R Alan Walks

Aestheticization and the cultural contradictions of neoliberal (sub)urbanism

the political, economic and socio-spatial characteristics of neoliberal urbanism have 1 received much attention.1 The cultural facets of neoliberalism, and the landscapes that are germane to it, however, have until recently remained less well understood. This is beginning to change as urban scholars begin to interrogate the cultural politics of contemporary urban development(s) and processes, and their effects on urban form. The aestheticization of politics-that is, the emergence of a politics driven by aesthetic motivations, delineated by aesthetic concerns and/or masked by aesthetic appeals- would appear to be an important component of neoliberal times. While a politics of the aesthetic plainly pre-dates neoliberalism, it would appear that neoliberalization and aestheticization are intertwined, emerging as a by-product of, and as a strategy for, social exclusion and the management of class and other social identities in the context

of deepening cultural reification. Yet, while evolving partly as a response to the contradictions of the contemporary (neoliberal) city, aestheticization processes produce new contradictions and amplify existing ones. This is born out, either explicitly or implicitly, in three important recent works, Landscapes ofprivilege, Behind the gates and Brave new neighbourhoods. The first, by James and Nancy Duncan, sets out the main theoretical arguments and demonstrates their validity and importance through a detailed case study of the town of Bedford, an elite suburb of New York City in Westchester County, New York State. Duncan and Duncan show how aesthetics have been politicized in Bedford, how this has worked to simultaneously enhance, naturalize and conceal class privilege there, and how this has become a tool for social exclusion. In Behind the gates, Setha Low seeks to understand the motivations and desires underpinning the demand for gated suburbs in the United States. While not concerned primarily about the aesthetic dimensions of her subject or their political implications, she nonetheless provides important evidence of the aesthetic context surrounding the social construction of gated communities as 'nice places to live', and thus of the role that aesthetics play in the rapid growth and social evolution of this neighbourhood form across the urban landscape. Finally, Margaret Kohn in Brave new neighbourhoods outlines how public spaces are being privatized across the United States through various means, and presents a number of arguments for why this inhibits and offends democratic principles. In so doing, she makes the case for why a truly public space matters. These works demonstrate how neoliberal urbanism, at least the forms present in the US (though with some comparison to other contexts), enforces socio-spatial divisions through appeals to aesthetic choices, all the while containing and concealing important cultural contradictions with implications for the neoliberal city and even for the very basis of urbanity.

Exclusionary landscapes, landscape exclusions

In a theoretical and empirical tour de force, Duncan and Duncan draw on their 30-plus years of research in and around Bedford, New York, a wealthy suburb an hour's drive from Manhattan, to map out how a politics of the aesthetic (built around 'the sensuous, passionate, apparently autonomous subjective experience of individuals')2 has developed there into a force for exclusion. It is not just tenants, the poor and non-whites that face exclusion, though these groups are the worst hit, but all forms of development that might impinge on the identity of Bedford as a pastoral, rural village with a very specific landscape aesthetic. The authors demonstrate how this landscape identity has been socially constructed and reconstructed through the continuous efforts and interventions of elites, town officials, residents and planners. Much political activity has been devoted to preserving the characteristics that provide Bedford with its particular status, its sense of place and its cultural codes, and thus underlie Bedford's value as a 'positional good'.3

These include its pastoral landscape (marked by dirt roads, country cottages, horse stables, giant trees on rolling lawns, iron gates and stone fences), the construction/ fabrication of Bedford Village as a historic/'authentic' New England village, and a romantic vision of rugged wilderness and forest untouched by urban sensibilities. Duncan and Duncan base their conclusions on the series of face-to-face interviews and multiple surveys they conducted with residents, real estate agents and town officials, and on the documented history of municipal planning statements, legal challenges and council minutes.

While the social reconstruction of Bedford as an idyllic pastoral village had been developing for a number of years, and Bedford had been home to a number of elite families since the late 1800s, it was with the influx of wealthy celebrities and professionals from New York City and Bedford's evolution into a commuter suburb from the late 1960s onwards that a particularly aestheticized politics of exclusion became dominant. Enacted in 1928, Bedford's four-acre minimum lot zoning (applying to four-fifths of Bedford's land area) was intended from the start to preserve the 'look of the land'. This was maintained in recent decades despite threats of exclusionary zoning lawsuits and the economic pressures resulting from high housing demand across the New York City region. Since the early 1970s, a number of ordinances relating to wetlands protection, tree preservation, protection of open space, historic buildings, dirt roads and popular views, and the establishment of nature preserves and land trusts, removed the majority of potentially developable or divisible lands from the market. This has worked to reproduce Bedford's pastoral aesthetic, all the while ensuring the exclusion of affordable and even middle-class suburban-style housing.

This particular planning practice, and the exclusions it produces, are routinely justified by local actors on aesthetic and environmental grounds, even in the face of evidence of deleterious environmental consequences. Duncan and Duncan draw on a number of theorists, among them David Harvey, Don Mitchell and Christopher Kutz,4 but mostly lean on Bourdieu5 in arguing that landscape tastes as cultivated in Bedford are a form of cultural capital that residents marshal for the protection of their identities, their residential investments, and for the performance of fine social distinctions and exclusions. The adoption and mobilization of such aesthetic tastes is not necessarily superficial or dishonest, employed to mask ulterior motives (though it sometimes is). More often, such landscapes and aesthetic dispositions are fundamental markers of identity (particularly among the elite), and as such provoke sincere emotions which inadvertently act to naturalize class privileges and tastes. The result is that socially exclusionary practices are not recognized as such. Indeed, Duncan and Duncan argue that the principle motive is not usually social exclusion, but landscape exclusion intended to preserve the aesthetic qualities of place.

The cultural contradictions of such an aestheticized politics are brought to the fore in the last empirical chapter, detailing the plight of the mostly Latino day labourers who are employed on Bedford's estates, hobby farms and the gardens of the 'country cottages', all of which require substantial maintenance. But the work is unskilled, casual and inconsistent, and the pay is low. The Latinos who find work there are forced to live in substandard and over-crowded conditions in the nearby suburb of Mount Kisco, since Bedford's aesthetic praxis has left virtually no affordable housing within its boundaries. Yet in Mount Kisco these workers faced another racialized form of aestheticized politics, this time concerning their appearance and deportment, which threatened nativist constructions of 'proper' middle-class white behaviour. In response, politicians at municipal and upper levels of government enacted legislation to prevent Latinos from congregating in parks or on the streets, and housing ordinances purported to reduce overcrowding and ensure fire safety. These laws had the effect of removing the day workers from view, often pushing them into other localities or forcing them into homelessness. Such acts have paradoxical consequences, for, as Duncan and Duncan note, the reproduction of white Anglo Bedford came to depend on Latino labour.

Imaginary/purified communities and the aesthetics of 'niceness' Perhaps no other neighbourhood form is more clearly associated with contemporary neoliberal times than gated communities. Particularly fast-growing in the US in the 1990s, gated communities are now being built across the developed and developing world in response to the class anxieties accompanying neoliberalization.6 Through a series of interviews in suburban Long Island, New York and San Antonio, Texas, Low gives voice to the reasons why people move into such places, why their residents like them and how they construct them as livable communities. While the traditional themes of fear, security and community feature prominently in her respondents' explanations, what stands out is how important the aesthetics of such places are for understanding how their residents relate to them. For example, a number of her respondents reported they valued the 'appearance' of security provided by the gates, and how the walled designs made them 'feel' less vulnerable, even in the face of evidence that the gates are not necessarily an effective deterrent against crime, and of their own realization after living in them of the ease with which would-be criminals could enter. Likewise, it was the aesthetic features of such neighbourhoods (including the lack of ethnic and class diversity) which made them feel 'like an old-fashioned neighbourhood', providing clues that such places were good places to live in and harking back to communities 'imagined from childhood'.7

However, it is Low's chapters on the construction of 'niceness' and fear of others that make the greatest contribution to our understanding of the cultural facets of this neighbourhood form. She argues (drawing on Brodkin8) that gated communities are culturally significant for reproducing 'whiteness'. This is accomplished through strict aesthetic controls both over the external environment, which keep it orderly and 'nice' according to the mores of white society, and over the social environment (which can thus be kept relatively more homogeneous and filled with 'nice' people), all of which act to naturalize the cultural preferences and codes of white privilege and assuage the anxieties of a white middle class beset by economic insecurity in a globalized world.9 Though she does not expand on it, Low at one point compares gated subdivisions to 'purified communities'.10 Contradictions arise, however, as these forms become accepted as 'normal' referents from which residents make sense of the world, allowing them to rationalize problems of race and class segregation, gender inequalities and paradoxical approaches toward immigration in US society. Importantly, she suggests that as gated communities become the norm, the dominant meaning of community itself may come to include notions of protective boundaries and constructions of otherness that preclude alternate definitions and forms of community (the public, racial integration, collective social justice, etc.).

Low's discussion of community boundaries is related to Kohn's concept of 'imaginary communities', which she defines as groups that are 'sustained by perceived similarities in lifestyle and absence of conflict.. . reinforced by similar patterns of consumption and cultural cues rather than shared activities and practices'."1 Yet Kohn is not talking about communities formed through the Internet or shared national histories, as in Anderson's 'imagined communities',12 but instead those formed through 'individuals identifying with a particular location'.13 In short, imaginary communities are those in which the aesthetic experience of living in a neighbourhood provides the basis of collective identity. Imaginary communities, argues Kohn, 'provide a way of avoiding the irreconcilable social antagonisms that pervade modern life'14 by making it easier to avoid encounters with strangers. The two examples Kohn provides are gated and new urbanist suburbs, where aesthetics can provide the necessary cultural cues to evoke a particular style of community without having to go through any face-to-face community-building work. When communities can be articulated through aesthetic means, it can be expected that community politics may increasingly revolve around the controlling and policing of aesthetic boundaries. The gated communities that Low examines are thus but one manifestation of this control.

Privatized landscapes, atomized culture

In Brave new neighbourhoods, Kohn tells us why all this matters. Focusing mostly on US federal and state court challenges in a series of case studies, Kohn documents how public spaces are being privatized through legislative action, and theorizes their implications for the future of democracy. Kohn's examples include not only the privately owned retail malls, which are legally allowed to exclude individuals from their premises, but also the various residential community associations (RCAs) and business improvement districts (BIDs) which are given unaccountable control over previously public spaces; public-private partnerships such as that which produced Battery Park City, which allow developers to produce private exclusionary spaces at public cost; municipalities who conspire to grant or sell public lands to private organizations whose self-interests are exclusionary (such as Salt Lake City's sale of a prime downtown street to the Mormon church); and the anti-panhandling, anti-leafleting and anti-protesting ordinances that work to exclude citizens from otherwise open public spaces such as sidewalks, public transit stops, benches, parks, etc. Yet Kohn's book ventures far beyond a mere discussion of empirical cases, to make an original statement of urban political theory that is historically informed (including an extensive discussion of the struggles of the 'wobblies'15 over public space during the Depression).

Regardless of whether it is accomplished through the granting of control over public and quasi-public spaces to private entities or through legislation constraining activities in publicly owned spaces, the main effect of such forms of privatization is to prevent those with an officially sanctioned claim to the use of the space from being inconvenienced by the presence or speech of strangers -in other words, to enhance the aesthetic experience of those groups deemed 'deserving' by removing the rights of the undeserving and unwashed. Typically, the ordinances and private actions cited by Kohn are aimed at preventing the mere sight and proximity of strange others, not physical violence, threat of injury or blocked access. (Don Mitchell has subsequently described similar legal developments in terms of an 'S.U.V. model of citizenship', characterized by the right to travel within a 'floating buffer zone' of protective legal space.16) In such a scenario, individuals in public can avoid being inconvenienced by the public, to avoid even having to view others or hear messages that might contradict their values and preferences.

