire ants problem (2)

•

  CompSust'09 -Cornell University -june 2009 HMRF model for fire ants problem (3) An initial arbitrary sampling (a 0 , o 0 ) is used for: • Parameters estimation: λ = (α, β, θ) approximate version of EM for HMRF (Simul field EM) -identification problem between α and θ -OK if θ known: use of expert values • Marginals computation: P i (x i |o 0 i Procedure: repeat 10 times -simulate hidden map x from P (x | α, β) (50 × 50 cells) -apply regular sampling (about 10% of area): a 0 -simulate o 0 from P a i (o i | x i , θ) (regular sampling plus passive search

CompSust'09 -Cornell University -june 2009

Spatial sampling optimization

The hidden variable x is a map

Approximate spatial sampling (2)

The approximation results from simplifying assumptions:

• Sampling actions are reliable

• No passive observations

• Joint probability approximated by one with idependent factors

Ongoing work

• Exact algorithms for small problems (Usman Farrokh): combining variable elimination and tree search

• "Random sets + kriging" approach (Mathieu Bonneau): development of a dedicated approximate method and comparison to the HMRF approach

• PhD thesis on adaptive spatial sampling for weeds mapping at the scale of an agricultural area (Sabrina Gaba, INRA-Dijon).

• Future? ⇒ Spatial partially observed Markov decision processes

Optimal sampling problem

Hidden variable model Where do we sample?

• No sampling in large empty areas

• Sampling preferably near detected occupied sites within low density areas

• If sampling ressources increase -SHS complete exploration until the whole area is covered -AHA can visit several times a site before extending exploration to another area


Patricia Gil

Renata Servan De Almeida

Saliha Hammoumi

Sophie Molia

Véronique Chevalier

H A Traore

K Samaké

Emmanuel Albina

Renata Servan

H A Traoré

K Samake

Isolation and molecular characterization of virulent Newcastle disease viruses in Mali in 2007 and 2008

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION MATERIAL AND METHODS

ewcastle disease virus (NDV) is the causal agent of a fatal respiratory and neurological disease that can result in 100% morbidity and mortality in chicken flocks. ND was in the former list A diseases of the World Organisation for Animal Health (OIE) and is still one of the most serious animal diseases in developing countries.

Although ND is endemic in Africa, little is known about the NDV strains circulating in this continent. The objective of this study was to detect, isolate and characterise the NDV strains circulating in Mali.

In this study, cloacal and tracheal swabs were collected on healthy chickens in two different regions in Mali in a framework of a surveillance programme implemented in 2007 and 2008. These samples were analysed by RT-PCR. Positive samples were inoculated into eggs and isolated viruses were sequenced. Nucleotide sequencing was achieved on 1659 pb of the F gene and on the full sequence of the HN gene. Phylogenetic analyses were carried out using the neighbor-joining method. Two motifs were identified for the cleavage sites were identified with at least three basic amino acids (112-RRRKR*FV-118 and 112-RRQKR*FI-118). The presence of V 118 associated with the cleavage site RRRKR*FV in two of these isolates was only reported recently, in the neighbour country Burkina Faso (1).

RESULTS AND DISCUSSION

Phylogenetic analysis based on a fragment of 356 nucleotides corresponding to position 48 to 422 of the F gene showed that these isolates are branched with the genotype VII (fig 2). This genotype represents the currently circulating genotype in Europe, Asia and Africa (2-3). Phylogenetic analysis performed on the full sequence of the F and HN genes confirmed that the Malian NDV isolates belong to the genotype VII. Two of the isolates clustered in the two subgenotypes (VIIg and VIIh) recently described by Snoeck et al in Burkina Faso, Niger and Nigeria (1). This suggest a circulation of the strains through animal trade between these countries. The two other isolates may define a new subgenotype (VIIi, fig 2).

These results show that the two survey regions were good sites for NDV ecology since they are major trade crossroads for domestic birds of different neighboring countries.

DV prevalence was 1.7% for domestic birds and 2.3% for wild birds. Four viruses were isolated from domestic birds after two passages into embryonated eggs. Unfortunately no isolates from wild birds could be obtained. The sequence of the cleavage site shows the presence of basic amino acids and thus characterise these isolates as virulent strains.

T his study reports on original NDV isolates detected in Mali during a surveillance campaign in domestic and wild birds over 2007 and 2008. Four strains, genetically characterise as virulent may define new subgenotypes. These virulent strains were surprisingly isolated from unvaccinated and apparently health poultry. Question about the real virulence of these strains warrants further experimental demonstration. N

  , F2AS (Wise et al, 2004-Pham et al 2005) -Sybrgreen QRT-PCR one-step Master mix kit (Stratagene) was done by a one-step RT-PCR targeting the F gene of NDV. All 1076 samples from domestic birds and all 1041 samples from wild birds were analysed and the positive ones were further inoculated into the allantoic cavities of 9 day-old to 11 day-old embryonated eggs. The complete procedure is represented in fig 1.

Fig2:Fig1:

  Fig2: Phylogenetic analysis of partial F sequences based on nucleotides 31-377. The tree was constructed using neighbor-joining method with 1000 bootstrap replicates. Only bootstrap values higher than 70 are shown. Malian NDV sequences are in red.