Kohn argues that these forms of privatization are a threat to the health and survival of public life and democracy. Her argument is different from the traditional defence of public space emanating from theories of deliberative democracy, which paints the public sphere as the domain within which the public can together reach a rational and inclusive consensus and discover universal truths.17 Instead, Kohn argues that public space is necessary because it establishes proximity between strangers, and this has the potential to transgress our values: 'We need free speech and public places not because they help us reach consensus, but because they disrupt the consensus that we have already reached too easily.'18 Physical proximity forces us to deal with our fears, to defend our rationalizations and to question our beliefs. Kohn reminds us that in the presence of another 'we cannot simply press delete ... the effect of a disconcerting encounter cannot be approximated through email'.19 Without a truly public realm characterized by the meaningful ability of strange people and strange messages to transgress our values and make us question our deeply held assumptions about the social world, individuals may lose the ability to empathize, to recognize and accept difference, to bridge divisions, and in so doing to care about others. Without a diverse public culture kept in check by constant interaction between strangers, individuals become atomized, privilege becomes naturalized, poverty and injustice become hidden, and the prevailing social order remains unquestioned.

The contradictions of neoliberal (sub)urbanism? Contemporary urbanism (neoliberal and postmodern) has been paradoxically char- acterized by the simultaneous celebration of diversity on the one hand, and increasing isolation, boundaries, and separation between social groups on the other.20 This is partly a result of the penetration of ethnic diversity discourses by market logics in the context of globalized inter-urban competition.21 Yet, as the three works reviewed here make plain, another part of the story relates to aestheticization processes which work to naturalize landscape tastes and to reify neighbourhood forms and cultural differences. Indeed, neoliberalism and aestheticization would appear mutually reinforcing, since both foreground 'choice' as an organizing principle in the field of consumption, and since neoliberalism provides the conditions -both greater social differentiation/class polarization and the market colonization of cultural goods -for realizing economic benefits from aesthetic distinctions.

This situation produces a number of contradictions. Duncan and Duncan show that the externalization to other locales of the costs of socially reproducing the labour on which depends the aesthetic grooming of places like Bedford is then articulated in the growth of a more virulent racialized politics that threatens to undermine regional social and economic cohesion. These underlying dynamics are concealed and mystified by the aesthetic focus in each case. Likewise, the forms of gated communities serve to demarcate new boundaries and new exclusions of others (and thus to limit choices), but these are concealed by their 'niceness' and by the aesthetic scaffolding upon which these and other imaginary communities are (socially) constructed. As Kohn's work suggests, the aesthetic exclusion of the Other from one's (public) experiences that is portended by the privatization of public space may have profound effects on contemporary political culture. In short, by concealing the casualties and irrationalities of the contemporary social order from view, society becomes less able to recognize them as such, and in turn to take corrective action. To the extent that neoliberalism accentuates aestheticization processes (and this has yet to be proven - the three books under review here do not directly tackle this question), it may be creating the conditions of its own mystification, and thus blocking its ability to learn and adapt.

Finally, there is the contradictory relation of an aestheticized neoliberalism to the city. Most urban scholars, going back to Georg Simmel and Louis Wirth, have construed the essence of urbanity as a high density of potential chance encounters with strangers, an essence which provides cities with some of their most valuable qualities (from agglomeration economies to dense networks of social capital). Yet, by enhancing an aestheticized praxis of exclusion that reifies and naturalizes existing landscapes and that works to exclude strange others from view, neoliberalism may be producing an urban realm that hampers such qualities. To be suburban literally means to be 'less than urban'. The vast majority of the cases examined in the three works under review are located in the 'suburbs' (as defined by North American custom), and this is not likely to be coincidental. However, Kohn also documents cases (like Battery Park City) in which spaces in the inner city are produced that exclude chance encounters. Thus, whether from the development of exclusive subdivisions in the 'suburbs' or gentrified high-rises in the inner city, the question arises: does an aestheticized neoliberalism entail an attack on urbanity itself, and its replacement with something 'less than urban'? Under such a cultural definition of 'sub-urban', the condominium developments and gentrification that have grown in the inner city in response to neoliberal urban policies, if they result in social 'tectonics' rather than social interaction,22 would constitute the sub-urbaniza- tion of the city as much as does the growth of gated communities at the edge. Neoliberal urbanism may thus be beset by a fundamental contradiction: the reification and revaluation of aesthetic forms may alter social practices in ways that undercut the propinquity of heterogeneity and mixing of strangers that provides urbanity's primary economic, social and cultural asset.

Conclusion

The place of aesthetics in the practice of exclusion has been until now under-theorized. The books examined here, despite differences in focus, all speak to the importance, and contradictions, of an aestheticized social and political praxis in the urban realm. These would appear to be growing in tandem with neoliberalism's colonization and privatization of public space, and its tendency to segment markets while privileging 'choice' in market discourses. Each of the three works under review sheds light on some aspect of this, and reaffirms (for me at least) that aestheticization processes are key to understanding the spatial articulations and exclusions of neoliberal urbanism. Yet the relation of aesthetics to the evolving policy at international/global scales and in other national contexts remains less well understood (all three of these works are primarily concerned with the US case), and the links to neoliberalism need to be made more explicit. Low could, furthermore, develop her theoretical arguments more fully, and could better organize and edit her discussion (indeed, she could be accused of over-aestheticizing her subject, since she invariably foregrounds her respondents' weight, hair colour and dress!). Nonetheless, each of the works under review makes a significant contribution to the literature, and grounds their theoretical statements in detailed empirical research. The books by Duncan and Duncan and by Kohn are particularly important interventions. The former is a seminal study that challenges many of the core assumptions of urban/suburban scholars, while the latter should have reverberations across the realm of urban, social, cultural, political and legal theory.

Notes

A number of authors have discerned similarities in the neoliberal urban experience. These include the shift to what Harvey calls the 'entrepreneurial city' and the naturalization of market logics which punish cities that fail to compete intemationally; the rescaling of the state through the adoption of deregulation, privatization and downloading, while at the same time 'rolling out' new meso-level layers of market rules; and the marginalization of (or outright attack on) the social welfare state, as evidenced in union-busting, criminalization of the poor and cutbacks to social programmes, welfare entitlements and tax rates (and the adoption of regressive taxation), while at the same time co-opting ethnic diversity and creativity discourses in the employ of intra-urban competition. The result is a form of urbanism beset by multiple contradictions and characterized by increasing social inequality, the impoverishment of longer-term infrastructure budgets, and zero-sum-game policy races to the bottom, all of which effectively act to narrow future policy choices and reinforce emerging crises which must remain hidden if the competitive benefits of urban spectacle are to do their magic. Of course, the geography of 'actually existing neoliberalism(s)' is quite uneven, and the local issues, policy tools, political struggles, social movements, mix of sub-cultures, regime strategies and contexts of partisanship (on behalf of the political parties and civil society organizations) in shaping political discourses differ from place to place. See D. Harvey, 'From managerialism to entrepreneurialism: the transformation of urban govemance in late


M Hamide

M Bellet

Adaptive Anisotropic Mesh Technique For Coupled Problems: Application To Welding Simulation

Keywords: finite elements, heat transfer, phase transformation, mesh adaptation, anisotropic metric, error estimation, welding

a major problem arising in finite element analysis of coupled problems, such as welding for instance, is the control of the mesh, that is an appropriate mastering of the spatial discretization to get accurate results in a minimum computer time. The present anisotropic adaptation procedure is controlled by a directional error estimator based on local interpolation error and recovery of the second derivatives of different fields involved in the finite element calculation. Error indicators are derived to define an anisotropic mesh metric field, which is an input of the pre existing 3D remeshing procedure. The mesh metric consists of a combination of several metrics, each corresponding to the error estimation associated with a selected field of the solution produced (temperature, phase fraction, stress component). Mesh modifications are used to anisotropically and continuously adapt the mesh. We demonstrate the efficiency of the method by applying it to a coupled thermal-mechanicalmetallurgical simulation of arc welding. We demonstrate that the use of an anisotropic adaptive finite element method can result in an order of magnitude reduction in computing time with no loss of accuracy compared to analyses obtained with isotropic meshes.

INTRODUCTION

The accuracy of a numerical solution obtained by the finite element method depends on the spatial discretization of the physical domain. In many physical problems, among them welding, the solution exhibits anisotropic features creating a demand for elements which are aligned with the solution's anisotropy. In realistic cases, the information required to compute the desired solution field to an acceptable level of accuracy is unknown a priori. An efficient approach to overcome this difficulty consists in applying an adaptative procedure in which the errors arising from spatial discretization are controlled within a specified tolerance.

The concentrated heat input that appears in most welding applications requires a refined discretization in the neighbourhood of the molten region below the moving electrodes. This requires some kind of remeshing capability in order to continuously maintain or regenerate a finely discretized region moving with the heat source. The initial work on an automated remeshing strategy for welding applications was performed by Lindgren et al. [START_REF] Lindgren | Automatic remeshing for three-dimensional finite element simulation of welding[END_REF]. Their work included remeshing of a moving region but did not use any error estimation to guide the remeshing scheme and control the accuracy of the solution produced. They prescribed the refinement/coarsening in the input file so that a smaller distance to the source gave smaller elements. Recently, Runnemalm and Hyun [START_REF] Runnemalm | Three-dimensional welding analysis using an adaptive mesh scheme[END_REF] proposed an adaptative strategy that evaluates both the thermal and the mechanical error distribution using a Zienkiewicz-Zhu error estimator [START_REF] Zienkiewicz | A simple error estimator and adaptive procedure for practical engineering analysis[END_REF]. It is combined with a hierarchic remeshing strategy using a so-called graded element. In this approach, the directionality of the error estimation is cannot be taken into account, resulting in isotropic adaptative remeshing.

In the present paper, following the approach initiated by D'Azevedo [START_REF] Azevedo | On optimal triangular meshes for minimizing the gradient error[END_REF], Fortin [START_REF] Fortin | Estimation d'erreur a posteriori et adaptation de maillages[END_REF] and Alauzet and Frey [START_REF] Alauzet | Anisotropic mesh adaptation for rayleigh-taylor instabilities[END_REF], we place particular emphasis on the anisotropic mesh adaptation process generated by a directional error estimator based on the recovery of the second derivatives of the different fields involved in the finite element solution. As shown hereafter, this approach allows us to refine the mesh, stretch and orient the elements in such a way that, along the adaptation process, accurate controlled solutions are obtained while keeping the number of unknowns affordably low. The organization of the paper is as follows. First, we briefly introduce the numerical model that is used to solve and describe the welding process. Then, we present the overall anisotropic mesh adaptation procedure: the anisotropic error estimator, together with the procedures to get the recovered Hessian matrix are described. In this section we discuss the Hessian strategy and review the concept of a mesh metric field. Finally, we apply the different anisotropic adaptative strategies to welding simulations and discuss the results.

WELDING ANALYSIS

During welding, the interaction of the heat source and the material leads to rapid heating, melting and the formation of the weld pool. When the heat source moves away, the weld pool cools and solidifies forming the weld joint. Depending on the welded alloys, as the temperature decreases, various solid-state phase transformations may take place resulting in the final microstructure of the weldment. The properties of a weldment, such as strength, ductility, toughness, and corrosion resistance are significantly affected by its microstructure.

In the present work, a so-called staggered approach was used to couple the thermal, mechanical and microstructural analysis, at each time increment, as shown in Fig. 1. The thermal field is first calculated followed by the microstructural evolution and finally the mechanical variables are determined. A series of phase transformations take place in both the fusion zone (FZ) and the heat-affected zone (HAZ) during welding of low alloyed steels. The calculation of austenite formation is based on the theory presented by Leblond et al. [START_REF] Leblond | A new kinetic model for anisothermal metallurgical transformations in steels including effect of austenite grain size[END_REF]. The modelling of the transformations during cooling is based on TTT diagrams, extended to non isothermal transformations by using an additivity principle and the notion of fictitious time [START_REF] Denis | Mathematical model coupling phase transformations and temperature evolutions in steels[END_REF].

The energy conservation in the multiphase material is written using spatial averaging. It is solved for the nodal enthalpies, from which temperature and liquid fraction can be calculated. This formulation takes into account the energy changes associated with the solid state phase changes, while taking advantage of the stability and robustness of the enthalpy formulation for the liquid-solid phase change [START_REF] Hamide | Adaptive mesh technique for thermal-metallurgical numerical simulation of arc welding processes[END_REF].

The mechanical model uses temperature history data from the thermal model to perform a thermal stress analysis. Along with the thermal stresses, the micro-structure evolution during solid state phase transformation induces some additional stresses. The total strain rate tensor can be written as the sum of the individual components of the strain rate as [10][11] :

ε = εe + ε p + εth + εtr + εt p (1)
The various components in this equation represent strain rates due to elastic, plastic and thermal loading, volumetric change and transformation plasticity, respectively. Momentum and mass conservation equations are solved using a velocity-pressure formulation. In the liquid and mushy states, steels are modelled as Newtonian and non Newtonian fluids, respectively, using a temperature dependent viscoplastic model. However, the liquid flow in the weld pool has not been simulated so far, using an arbitrary high viscosity alone the solidus temperature. In the solid state a temperature dependent elasticviscoplastic model is used [START_REF] Bellet | An ale-fem approch to the thermomechanics of solidification processes with application to the prediction of pipe shrinkage[END_REF].

MESH ADAPTATION

In this section, we will briefly present the anisotropic error estimate of the interpolation error based on the second derivatives of the solutions presented since the details can be found in references [START_REF] Alauzet | Anisotropic mesh adaptation for rayleigh-taylor instabilities[END_REF][4] [START_REF] Garcia | Directional adaptive surface triangulation[END_REF] The a posteriori error estimator

It has been proved that for elliptic problems, the finite element error can be bounded by the interpolation error (Céa's lemma [START_REF] Ciarlet | Basic error estimates for elliptic problems[END_REF]) :

u -u h ≤ c u -Π h u (2)
where Π h u is the linear interpolate of u on the mesh and c is a constant independent of the current mesh.

Actually, similar analysis based on the interpolation error show (practically) that the link between the interpolation error and the approximation error is even stronger than the bound given by Céa's lemma [START_REF] Ciarlet | Basic error estimates for elliptic problems[END_REF]. Therefore, the interpolation error appears a reasonable way of defining an error estimate according to [START_REF] Fortin | Estimation d'erreur a posteriori et adaptation de maillages[END_REF]. Recall that for each element K, the anisotropic error interpolation bound involves the second derivatives of the variable u ( [START_REF] Alauzet | Anisotropic mesh adaptation for rayleigh-taylor instabilities[END_REF]):

e K ∞ ≤ cmax x∈K max e ( e • |H u (x)| e) (3)
where e denotes one of the 6 tetrahedron edges. c = 9/32, v is any vector joining two interior points in K and |H u | is the absolute value of the Hessian matrix of the solution (i.e., consisting of absolute eigenvalues). The Hessian strategy involves the computation of the symmetric matrix of second derivatives that can be decomposed as |H u | = R|Λ|R T , where R is the eigenvectors matrix and Λ = diag(λ k ) is the matrix of eigenvalues.

Metric definition

The stated goal of the mesh adaptation algorithm is to yield a mesh with regular elements in the metric space where each edge e must satisfy the following relation (see, for example, [START_REF] Gruau | 3d tetrahedal unstructured and anisotropic mesh generation with adaptation to natural and multidomain metric[END_REF]):

( e • M e) = 1 (4)
A mesh with all its edges satisfying the above relationship is commonly referred as a unit mesh.

To achieve a suitable mesh resolution in different directions, a uniform distribution of local errors is applied in the principal directions which leads to ch 2 k λk = ε, where ε is the user specified tolerance for the error and h k is the desired size in the k th principal direction. So, the edges e of the adapted mesh must check ε = c ( e • M e). A mesh metric tensor M is then obtained at each node by calculating a scaled eigenspace of the recovered Hessian matrix as M = R ΛR T , where R is the eigenvector matrix and Λ = c ε Λ is the diagonal matrix of scaled eigenvalues. When several variables fields u are considered concurrently, the previous approach leads to a metric for each variable and we chose to take the intersection of the different metrics. In practice the intersection of metrics is achieved by the simultaneous reduction of two quadratic forms which is a valid operation since the metric tensors are positive definite [START_REF] Alauzet | Anisotropic mesh adaptation for rayleigh-taylor instabilities[END_REF].

Hessian evaluation

To compute the Hessian matrix H u of the P1 field u, we reconstruct in two steps the second derivatives at each node P by using the computed solution from the patch S of all elements K surrounding node P. This is done as follows:

In a first step we recover the gradient at node P by taking the volume weighted average of gradients on elements in the patch S. Note that u being a P1 field, its gradient is constant on each element K:

Π h (∇u h )(P) = 1 ∑ K∈S |K| ∑ K∈S |K| (∇u h ) |K (5)
It can also be noticed that this is equivalent to a lumped-mass approximation of a least squares reconstruction of the gradient for linear elements. In a second step, the same procedure is now applied to the P1 field Π h (∇u h ) to obtain the recovered Hessian matrix.

APPLICATION TO WELDING SIMULATION

Welding conditions and material properties

We consider a thick plate of A508 steel, the dimensions of which are given by Fig. 2(a). The temperature dependent thermophysical properties are given in Fig. 3. The welding parameters chosen for this analysis are as follows. Welding process: gas tungsten-arc welding (GTAW); welding current I = 150A, welding voltage V = 10V and travelling speed of 1mm.s -1 . The weld efficiency is assumed to be χ = 0.65 . The associated heat input, I ×V × χ, moving with the electrode, is simulated by a simple surfacic uniform distribution within a disc of radius 5mm.

Finite element model

For the simulation study, only one-half of the plate is analyzed due to symmetry. The boundary conditions of the welding process incorporate heat transfer boundary conditions. The symmetry surface is defined as under adiabatic boundary conditions. On all other surfaces boundary conditions of convection and radiation with the environment are applied with a convective coefficient h = 12W m -2 K -1 an emissivity coefficient ε = 0.75 and an external temperature T ext = 25/C.

To evaluate the efficiency of our adaptative procedure we first obtain results on a fine mesh (Fig. 2(b)). The result obtained is then used for purpose of comparison. Several simulations have been performed with adaptative remeshing. The calculations differ by the prescribed tolerance error, all other conditions being identical. Three types of simulation with remeshing have been carried out:

• Thermal-driven mesh adaptation : the adaptative technique is based on the thermal error distribution • Thermometallurgical-driven mesh adaptation : the automatic mesh refinement is based on both thermal and metallurgical error distributions • Thermomechanical-driven mesh adaptation : the automatic mesh refinement is based on temperature, longitudinal and transversal stresses.

Thermal-based mesh adaptation

The reference FE-model without remeshing consists of 14329 nodes and 68891 linear tetrahedral elements and is presented in Fig. 2. The initial FE-model used in calculations with remeshing consists of 6842 tetrahedral elements (1683 nodes). The remeshing is performed at each time step (dt = 1s). See Figs. 2(b) and 4(a) for the FE-mesh. As expected, the adaptative remeshing generates more refined elements in the neighbourhood of the thermal source and coarser elements in its trail. It can be seen also in Fig. 4 that anisotropic elements aligned with the heat flow are created around the fusion zone.

As shown in Tab. 1, the calculation on the fine reference mesh (without remeshing) required 6h and 25min of CPU-time. Two calculations with remeshing have been performed, one with a prescribed error ε = 0.01 and a second one with ε = 0.005. The CPU-time was 1h and 1min for ε = 0.01 and 1h and 52min for ε = 0.005. As expected, the calculation with ε = 0.005 generates a larger refined zone in the neighbourhood of the thermal source than the calculation with ε = 0.01.

The temperature evolutions at a given point A (Fig. 2)in the different analyses are shown in Fig. 5. The first observation from the plots in Fig. 5 is that the results are significantly smoother in the time domain. This illustrates that the spatial noise associated with the Hessian recovery does not globally pollute the solution in time, suggesting that the primary fields (temperature and phase fractions) remain unaffected. We observe in Fig. 5 that the temperature evolution à Point A converges to the reference temperature evolution when reducing the prescribed error ε. From Tab. 1 and Fig. 5, it can be seen that for a comparable accuracy of the results, the use of an adaptative meshing procedure reduces CPU costs by a factor six, which demonstrates the efficiency of the proposed approach.

Thermo-metallurgical based mesh adaptation

Comparing Fig. 4 and Fig. 6 a clear difference between the obtained two meshes when using only the thermal error distribution Fig. 4 or both the thermal and the metallurgical error distributions is evidenced. A distinct behaviour is found when guiding the mesh adaptation with both phase proportion (in the present case, the bainite volume fraction) and temperature. In this case, the mesh is much denser in the wake of the heat source in order to provide a better representation of the steep gradients of phase fraction. It can be seen that the thermometallurgical-driven mesh generation produces significantly more elements than the thermal-driven one (see Tab. 2, Figs. 4 and6). This is due to the residual gradient of phases fractions that remains in the welded component after welding. Fig. 7 shows the residual profile of the bainite volume fraction in a transverse section, using different meshes. It can be seen that the phase distribution is much more accurate than with an adaptation based only on the thermal error distribution.

Thermomechanical based mesh adaptation

Mechanical properties applied in the thermomechanical model are shown in Fig. 3. In order to simplify the analysis, we don't take into account the effect of metallurgical transformation here. The conditions of thermal calculation are unchanged. As to mechanical boundary conditions when welding, the plate has been fixed at both lateral faces (parallel to the welding direction). The symmetric simplification is not considered, the whole plate has been modelled. The mesh adaptation is based on three fields: temperature, longitudinal and transversal stresses.

The reference mesh and the mesh refinement obtained by using temperature and stress as error indication are shown in Fig. 8. A distinct behavior is found when using a purely thermal mesh adaptation (Fig. 4) which gives much lower number of elements in the model. The evolution of the calculated longitudinal and transverse stresses is shown in Fig. 8 to 10, it can be seen that the simulation using remeshing gives similar results to the simulation on reference mesh. The reduction in size of the models decreases the required CPU-time by almost one-third (Tab.3). Figure 10 shows that the stress change during welding is complex and depends on geometry and constraint. Thus the thermal-mechanical adaptation is necessary when modelling complex three-dimensional welding problems as even an experienced user may find it difficult to mesh the model efficiently.

CONCLUSION

In this paper, adaptive remeshing procedures have been presented and applied in the context of coupled thermalmechanical-metallurgical simulations of welding. The method is based on anisotropic mesh adaptivity dictated by directional error estimators. Those estimators, based on the Hessian recovery of P1 field, are used to construct a mesh metric field that provides information on the local mesh resolution desired in different directions. The method allows to easily combine metric tensors for variables of different type and nature. In practice, the calculation results show that the temperature field, the distributions of phase fractions and stress obtained with adaptive mesh converge to the results obtained with reference meshes. For the cases tested here, the calculation time comparison shows that the adaptive mesh technique can reduce the calculation time by almost one-third. It also reduces the data-storage requirement substantially. For some applications, both points are key-factors in determining whether a successful FE-simulation can be completed or not.

FIGURE 1 .

 1 FIGURE 1. Coupling between thermal field, microstructural evolution and mechanical field. T is temperature, g k volume phase fraction of the different phases (index k)

FIGURE 2 .FIGURE 3 .

 23 FIGURE 2. a) Geometry of the specimen (dimensions in mm) and comparison point A, b) Reference FEM mesh of one half of the plate

FIGURE 4 .

 4 FIGURE 4. Thermal-based mesh adaptation (ε = 0.01), a) Adapted FEM mesh b) Zoom on refined zone

FIGURE 5 .

 5 FIGURE 5. Thermal-based mesh adaptation (ε = 0.01), Temperature evolution at Point A (2), [K]

FIGURE 6 .FIGURE 7 .

 67 FIGURE 6. Thermo-metallurgical based mesh adaptation (ε = 0.01), a) Adapted FEM mesh, b) Zoom on refined zone

FIGURE 8 .FIGURE 9 .FIGURE 10 .

 8910 FIGURE 8. a) Reference mesh, b) adapted mesh based on temperature and stress; Transverse stress (in Pa) on reference mesh (c) and on adapted mesh (d)

TABLE 1 .

 1 Refinement parameters (Nbe: denotes the number of elements. Calculation run on a Pentium 4 PC, 2GHz with 2Gb RAM.

		Initial	Final	CPU-
		Nbe	Nbe	Time
	Fine ref.	68891	68891	6h25min
	Coarse ref.	11439	11439	58 min
	Adapt. ε = 0.01	6842	5866	1h1min
	Adapt. ε = 0.005	6842	11012	1h52min

TABLE 2 .

 2 Refinement parameters and results for thermalmetallurgical adaptation. Calculation run on a Pentium 4 PC, 2GHz with 2Gb RAM.

		Initial Nbe	Final Nbe CPU-Time
	Fine ref.	68891	68891	6h 25min
	Adapt. ε = 0.01 6842	15816	2h 22min

TABLE 3 .

 3 Refinement parameters and results for thermalmetallurgical adaptation. Calculation run on a Pentium 4 PC, with 512Mb RAM.

		Initial	Final	CPU-Time
		Nbe	Nbe
	Fine ref. mesh	194263 194263 81h07 min
	Adapt. ε = 0.005	7152	27447	26h10 min


A Chrysochoos
email: chryso@lmgc.univ-montp2.fr

MATERIAL TRANSFORMATION STUDIES WITH FULL-FIELD THERMAL AND DEFORMATION MEASUREMENTS

INTRODUCTION

Optical techniques associated with quantitative imaging methods have been developed considerably over the last twenty years. During a deformation process, these techniques enable us to obtain discrete field measurements, invaluable not only to characterize and identify material behaviors but also for better distinguishing "material" effects from "structure" effects at an observation scale imposed by the optical systems. For example, digital image correlation (DIC) gives access to kinematical data and allows considering the homogeneous character of deformation fields; infrared thermography provides temperature fields and gives information about the dissipative nature of deformation mechanisms (viscosity, plasticity, damage, etc.). It also enables us to show the existence of heat coupling sources which translate a strong interaction between thermal and mechanical states (thermodilatability, first order phase change, rubber effects, etc.). Through this presentation, we wish to show that, parallel to theoretical and numerical work concerning the multiscale analysis of material behavior, it may be interesting and complementary to combine quantitative imaging techniques. During the presentation, we will briefly recall the definition of the different heat sources and the form of the energy balance associated with the deformation process; then, we will evoke the main characteristics of the experimental setup and that of the electronic synchronization system linking visible with infrared cameras; finally, we will seek to illustrate the potentialities of the combined imaging approach through results extracted from many applications: hardening and damage effects in metals, strain localization in polymers, phase change front tracking in SMAs, etc.. We will emphasize the role of the space observation scale in the definition of some behavioural properties of the material.

HEAT SOURCES AND ENERGY BALANCE

From a thermomechanical standpoint, deformation process is considered as a dissipative quasi-static process. The equilibrium state of each volume material element is then described using a set of N state variables: the absolute temperature T, a strain tensor ε and an N-2 element vector α which pools the so-called internal variables describing the macroscopic effects of complex, coupled microstructural phenomena. The rate of deformation energy is classically defined by def :

w D • = σ
, where σ is the Cauchy stress tensor and D the strain rate tensor. The symbol ( ) • signifies that the time variation of ( ) is path-dependent. The local heat diffusion equation is written as:

1 the ic ext div( grad ) CT K T d s s r ρ - = + + + ɺ ( 1
)
where ρ is mass density, C the specific heat, K the conduction tensor. The heat sources are the intrinsic dissipation d1, the thermomechanical coupling sources that pool the thermoelastic source sthe and the other internal coupling sources sic (e.g. latent heat rate), and the external volume heat supply rext. With the specific free energy ( , , ) T ψ ε α , the volume heat sources d1, sthe and sic are defined by:

1 2 2 the ic : . . 0 : . d D s s T T T T = σ -ρ ∂ψ ∂ε ε -ρ ∂ψ ∂α α ≥    + = ρ ∂ ψ ∂ ∂ε ε + ρ ∂ ψ ∂ ∂α α   ɺ ɺ ɺ ɺ (2)
3.

EXPERIMENTAL SETUP

Figure 1 illustrates the experimental set-up designed for this purpose. It involves a MTS hydraulic testing machine (frame: 100 kN, load cell: 25 kN), a Cedip Jade III infrared camera and a Camelia 8M high resolution CCD camera. The optical axis of both cameras was set perpendicularly to the frame of the testing machine, and remained fixed during the test. A simultaneous record of infrared and visible images was performed on each side of the sample surface, and we therefore used thin, flat samples. Each camera was controlled by a separate computer. A specific electronic device was designed to synchronise the frame grabbing of the two cameras. Reference speckle and infrared images were mapped using a calibration target. Comparing the visible and infrared images of the target, we determined the rigid body movements and the scale factor ratios between the two cameras. For each acquisition time, the thermal data given by the IR camera (measured in the current, Eulerian configuration) were linearly interpolated spatiotemporally using the positions of the deformed configuration given by the DIC computation.

Figure 1-Experimental setup

This operation allowed us to track the material surface element (mse) associated with the DIC mesh, and thus enabled us to compute and check the importance of the volume heat rate associated with the convective terms grad .

C T v ρ

, where v is the velocity vector of mse.

EXAMPLE OF APPLICATION

Results extracted from an application example are shown here to give a brief overview of the presentation. The example given relates to the hardening and damage development during a monotonous straining of steel classically used in metal forming. The test was performed at constant velocity (0.25 mm.s -1

). Fig. 2a shows the time course of the longitudinal profile of the acceleration component in the loading direction, while Fig. 2b shows the corresponding evolution of the intrinsic dissipation profile. In the figures above, the horizontal axis represents the time while the vertical axis represents the longitudinal sample axis in the initial configuration. The load vs. time curve was superimposed in order to link the local data to the overall mechanical loading. The paths of three mse (A, B and C) are plotted to illustrate the material flows. Element A is quite specific as it represents the fracture point. Elements B and C were a posteriori chosen to be respectively placed at the boundary and clearly outside the localization zone when the crack occurred. Naturally, as material particles remain fixed in the Lagrangian configuration, their paths are simple horizontal lines. Figure 2a shows that acceleration amplitude is negligible compared with the gravity acceleration (≈10 -5 m.s -2

). A steady concentration of level curves of positive and negative acceleration, distributed on both sides of the necking zones, is clearly observed. The section where the fracture will take place (i.e. mse A) is then characterized by zero acceleration. Figure 2b presents the space-time distribution of intrinsic dissipation during strain hardening. Data processing was stopped before rapid growth of localized necking occurred to take into account only relatively small stress and strain-rate gradients and slight triaxiality effects (plane stress). The early narrowing of the level curves reveals a gradual development of localization, long before the maximum load is reached.

5.


Samy Skander Bahoura

UNIQUENESS TYPE RESULT IN DIMENSION 3

We give some estimates of type sup × inf on Riemannian manifold of dimension 3 for the prescribed curvature type equation. As a consequence, we derive an uniqueness type result.

INTRODUCTION AND MAIN RESULTS

In this paper, we deal with the following prescribed scalar curvature type equation in dimension 3: ∆u + h(x)u = V (x)u 5 , u > 0.

(E) Where h, V are two continuous functions. In the case 8h = R g the scalar curvature, we call V the prescribed scalar curvature. Here, we assume h a bounded function and h 0 = ||h|| L ∞ (M) .

We consider three positive real number a, b, A and we suppose V lipschitzian:

0 < a ≤ V (x) ≤ b < +∞ and ||∇V || L ∞ (M) ≤ A. (C)
The equation (E) was studied a lot, when M = Ω ⊂ R n or M = S n see for example, [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF][START_REF] Bahoura | Harnack inequalities for Yamabe type equations[END_REF][START_REF] Bahoura | Lower bounds for sup+inf and sup × inf and an extension of Chen-Lin result in dimension 3[END_REF], [START_REF] Chen | Estimates of the conformal scalar curvature equation via the method of moving planes[END_REF], [START_REF] Yy | Prescribing scalar curvature on Sn and related Problems[END_REF]. In this case we have a sup × inf inequality.

The corresponding equation in two dimensions on open set Ω of R 2 , is:

∆u = V (x)e u
, (E ′ ) The equation (E ′ ) was studed by many authors and we can find very important result about a priori estimates in [START_REF] Brezis | A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities[END_REF], [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF], [START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF], [START_REF] Yy | Harnack Type Inequality: the Method of Moving Planes[END_REF], and [START_REF] Shafrir | A sup+inf inequality for the equation -∆u = V e u[END_REF]. In particular in [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF] we have the following interior estimate:

sup K u ≤ c = c(inf Ω V, ||V || L ∞ (Ω) , inf Ω u, K, Ω).
And, precisely, in [START_REF] Brezis | A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities[END_REF], [START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF], [START_REF] Yy | Harnack Type Inequality: the Method of Moving Planes[END_REF], and [START_REF] Shafrir | A sup+inf inequality for the equation -∆u = V e u[END_REF], we have:

C sup K u + inf Ω u ≤ c = c(inf Ω V, ||V || L ∞ (Ω) , K, Ω),
and,

sup K u + inf Ω u ≤ c = c(inf Ω V, ||V || C α (Ω) , K, Ω).
where K is a compact subset of Ω, C is a positive constant which depends on inf Ω V sup Ω V , and,

α ∈ (0, 1].
In the case V ≡ 1 and M compact, the equation (E) is Yamabe equation. Yamabe has tried to solve problem but he could not, see [START_REF] Yamabe | On a deformation of Riemannian structures on compact manifolds[END_REF]. T.Aubin and R.Schoen have proved the existence of solution in this case, see for example [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF] and [START_REF] Lee | The Yamabe problem[END_REF] for a complete and detailed summary.

When M is a compact Riemannian manifold, there exist some compactness result for equation (E) see [START_REF] Yy | Yamabe Type Equations On Three Dimensional Riemannian Manifolds[END_REF]. Li and Zhu see [START_REF] Yy | Yamabe Type Equations On Three Dimensional Riemannian Manifolds[END_REF], proved that the energy is bounded and if we suppose M not diffeormorfic to the three sphere, the solutions are uniformly bounded. To have this result they use the positive mass theorem. Now, if we suppose M Riemannian manifold (not necessarily compact) and V ≡ 1, Li and Zhang [START_REF] Yy | A Harnack type inequality for the Yamabe equation in low dimensions[END_REF] proved that the product sup × inf is bounded. Here we extend the result of [START_REF] Bahoura | sup × inf inequality on manifold of dimension 3[END_REF].

Our proof is an extension of Brezis-Li and Li-Zhang result in dimension 3, see [START_REF] Brezis | Some nonlinear elliptic equations have only constant solutions[END_REF] and [START_REF] Yy | A Harnack type inequality for the Yamabe equation in low dimensions[END_REF], and, the moving-plane method is used to have this estimate. We refer to Gidas-Ni-Nirenberg for the moving-plane method, see [START_REF] Gidas | Symmetry and related properties via the maximum principle[END_REF]. Also, we can see in [START_REF] Caffarelli | Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[END_REF], one of the application of this method.

Here, we give an equality of type sup × inf for the equation (E) with general conditions (C). Note that, in our proof, we do not need a classification result for some particular elliptic PDEs on R 3 .

In dimension greater than 3 we have other type of estimates by using moving-plane method, see for example [START_REF] Bahoura | Harnack inequalities for Yamabe type equations[END_REF][START_REF] Bahoura | Estimations uniformes pour l'quation de Yamabe en dimensions 5 et 6[END_REF].

There are other estimates of type sup + inf on complex Monge-Ampere equation on compact manifolds, see [START_REF] Siu | The existence of Kahler-Einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group[END_REF][START_REF] Tian | A Harnack type inequality for certain complex Monge-Ampre equations[END_REF] . They consider, on compact Kahler manifold (M, g), the following equation:

(ω g + ∂ ∂ϕ) n = e f -tϕ ω n g , ω g + ∂ ∂ϕ > 0 on M
And, they prove some estimates of type sup M +m inf M ≤ C or sup M +m inf M ≥ C under the positivity of the first Chern class of M.

Here, we have, Theorem 1.1. For all compact set K of M and all positive numbers a, b, A, h 0 there is a positive constant c, which depends only on, a, b, A, h 0 , K, M, g such that:

sup K u × inf M u ≤ c,
for all u solution of (E) with conditions (C).

This theorem generalise Li-Zhang result, see [START_REF] Yy | A Harnack type inequality for the Yamabe equation in low dimensions[END_REF] in the case V ≡ 1. Here, we use Li and Zhang method in [START_REF] Yy | A Harnack type inequality for the Yamabe equation in low dimensions[END_REF].

In the case h ≡ ǫ ∈ (0, 1) and u ǫ solution of :

∆u ǫ + ǫu ǫ = V ǫ u 5 ǫ , u ǫ > 0. (E ǫ )
We have:

Corollary 1.2. For all compact set K of M and all positive numbers a, b, A there is a positive constant c, which depends only on, a, b, A, K, M, g such that:

sup K u ǫ × inf M u ǫ ≤ c,
for all u solution of (E ǫ ) with conditions (C). Now, if we assume M a compact riemannian manifold and 0 < a ≤ V ǫ ≤ b < +∞ we have: Theorem 1.3. (see [START_REF] Bahoura | Harnack inequalities for Yamabe type equations[END_REF]). For all positive numbers a, b, m there is a positive constant c, which depends only on, a, b, m, M, g such that:

ǫ sup M u ǫ × inf M u ǫ ≥ c, for all u ǫ solution of (E ǫ ) with max M u ǫ ≥ m > 0 .
As a consequence of the two previous theorems, we have: Theorem 1.4. For all positive numbers a, b, A we have:

max M u ǫ → 0,
and (up to a subsequence),

max M u ǫ ǫ 1/4
→ w 0 > 0, and, min M u ǫ ǫ 1/4 → w 0 > 0.

Remarks:

• It is not necessary to have u ǫ ≡ w 0 ǫ 1/4 , because if we take a nonconsant function V , we can find by the variational approach a non constant positive solution of the subcritical equation:

∆u ǫ + ǫu ǫ = µ ǫ V u 5-ǫ ǫ
, with µ ǫ , u ǫ > 0. In this case (subcritical which tends to the critical) we also have the sup × inf inequalities and the uniqueness type theorem.

• In fact, we prove, up to a subsequence that u ǫ ǫ 1/4 converge to a constant which depends on a, b and A.

PROOF OF THE THEOREMS

Proof of theorem 1.1:

We want to prove that:

ǫ max B(0,ǫ) u × min B(0,4ǫ) u ≤ c = c(a, b, A, M, g) (1)
We argue by contradiction and we assume that:

max B(0,ǫ k ) u k × min B(0,4ǫ k ) u k ≥ kǫ k -1 (2)
Step 1: The blow-up analysis

The blow-up analysis gives us : For some xk ∈ B(0, ǫ k ), u k (x k ) = max B(0,ǫ k ) u k , and, from the hypothesis,

u k (x k ) 2 ǫ k → +∞. By a standard selection process, we can find x k ∈ B(x k , ǫ k /2) and σ k ∈ (0, ǫ k /4) satisfying, u k (x k ) 2 σ k → +∞, (3)
u k (x k ) ≥ u k (x k ), (4) and,
u k (x) ≤ C 1 u k (x k ), in B(x k , σ k ),
(5) where C 1 is some universal constant. It follows from above ( (2), ( 4)) that:

u k (x k ) × min ∂B(x k ,2ǫ k ) u k ǫ k ≥ u k (x k ) × min B(0,4ǫ k ) u k ǫ k ≥ k → +∞. ( 6
)
We use {z 1 , . . . , z n } to denote some geodesic normal coordinates centered at x k (we use the exponential map). In the geodesic normal coordinates, g = g ij (z)dzdz j ,

g ij (z) = δ ij = O(r 2 ), g := det(g ij (z)) = 1 + O(r 2 ), h(z) = O(1), (7)
where r = |z|. Thus,

∆ g u = 1 √ g ∂ i ( √ gg ij ∂ j u) = ∆u + b i ∂ i u + d ij ∂ ij u, where b j = O(r), ∂ ij = O(r 2 ) (8) We have a new function: v k (y) = M -1 k u k (M -2 k y) for |y| ≤ 3ǫ k M 2 k
where M k = u k (0). From (5), ( 6), we have:

         ∆v k + bi ∂ i v k + dij ∂ ij v k -cv k + v k 5 = 0 for |y| ≤ 3ǫ k M 2 k v k (0) = 1 v k (y) ≤ C 1 for |y| ≤ σ k M 2 k lim k→+∞ min |y|=2ǫ k M 2 k (v k (y)|y|) = +∞ (9) where C 1 is a universal constant and bi (y) = M -2 k b i (M -2 k y), dij (y) = d ij (M -2 k y) (10) and, c(y) = M -4 k h(M -2 k y) (11)
We can see that for |y| ≤ 3ǫ k M 2 k , we have:

| bi (y)| ≤ CM -4 k |y|, | dij (y)| ≤ CM -4 k |y| 2 , |c(y)| ≤ CM -4 k ( 12
)
where C depends on n, M, g.

It follows from ( 9), ( 10), ( 11), ( 12) and the elliptic estimates, that, along a subsequence, v k converges in C 2 norm on any compact subset of R 2 to a positive function U satisfying:

∆U + U 5 = 0 in R 2 U (0) = 1, 0 < U ≤ C 1 (13)
Step 2: The Kelvin transform and moving-plane method For x ∈ R 2 and λ > 0, let,

v λ,x k (y) := λ |y -x| v k x + λ 2 (y -x)
|y -x| 2 denote the Kelvin transformation of v k with respect to the ball centered at x and of radius λ.

We want to compare for fixed x, v k and v λ,x k . For simplicity we assume x = 0. We have:

v λ k (y) := λ |y| v k (y λ ), with y λ = λ 2 y |y| 2 For λ > 0, we set, Σ λ = B 0, ǫ k M k 2 -B(0, λ).
The boundary condition, (9), become:

lim k→+∞ min |y|=ǫ k M 2 k (v k (y)|y|) = lim k→+∞ min |y|=2ǫ k M 2 k (v k (y)|y|) = +∞ (14)
We have:

∆v λ k + V λ k (v λ k ) 5 = E 1 (y) for y ∈ Σ λ (15) where, E 1 (y) = - λ |y| 5 bi (y λ )∂ i v k (y λ ) + dij (y λ )∂ ij v k (y λ ) -c(y λ )v k (y λ ) . (16)
Clearly, from (10), [START_REF] Chen | Estimates of the conformal scalar curvature equation via the method of moving planes[END_REF], there exists

C 2 = C 2 (λ 1 ) such that, |E 1 (y)| ≤ C 2 λ 5 M -4 k |y| -5 for y ∈ Σ λ (17)
Let,

w λ = v k -v λ k .
Here, we have, for simplicity, omitted k. We observe that by (9), [START_REF] Yy | Prescribing scalar curvature on Sn and related Problems[END_REF]:

∆w λ + bi ∂ i w λ + dij ∂ ij w λ -cw λ + 5ξ 4 V k w λ = E λ in Σ λ (18)
where ξ stay between v k and v λ k , and,

E λ = -bi ∂ i v λ k + dij ∂ ij v λ k + cv λ k -E 1 -(V k -V λ k )(v λ k ) 5 . (19)
A computations give us the following two estimates:

|∂ i v λ k (y)| ≤ Cλ|y| -2 , and |∂ ij v λ k (y)| ≤ Cλ|y| -3 in Σ λ (20)
From ( 10), [START_REF] Chen | Estimates of the conformal scalar curvature equation via the method of moving planes[END_REF], [START_REF] Siu | The existence of Kahler-Einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group[END_REF], we have,

Lemma 2.1. . For somme constant C 3 = C 3 (λ) |E λ | ≤ C 3 λM -4 k |y| -1 + C 3 λ 5 M -2 k |y| -4 in Σ λ (21)
we consider the following auxiliary function:

h λ = -C 1 AM -2 k λ 2 1 - λ |y| + C 2 AM -2 k λ 3 1 - λ |y| 2 -C 3 M -4 k λ(|y| -λ),
where C 1 , C 2 and C 3 are three positive numbers.

Lemma 2.2. . We have,

w λ + h λ ≥ 0, in Σ λ ∀0 < λ ≤ λ 1 (22)
Proof of Lemma 2.2. We divide the proof into two steps.

Step 1. There exists λ 0,k > 0 such that (22) holds :

w λ + h λ ≥ 0, in Σ λ ∀0 < λ ≤ λ 0,k .
To see this, we write:

w λ = v k (y) -v λ k (y) = 1 |y| |y|v k (y) -|y λ |v k (y λ ) .
Note that y and y λ are on the same ray starting from the origin. Let, in polar coordinates,

f (r, θ) = √ rv k (r, θ).
From the uniform convergence of v k , there exists r 0 > 0 and C > 0 independant of k such that, ∂f ∂r (r, θ) > Cr -1/2 for 0 < r < r 0 .

Consequently, for 0 < λ < |y| < r 0 , we have:

w λ (y) + h λ (y) = v k (y) -v λ k (y) + h λ (y), > 1 √ r 0 C √ r 0 -1/2 (|y| -|y λ |) + h λ (y) > ( C r 0 -C 3 λM -2 k )(|y| -λ) since |y| -|y λ | > |y| -λ > 0. ( 23
)
Since,

|h λ (y)| + v λ k (y) ≤ C(k, r 0 )λ, r 0 ≤ |y| ≤ ǫ k M k 2 ,
we can pick small λ 0,k ∈ (0, r 0 ) such that for all 0 < λ ≤ λ 0,k we have,

w λ (y) + h λ (y) ≥ min B(0,ǫ k M 2 k ) v k -C(k, r 0 )λ 0,k > 0 ∀ r 0 ≤ |y| ≤ ǫ k M k 2
Step 1 follows from (23).

Let,

λk = sup{0 < λ ≤ λ 1 , w µ + h µ ≥ 0, in Σ µ ∀0 < µ ≤ λ} (24)
Step 2. λk = λ 1 , (22) holds.

For this, the main estimate needed is:

(∆ + bi ∂ i + dij ∂ ij -c + 5ξ 4 V k )(w λ + h λ ) ≤ 0 in Σ λ (25)
Thus,

∆h λ + bi ∂ i h λ + dij ∂ ij h λ + (-c + 5ξ 4 V k )h λ + E λ ≤ 0 in Σ λ . (26)
We have h λ < 0, and, (12) and a computation give us, From (25) and above we conclude that λk = λ 1 and lemma 2.2 is proved.

|ch λ | ≤ C 3 λM -4 k |y| -1 + C 3 λ 2 M -6 k ≤ C 3 λM -4 k |y| -1 , and, | bi ∂ i h λ | + | dij ∂ ij h λ | ≤ C 3 λM -8 k |y| + C 3 λ 3 M -6 k |y| -1 + C 3 λ 5 M -6 k |y| -2 , ≤ C 3 λM -4 k |y| -1 + C 3 λ 5 M -2 k |y| -4 Thus, | bi ∂ i h λ | + | dij ∂ ij h λ | + |ch λ | ≤ C 3 λM

-4 k |y| - 1 + 2 ,

 12 C 3 λ 5 M -2 k |y| -4 in Σ λ Thus, by[START_REF] Tian | A Harnack type inequality for certain complex Monge-Ampre equations[END_REF],∆h λ + bi ∂ i h λ + dij ∂ ij h λ + (-c + 5ξ 4 V k )h λ + E λ ≤ ≤ ∆h λ + C 3 λM -4 k |y| -1 + C 3 λ 5 M -2 k |y| -4 + |E λ | ≤ 0, because, ∆h λ = -2C 3 λM -4 k |y| -1 -2C 3 λ 5 M -2 k |y| -4 .From the boundary condition and the definition of v λk and h λ , we have:|h λ (y)| + v λ k (y) ≤ C(λ 1 ) |y| , ∀ |y| = ǫ k M k 2 ,and, thus,wλk (y) + hλk (y) > 0 ∀ |y| = ǫ k M kWe can use the maximum principal and the Hopf lemma to have:wλk + hλk > 0, in Σ λ ,and, ∂ ∂ν (wλk + hλk ) > 0, in Σ λ .

Given any λ > 0, since the sequence v k converges to U and hλk converges to 0 on any compact subset of R 2 , we have:

Since λ 1 > 0 is arbitrary, and since we can apply the same argument to compare v k and v λ,x k , we have:

Thus implies that U is a constant which is a contradiction.

Proof of theorem 1.4:

From theorem 2.1 (see [START_REF] Bahoura | Harnack inequalities for Yamabe type equations[END_REF]), we have:

We conclude with the aid of the elliptic estimates and the classical Harnack inequality that:

where C is a positive constant independant of ǫ.

Let G ǫ the Green function of the operator ∆ + ǫ, we have,

We write:

With the similar argument, we have :

Finaly, we have:

Where C 1 and C 2 are two positive constant independant of ǫ.

We set w ǫ = u ǫ ǫ 1/4 , then, ∆w ǫ + ǫw ǫ = ǫV ǫ w 5 ǫ . The theorem follow from the standard elliptic estimate, the Green function of the lapalcian and the Green representation formula for the solutions of the previous equation.


Keywords: Automorphisme, fonction génératrice, genre, marche aléatoire homogène par morceaux, quart de plan, fonction elliptique de Weierstrass

Marches aléatoires dans le quart de plan avec dérive nulle: un critère explicite de finitude pour le groupe associé

Résumé : Dans plusieurs études récentes sur les marches aléatoires dans le quart de plan avec des sauts vers les huit plus proches voisins, il apparat que le comportement de certaines quantits d'intérêt est directement lié au groupe de la marche, notamment à la finitude de son ordre. Pour les marches dérive nulle, nous donnons une formule pour l'ordre de ce groupe, en fonction explicite des probabilits de saut. De façon générale, lorsque le genre de la courbe algébrique définie par le noyau est 0, le groupe est toujours infini, sauf précisément lorsque le saut moyen est nul, auquel cas la finitude est parfaitement possible.

Introduction and main results

For several decades, lattice random walks in the Euclidean quarter plane with unit jumps (or steps) have been at the crossroads of several mathematical areas: probability, complex analysis and, more recently, combinatorics. In this context, one of the basic problems often amounts to solve functional equations of two complex variables x, y, which have the following typical form K(x, y)Q(x, y) = k(x, y)Q(x, 0) + k(x, y)Q(0, y) + k 0 (x, y)Q(0, 0) + κ(x, y),

where

• K(x, y) [usually called the kernel], k(x, y), k(x, y), k 0 (x, y), and κ(x, y) are known functions;

• Q(x, y) is sought to be analytic in the region {|x|, |y| 1} and continuous up to the boundary.

We briefly describle two situations where equations of type (1.1) appear.

Example 1.1. This example deals with the random walks with jumps of unit length inside the quarter plane, but arbitrary big on the axes. The question is then to calculate the related invariant measure {π i,j } i,j 0 as well as to provide conditions for its existence. Letting Q(x, y) = i,j 0 π i,j x i y j , the classical Kolmogorov's equations yield (see, e.g., [START_REF] Fayolle | Random walks in the quarter plane[END_REF][START_REF] Fayolle | Topic in the constructive theory of countable Markov chains[END_REF]) equation (1.1) with κ(x, y) = 0, whereas k(x, y), k(x, y) and k 0 (x, y) correspond to the generating functions of the jump probabilities on the horizontal axis, on the vertical axis and at (0, 0), respectively. Letting {p i,j } -1 i,j 1 denote the jumps in Z 2 + , the kernel is then given by K(x, y) = xy[ -1 i,j 1 p i,j x i y j -1].

(1.2)

Example 1.2. The enumeration of planar lattice walks-a classical topic in combinatoricsis the topic of this second example. For a given set S of allowed jumps, it is a matter of counting the number of paths of a given length, starting from a fixed origin and ending at an arbitrary point, and possibly restricted to certain regions of the plane. A first basic and natural question arises: how many such paths exist? If the paths are confined to Z 2 + = {0, 1, . . .} 2 and if S is included in the set of the eight nearest neighbors, let q(i, j; n) denote the number of paths of length n, which start from (0, 0) and end at (i, j). Then Q(x, y) = i,j,n 0 q(i, j; n)x i y j z n does satisfy equation (1.1), see [START_REF] Bousquet-Mélou | Walks with small steps in the quarter plane[END_REF], where the kernel is now equal to

K(x, y) = xyz[ (i,j)∈S x i y j -1/z],
k(x, y) = K(x, 0), k(x, y) = K(0, y), k 0 (x, y) = -K(0, 0) and κ(x, y) = -xy. Here Q(x, y) does exist at least in the domain {|x|, |y| 1; |z| 1/|S|}, and the third variable z essentially plays the role of a parameter. This situation would also occur in the context of example 1.1, provided we would be interested in the transient behavior of the process.

Throughout this paper, we shall essentially remain in the framework of example 1.1. An exhaustive original method of solution of equation (1.1) has been given in the book [START_REF] Fayolle | Random walks in the quarter plane[END_REF], allowing to get explicit expressions for Q(x, y). It mainly resorts to a reduction to boundary value problems of Riemann-Hilbert type as in [START_REF] Fayolle | Two coupled processors: the reduction to a Riemann-Hilbert problem[END_REF], together with the analysis of a certain group of Galois automorphisms introduced in [START_REF] Malyshev | Positive random walks and Galois theory[END_REF] and acting on the algebraic curve (and its related Riemann surface)

K = {(x, y) ∈ C 2 : K(x, y) = 0}. (1.3)
Let C (x, y) be the field of rational functions in x, y over C. Since we shall assume that K, given by (1.2), is irreducible (see [START_REF] Fayolle | Random walks in the quarter plane[END_REF]Lemma 2.3.2] for an interpretation of this hypothesis in terms of the parameters {p i,j } -1 i,j 1 ), the quotient field C(x, y) denoted by C Q (x, y) is also a field. Definition 1.3. The group of the walk is the Galois group W = ξ, η of automorphisms of C Q (x, y) generated by ξ and η given by

ξ(x, y) = x, 1 y -1 i 1 p i,-1 x i -1 i 1 p i,+1 x i , η(x, y) = 1 x -1 j 1 p -1,j y j -1 j 1 p +1,j y j , y . Let δ = η • ξ. (1.4)
Then W has a normal cyclic subgroup W 0 = {δ ℓ , ℓ ∈ Z }, which is possibly infinite, and such that W/W 0 is a group of order 2. Thus, when W is finite, say of order 2n, δ n is the identity.

In many studies, stemming either from probability theory [START_REF] Fayolle | Random walks in the quarter plane[END_REF][START_REF] Kurkova | Random walks in Z 2 + with non-zero drift absorbed at the axes[END_REF][START_REF] Raschel | Green functions for killed random walks in the Weyl chamber of Sp(4)[END_REF] and, more recently, from combinatorics [START_REF] Bostan | The complete generating function for Gessel's walk is algebraic[END_REF][START_REF] Bousquet-Mélou | Walks with small steps in the quarter plane[END_REF][START_REF] Fayolle | On the holonomy or algebraicity of generating functions counting lattice walks in the quarter plane. Markov Process[END_REF][START_REF] Mishna | Two non-holonomic lattice walks in the quarter plane[END_REF], the analysis of type (1.1) equations is often crucial, and the importance of the group W has repeatedly been noticed. Broadly speaking, it appears that the question of the finiteness of W can be very useful in the three following respects.

• To find an explicit form of the generating function Q(x, y), see [START_REF] Bostan | The complete generating function for Gessel's walk is algebraic[END_REF][START_REF] Bousquet-Mélou | Walks with small steps in the quarter plane[END_REF][START_REF] Fayolle | Random walks in the quarter plane[END_REF].

• To prove the holonomy-or even the algebraicity-of Q(x, y), see [START_REF] Bostan | The complete generating function for Gessel's walk is algebraic[END_REF][START_REF] Bousquet-Mélou | Walks with small steps in the quarter plane[END_REF][START_REF] Fayolle | Random walks in the quarter plane[END_REF][START_REF] Fayolle | On the holonomy or algebraicity of generating functions counting lattice walks in the quarter plane. Markov Process[END_REF].

• To derive asymptotics of the Taylor coefficients of Q(x, y), see [START_REF] Kurkova | Random walks in Z 2 + with non-zero drift absorbed at the axes[END_REF][START_REF] Raschel | Green functions for killed random walks in the Weyl chamber of Sp(4)[END_REF].

Brief overview

In spite of the facts recalled above, very few criteria exist in the literature to decide about the finiteness of the group W . Up the knowledge of the authors, the main known results can be found in [4, Chapters 3 and 4]. For instance, it is proved that the group W has order 4 if and only if

∆ = p 1,1 p 1,0 p 1,-1 p 0,1 p 0,0 -1 p 0,-1 p -1,1 p -1,0 p -1,-1 = 0. (1.5)
For groups of order 6, a criterion, still in terms of certain determinants, is also given in [4, Section 4. On other hand, this nice theoretical formula is not easy to rewrite concretely in terms of the parameters {p i,j } -1 i,j 1 . For instance, an expression (if any!) by means of determinants, as in (1.5), is absolutely unclear.

As for the genus 0 case, although the solution of equation (1.1) was constructed in [START_REF] Fayolle | Random walks in the quarter plane[END_REF]Chapter 6], no criterion was yet provided to decide about the finiteness of the group: this is precisely the subject of this paper, where we give an explicit criterion for W to be finite. The method relies mainly on a continuity argument from the genus 1 case, in a sense to be made precise in Sections 2 and 3.

A criterion for the finiteness of W in the genus 0 case

Now we state our main result, according to the classification of Lemma A.2 listed in the appendix. As the reader will realize, most of the analysis will be devoted to walks having a zero drift, i.e., -→ M = 0, since it appears to be the most difficult (and maybe interesting!) situation. Introduce the correlation coefficient

R = -1 i,j 1 ijp i,j [ -1 i,j 1 i 2 p i,j ] 1/2 • [ -1 i,j 1 j 2 p i,j ] 1/2 ,
and define the angle θ = arccos(-R).

(1.7)

Theorem 1.4.

(I) When -→ M = 0, the group W is finite if, and only if, θ/π is rational, in which case the order of W is equal to

2 inf{ℓ ∈ Z * + : ℓθ/π ∈ Z}. (1.8)
(II) When -→ M = 0, the order of W is always infinite in the four cases (A.1), (A.2), (A.3) and (A.4).

In Section 2.4, we shall propose another theoretical form of the angle (1.7), merely involving characteristic values related to the uniformization of curves of genus 0.

Proof of Part (I) of Theorem 1.4

The proof proceeds in stages, the key idea being to consider the genus 0 case as a continuous limit of the genus 1 case. In particular, we shall consider ad hoc limit conformal gluing and limit uniformizing functions, allowing to connect the ratio of certain limit periods with the finiteness of the limit group.

Basic properties of the kernel

To render the paper as self-contained as possible, we recall hereafter some important results proved in [START_REF] Fayolle | Random walks in the quarter plane[END_REF]. They are needed for our purpose and they will be stated without further comment.

First, let us rewrite the kernel (1.2) in the two equivalent forms

K(x, y) = a(x)y 2 + b(x)y + c(x) = a(y)x 2 + b(y)x + c(y), where a(x) = p 1,1 x 2 + p 0,1 x+ p -1,1 , b(x) = p 1,0 x 2 + (p 0,0 -1)x+ p -1,0 , c(x) = p 1,-1 x 2 + p 0,-1 x+ p -1,-1 , a(y) = p 1,1 y 2 + p 1,0 y+ p 1,-1 , b(y) = p 0,1 y 2 + (p 0,0 -1)y+ p 0,-1 , c(y) = p -1,1 y 2 + p -1,0 y+ p -1,-1 .
Then we set

D(x) = b 2 (x) -4a(x)c(x), D(y) = b 2 (y) -4 a(y) c(y).
The polynomials D and D are of degree 4, with respective dominant coefficients On the cut [y 1 y 2 ] ∪ [y 3 y 4 ], the roots X 0 and X 1 are complex conjugate. Then we consider the two components of the quartic curve

C = p 2 1,0 -4p 1,1 p 1,-1 , C = p 2 0,1 -4p 1,1 p -1,1 . (2.
   M 1 = X 0 ([ --→ y 1 y 2 ← -- ]) = X 1 ([ ← -- y 1 y 2 --→ ]), M 2 = X 0 ([ --→ y 3 y 4 ← -- ]) = X 1 ([ ← -- y 3 y 4 --→ ]),
where --→ y 1 y 2 ← --stands for the contour [y 1 y 2 ], traversed from y 1 to y 2 along the upper edge of the slit [y 1 y 2 ], and then back to y 1 along the lower edge of the slit. Similarly, ← -y 1 y 2 --→ is defined by exchanging "upper" and "lower". In particular, the simply connected domain D 1 bounded by M 1 will play here a fundamental role, for reasons explained in Section 2.1.2.

Uniformization in the genus 1 case

When the associated Riemann surface is of genus 1, the algebraic curve K(x, y) = 0 admits a uniformization in terms of the Weierstrass ℘ function with periods ω 1 , ω 2 (see equation (2.4)) and its derivatives. Indeed, setting

D(x) = b 2 (x) -4a(x)c(x) = d 4 x 4 + d 3 x 3 + d 2 x 2 + d 1 x + d 0 , z = 2a(x)y + b(x),
the following formulae hold [noting that for notational convenience d 4 corresponds to C defined in equation (2.1)].

• If d 4 = 0 (four finite branch points {x ℓ } 1 ℓ 4 ), then D ′ (x 4 ) > 0 and          x(ω) = x 4 + D ′ (x 4 ) ℘(ω) -D ′′ (x 4 )/6 , z(ω) = D ′ (x 4 )℘ ′ (ω) 2[℘(ω) -D ′′ (x 4 )/6] 2 . (2.2) INRIA • If d 4 = 0 (three finite branch points {x ℓ } 1 ℓ 3 and x 4 = ∞), then        x(ω) = ℘(ω) -d 2 /3 d 3 , z(ω) = - ℘ ′ (ω) 2d 3 .
(2.3)

Let us now introduce the three key quantities

ω 1 = 2i x 2 x 1 dx -D(x) , ω 2 = 2 x 3 x 2 dx D(x) , ω 3 = 2 x 1 X(y 1 ) dx D(x) .
(2.4)

The period ω 1 is clearly purely imaginary, and 0 < ω 3 < ω 2 , see [START_REF] Fayolle | Random walks in the quarter plane[END_REF]Lemma 3.3.3]. In this respect, for ℓ ∈ {2, 3}, it will be convenient to denote by ℘ 1,ℓ the Weierstrass elliptic function with periods ω 1 , ω ℓ and series expansion

℘ 1,ℓ (ω) = 1 ω 2 + (p 1 ,p ℓ )∈Z 2 \{(0,0)} 1 (ω -p 1 ω 1 -p ℓ ω ℓ ) 2 - 1 (p 1 ω 1 + p ℓ ω ℓ ) 2 .
(2.5)

On the universal covering, we have δ(ω • w establishes a conformal mapping of D onto the complex plane cut along a segment;

+ ω 3 ) = δ(ω),
• For all t in the boundary of D, w(t) = w(t).

For instance, w(t) = t + 1/t is a CGF for the unit disc, and any non-degenerate linear transformation of w(t), namely ew(t) + f gw(t) + h , (e, g) = (0, 0), ehf g = 0 is also a CGF for the unit disc. Incidentally, this implies one can choose arbitrarily the pole of w within the unit disc-in particular, taking e = 0, f = 1, g = 1 and h = -2, we get the CGF t/(t -1) 2 with a pole at 1. Conversely, two CGFs for a same domain are fractional linear transformations of each other, see [START_REF] Litvinchuk | Solvability theory of boundary value problems and singular integral equations with shift[END_REF].

Some topological facts in the genus 0 case

As -→ M = 0, the branch points x 2 and x 3 coincide and we have

x 2 = x 3 = 1, x 1 ∈ [-1, 1), x 4 ∈ (1, ∞) ∪ {∞} ∪ (-∞, -1],
Similar results hold for the {y ℓ } 1 ℓ 4 .

Here the curve M 1 and M 2 intersect at the point x = 1, which is a corner point. In the figure 2.1, borrowed from [4, Chapter 6], the dotted curve is the unit circle, and one has drawn the contour M 1 ∪ M 2 , which has a self-intersection and is the image of the cut [ --→ y 1 y 4 ← --] by the mapping y → X(y), remembering that X 0 ([ --→

y 1 y 4 ← -- ]) = X 1 ([ ← -- y 1 y 4 --→ ]).
In the sequel, we shall prove that the angle of the tangent line at the corner point is indeed deeply related to the order of the group.

X(y 1 ) -1 M 2 1 0 X(y 4 ) M 1 Fig. 2.1: The contour M 1 ∪ M 2 for R < 0

Limit conformal gluing when passing from genus 1 to genus 0

Our approach resides in viewing genus 0 as a topological deformation of genus 1, when the parameters are modified in such a way that -→ M → 0. According to the uniformization (2.2) or (2.3), and together with the notations of Section 2.1.1, let

f (t) =      D ′′ (x 4 ) 6 + D ′ (x 4 ) t -x 4 if x 4 = ∞, 1 6 (D ′′ (0) + D ′′′ (0)t) if x 4 = ∞.
To avoid technicalities of minor importance, we shall assume in the sequel C = 0, with C given by (2.1), so that x 4 = ∞. Then, in the genus 1 case, it was shown in [START_REF] Fayolle | Random walks in the quarter plane[END_REF] that a CGF for the domain D 1 bounded by M 1 is obtained via the function

w(t) = ℘ 1,3 (℘ -1 1,2 (f (t)) -[ω 1 + ω 2 ]/2).
Since any fractional linear transformation of a CGF is again a CGF, the well-known addition theorem for ℘ 1,3 involving translation by the half-period ω 1 /2 entails that

℘ 1,3 (℘ -1 1,2 (f (t)) -ω 2 /2) (2.6)
is also a CGF for D 1 . Now letting -→ M → 0, so that x 2 , x 3 → 1, and by using the continuity with respect to the parameters {p i,j } -1 i,j 1 , a direct calculation in the integral formulae (2.4) gives that ω 1 → i∞ and that ω 2 , ω 3 converge to certain non-degenerate quantities as below:

               ω 1 → i∞, ω 2 → α 2 = π [C(x 4 -1)(1 -x 1 )] 1/2 , ω 3 → α 3 = x 1 X 0 (y 1 ) dx (1 -x)[C(x -x 1 )(x -x 4 )] 1/2 .
(2.7)

INRIA Lemma 2.2. Letting -→ M → 0, we have θ π = lim -→ M→0 ω 2 ω 3 = α 2 α 3 .
Proof. At the corner point of the curve M 1 , we shall calculate the angle of the tangent line with the horizontal axis in two different ways.

First, when -→ M → 0, the derivative X ′ 0 (1) is a root of a second degree polynomial, since

a(1)X ′ 0 2 (1) + RX ′ 0 (1) + a(1) = 0, see [4, equation (6.5.3)]. Since 2a(1) = -1 i,j 1 j 2 p i,j , 2 a(1) = -1 i,j 1 i 2 p i,j , it is quite easy to check that arg(X ′ 0 (1)) = ±θ, (2.8)
where θ is given by equation (1.7).

The second way of computing θ relies on the construction of a convenient CGF, according to the definition given in Section 2.1.2. Here we shall choose w with w(1) = ∞. As we shall see, this implies that, in the neighborhood of t = 1, w(t) has the form

w(t) = [α + o(1)]/[1 -t] χ , (2.9)
for some constant α = 0 and χ > 0. In addition, the exponent χ must satisfy the relation

χ = π/θ. (2.10)
Indeed, equation (2.8) together with the symmetry of the CGF yield exp(iθχ) = exp(-iθχ), whence θχ/π is a positive integer. On the other hand, if θχ/π 2, then w would not be one-to-one. Identity (2.10) follows. Thus we are left with the proof of expansion (2.9). In order to obtain a CGF in the zero drift case, we could use [START_REF] Fayolle | Random walks in the quarter plane[END_REF]Lemma 6.5.5], but it will be more convenient to use here the CGF obtained from the last paragraph, and then to let the drift go to zero. When -→ M → 0, we know from (2.4) that ω 1 → i∞. This way, using (2.5) as well as the well-known identity

p∈Z 1 (Ω + p) 2 = π 2 sin 2 (πΩ) ,
we obtain, for ℓ ∈ {2, 3}, that uniformly in ω,

℘ 1,ℓ (ω) → π α ℓ 2 1 sin 2 (πω/α ℓ ) - 1 3 .
In particular, setting

u(t) = sin 2 α 2 α 3 arcsin 1 3 + f (t) ω 2 π 2 -1/2 - π 2 ,
and taking the limit in (2.6), it follows that an admissible CGF for M 1 is given by

π α 3 2 1 u(t) - 1 3 .
Since any linear transformation of a CGF is a CGF, u(t) itself is a CGF for M 1 . We shall now show the existence of α = 0 such that in the neighborhood of t = 1,

u(t) = [α + o(1)]/[1 -t] α 2 /α 3 . (2.11)
By a direct calculation, it can be seen that for t ∈ [x 1 , 1],

1 3 + f (t) α 2 π 2 (2.12)
belongs to the segment [0, 1], equals 1 at x 1 and 0 at 1. In other words, in order to understand the behavior of u(t) near t = 1, it is necessary to analyze the asymptotics, as T → ∞, of the function

sin 2 α 2 α 3 arcsin{T } - π 2 .
For T 1, we have

arcsin{T } = 1 0 du (1 -u 2 ) 1/2 ± i T 1 du (u 2 -1) 1/2 = π 2 ± i ln T + (T 2 -1) 1/2 .
Hence, with sin(ix) = i sinh(x), we can write

sin 2 α 2 α 3 arcsin {T } - π 2 = -sinh 2 α 2 α 3 ln T + (T 2 -1) 1/2 = - 1 4 T + (T 2 -1) 1/2 2α 2 /α 3 + T -(T 2 -1) 1/2 2α 2 /α 3 -2 . When T → ∞, T + (T 2 -1) 1/2 = 2T + O(1/T ) and T -(T 2 -1) 1/2 = O(1/T ), so that u(t) = - 1 4 2 1 3 + f (t) α 2 π 2 -1/2 2α 2 /α 3 + O(1) .
As remarked earlier, the function defined in (2.12) has a zero at 1, which is simple. Equation (2.11) follows immediately, and, since conformal mapping conserves angles, the proof of the lemma is concluded.

Limit of the uniformization when passing from genus 1 to genus 0

The last step consists in connecting the angle θ to the group W introduced in Definition 1.3. In fact, we shall deal with the ratio α 3 /α 2 and prove that, when -→ M = 0, W can be interpreted as the group of transformations of C/(α 2 Z)

ω → -ω + α 2 , ω → -ω + α 2 + α 3 , which is of order 2 inf{ℓ ∈ Z * + : ℓα 3 /α 2 ∈ Z}.

As

-→ M → 0, the limit of the uniformization (2.2), after some algebra using the continuity with respect to the parameters {p i,j } -1 i,j 1 , becomes

         x(ω) = 1 + (x 4 -1)(1 -x 1 ) 1 -x 1 -(x 4 -x 1 ) sin 2 (πω/α 2 ) , z(ω) = π α 2 (x 4 -x 1 )(x 4 -1)(x 1 -1) sin(2πω/α 2 ) [1 -x 1 -(x 4 -x 1 ) sin 2 (πω/α 2 )] 2 ,
(2.13)

INRIA
where we have used the expression of α 2 given by equation (2.7). Setting u = exp(2iπω/α 2 ), the obvious identity sin 2 (πω/α 2 ) = -(u + 1/u -2)/4 allows to rewrite in (2.13) as x(ω) = x(u), where

x(u) = (u -z 1 )(u -1/z 1 ) (u -z 0 )(u -1/z 0 ) , (2.14)
and z 0 , z 1 are complex numbers given by

       z 0 = 2 -(x 1 + x 4 ) ± 2[(1 -x 1 )(1 -x 4 )] 1/2 x 4 -x 1 , z 1 = x 1 + x 4 -2x 1 x 4 ± 2[x 1 x 4 (1 -x 1 )(1 -x 4 )] 1/2 x 4 -x 1 .
Clearly, y(u) will be also a rational function of the same form as x(u), but its explicit computation hinges on some general properties of Riemann surfaces. the functions (2.14) and y(u) are automorphic and provide a rational uniformization of the algebraic curve K defined in (1.3), which is here of genus 0. Moreover, this unifomization is unique up to a fractional linear transformation (see, e.g., [START_REF] Springer | Introduction to Riemann Surfaces[END_REF]). Before deriving the formula for y(u), let us quote right away a first limit automorphism, namely ω → -ω + α 2 , since x(ω) = x(-ω + α 2 ).

Equivalently, we have

x(u) = x(1/u),
which corresponds precisely to the automorphism of C ξ(u) = 1 u .

(2.15)

Similarly, exchanging the roles of x and y, a continuity argument when -→ M → 0 in the general uniformization given in (2.2) or (2.3) yields y(ω) = y(-ω + α 2 + α 3 ).

In particular, ω → -ω + α 2 + α 3 is the second automorphism. Indeed, z(ω) = -z(-ω), so that y(ω) = y(-ω). Moreover x(ω) = x(-ω + α 3 ), since, by taking the limit of the genus 1 in (2.4), we get α 3 < α 2 with a strict inequality (see formula (2.21) in Section 2.4). To derive y(u) quickly (and nicely!), we shall make two observations.

• Exchanging the roles of x and y would lead to another rational uniformization with some parameter v, such that

y(v) = (v -z 3 )(v -1/z 3 ) (v -z 2 )(v -1/z 2 ) , (2.16)
where z 2 , z 3 are obtained from z 0 , z 1 , just replacing x 1 , x 4 by y 1 , y 4 , respectively.

• From the above remark, we necessarily have

y(u) = y(σ(u)),
where σ(u) = eu + f gu + h , (e, g) = (0, 0), ehf g = 0 is a linear transformation which will be completely determined.

Since x(0) = x(∞) = 1, see (2.14), we must have y(0) = y(∞) = 1 (in the case -→ M = 0, Y 0 (1) = Y 1 (1) = 1, see Section 2.1). Similarly, (2.16) yields y(0) = y(∞) = 1. These simple equalities entail at once σ(u) = ρu, where ρ is a complex number, to be calculated later. Then, by (2.16), we obtain

y(u) = (ρu -z 3 )(ρu -1/z ) (ρu -z 2 )(ρu -1/z 2 ) , ( 2
.17) and the second limit automorphism η becomes on C

η(u) = 1 ρ 2 u . ( 2
(u) = (η • ξ)(u) = u ρ 2 .
In the genus 0 case, the group W therefore has the order (possibly infinite)

2 inf{ℓ ∈ Z * + : ρ 2ℓ = 1}.
At that moment, we make a brief detour to connect (2.14) with the uniformization proposed in [4, equation 6.5.11], namely

y(t) = y 1 + y 4 2 + y 4 -y 1 4 t + 1 t . Setting T (v) = vz 2 -1 v -z 2 ,
one easily checks the identity y(v) = y(T (v)), which by (2.17) yields in particular y(1) = y(-1) = y 1 = y(1/ρ), whence, using the properties of the algebraic curve, x(1/ρ) = X(y 1 ). Finally, by (2.14), we obtain that ρ is a root of the second degree equation

ρ + 1 ρ = 2 x 1 + x 4 -2x 1 x 4 + (x 1 + x 4 -2)X(y 1 ) (x 4 -x 1 )(1 -X(y 1 )) . ( 2

.19)

The roots of (2.19) are complex conjugate and of modulus one. Choosing then the root satisfying arg(ρ) ∈ [0, π] and letting

Λ = x 1 + x 4 -2x 1 x 4 + (x 1 + x 4 -2)X(y 1 ) 2[(X(y 1 ) -x 1 )(X(y 1 ) -x 4 )(1 -x 1 )(x 4 -1)] 1/2 , (2.20) a direct algebra yields arg(ρ) = π 2 -arctan(Λ).
To conclude the proof of Part (I) of Theorem 1.4, it suffices to apply equation (2.21), derived in Proposition 2.4 below, which makes the link between arctan(Λ) and the ratio α 3 /α 2 . All conclusions so far obtained remain true, and even easier to prove, when C = 0 (see (2.1) and the assumption made at the beginning of Section 2.2). Proof. A straightforward calculation carried out along the same lines as for the derivation of formulae (2.7) yields, for any t < x 1 , .

x 1 t dx (1 -x)[C(x -x 1 )(x -x 4 )] 1/2 = 1 [C(x 4 -1)(1 -x 1 )]
Instantiating now t = X(y 1 ) < x 1 in the last formula, we get exactly

α 3 α 2 = 1 2 - arctan(Λ) π .
(2.21)

3 Proof of Part (II) of Theorem 1.4

In this part we prove that in all situations (A.1), (A.2), (A.3) and (A.4) listed in the appendix, the group W is infinite. In fact, it is enough to prove this result for only one case, since two groups corresponding to jump probability sets obtained one another by one of the eight symmetries of the square are necessarily isomorphic, see [START_REF] Fayolle | Random walks in the quarter plane[END_REF]Section 2.4]. We choose to focus on the particular case (A.2).

Like for the analysis of the case -→ M = 0, we consider the genus 0 case (A.2) as a continuous limit of the genus 1 case. All the results of Section 2.1 still hold, with x 3 = x 4 = ∞. Accordingly, letting the parameters {p i,j } -1 i,j 1 be distorted so as to yield (A.2), we get from equations (2.4)

   ω 1 → iα 1 , with α 1 ∈ (0, ∞), ω 2 → ∞, ω 3 → α 3 ∈ (0, ∞).
In particular, for the same reasons as in Section 2.3, the limit group can be interpreted as the group of transformations ω → -ω, ω → -ω + α 3 on C/(α 1 Z). This group is obviously infinite, and so is W .

The proof of Theorem 1.4 is terminated.

RR n°7555

1 )

 1 Let X(y) [resp. Y (x)] be the algebraic function defined by (1.2), so that K(X(y), y) = 0 [resp. K(x, Y (x)) = 0]. This function has two branches, say X 0 and X 1 [resp. Y 0 and Y 1 ], and we fix the notation by taking |X 0 | |X 1 | [resp. |Y 0 | |Y 1 |]. Denote by {y ℓ } 1 ℓ 4 the four roots of D(y)-they are branch points of the Riemann surface (1.3). They are enumerated in such a way that |y 1 | |y 2 | |y 3 | |y 4 |. Moreover y 1 y 2 , [y 1 y 2 ] ⊂ [-1, +1] and 0 y 2 y 3 . The branches X 0 and X 1 are meromorphic on the complex plane cut along [y 1 y 2 ] ∪ [y 3 y 4 ]. Similar results hold for D(x), exchanging y and x.

INRIA 2 . 4 AProposition 2 . 3 .= π 2 -

 24232 second form of the criterion for the zero drift case By computing the limit ratio α 3 /α 2 in a different manner, we shall derive another expression of the angle θ defined in equation (1.7). The following proposition holds. With Λ defined in (2.20) and its equivalent Λ obtained from Λ by exchanging x and y, we have θ arctan(Λ), so that Λ = Λ and the order of the group W equals 2 inf{ℓ ∈ Z * + : ℓ[1/2arctan(Λ)/π] ∈ Z}.

  see(1.4) and[4, equation (3.1.10)], and the necessary and sufficient condition for the group to be finite has been quoted in(1.6).

2.1.2 Conformal gluing in the genus 1 case Definition 2.1. Let D ⊂ C ∪ {∞} be an open and simply connected set, symmetrical with respect to the real axis, and different from ∅, C and C ∪ {∞}. A function w is said to be a conformal gluing function (CGF) for the set D if • w is meromorphic in D;

  + x 4 -2x 1 x 4 + (x 1 + x 4 -2)t 2[(tx 1 )(tx 4 )(x 4 -1)(1x 1 )] 1/2

	1/2	π 2	-arctan	x 1

RR n°7555

Acknowledgments. K. Raschel would like to thank I. Kurkova and J.S.H van Leeuwaarden for stimulating discussions. His work was supported by CRC 701, Spectral Structures and Topological Methods in Mathematics at the University of Bielefeld. INRIA

Miscellaneous remarks

For the sake of completeness, we quote hereaftert some facts related to existing works.

• Theorem 1.4 is quite simple to check, and therefore provides a really effective criterion.

• A direct calculation gives ∆ = -a(1) a(1) -1 i,j 1 ijp i,j , with ∆ defined in (1.5).

Hence, for the group of order 4, our criterion clearly agrees with that of [START_REF] Fayolle | Random walks in the quarter plane[END_REF].

• From Theorem 1.4, it becomes clear that the famous Gessel's walk (i.e. with jump probabilities satisfying p 1,0 = p 1,1 = p -1,0 = p -1,-1 = 1/4) has a group of order 8. More generally, Theorem 1.4 leads to another proof of the (non-)finiteness of the group for all combinatorial models (with an underlying genus 0) appearing in [START_REF] Bostan | The complete generating function for Gessel's walk is algebraic[END_REF][START_REF] Bousquet-Mélou | Walks with small steps in the quarter plane[END_REF].

• In [2, Section 3], the non-finiteness of the group for some models (including particular instances of case (A.4)) has been proved via two approaches based on valuation and fixed point arguments, but no general criterion was really obtained. It is also worth noting that there the group was not restricted to the algebraic curve, so that Theorem 1.4 is, in a sense, more precise.

• As an other straightforward consequence of Theorem 1.4, one can check the random walk considered in [START_REF] Raschel | Green functions for killed random walks in the Weyl chamber of Sp(4)[END_REF], with

has a group of order 2n, for all n 3.

• The angle θ defined in (1.7) gives the angle of the cone in which, after a suitable linear transformation, the random walk with transitions {p i,j } -1 i,j 1 has a covariance which equals to some multiple of the identity.

A About the genus

Introduce the drift (mean jump vector) -→ M = 0, p 0,1 = p -1,0 = p -1,1 = 0, (A.1) p 1,0 = p 1,-1 = p 0,-1 = 0, (A.2) p 1,0 = p 0,1 = p 1,1 = 0, (A.3) p 0,-1 = p -1,0 = p -1,-1 = 0.

(A.4)


Jean Michel Muracciole

Bertrand Wattrisse

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