Lom Messan Hillah

Rodrigo Assad
email: assad@usto.re

Antonia Bertolino
email: antonia.bertolino@isti.cnr.it

Marcio Delamaro
email: delamaro@icmc.usp.br

Fabio De Rosa
email: fabio.de-rosa@simple-eng.com

Vinicius Garcia

Francesca Lonetti
email: francesca.lonetti@isti.cnr.it

Ariele-Paolo Maesano
email: ariele.maesano@simple-eng.com

Libero Maesano
email: libero.maesano@simple-eng.com

Eda Marchetti
email: eda.marchetti@isti.cnr.it

Breno Miranda

Auri Vincenzi

Juliano Iyoda

Towards automated deployment of self-adaptive applications on hybrid clouds

Cloud computing promises high dynamism, flexibility, and elasticity of applications at lower infrastructure costs. However, resource management, portability, and interoperability remain a challenge for cloud application users, since the current major cloud application providers have not converged to a standard interface, and the deployment supporting tools are highly heterogeneous. Besides, by their very nature, cloud applications bring serious traceability, security and privacy issues. This position paper describes a research thread on an extensible Domain Specific Language (DSL), a platform for the automated deployment, and a generic architecture of an ops application manager for selfadaptive distributed applications on hybrid cloud infrastructures. The idea is to overcome the cited limitations by empowering the cloud applications with self-configuration, self-healing, and self-protection capabilities. Such autonomous governance can be achieved by letting cloud users define their policies concerning security, data protection, dependability and functional compliance behavior using the proposed DSL. Real world trials in different application domains are discussed

Introduction

Cloud computing is used to provision the physical resources (servers, storage, network) of the digital ecosystem, allowing a substantial optimization of the operating costs. However, cloud computing is more than a cost-optimizing technology. It bears to users significant features -virtualization, job scheduling, and programmability -allowing the sustainable implementation of robust scalability, availability, and serviceability requirements on commodity hardware.

However, the most notable features of cloud computing, such as virtualization, come at the price of increased security and data protection risks. Moving to a virtualized environment does not free from the security risks already faced in the physical environment, but rather introduces new ones, related to virtual machine and network management, resource exhaustion, hypervisor vulnerabilities, multi-tenancy handling, and cloud access control.

DevOps is an emerging paradigm of integration of the development process within the production stage. To adopt DevOps effectively, cloud application developers have yet to find solutions to tough problems: i) how to design, develop, deploy, and operate efficiently applications that fulfill, on one side, stringent scalability, availability and serviceability needs, and, on the other side, strict security and data protection obligations; ii) how to cope with the heterogeneity and lack of interoperability of cloud infrastructures and the consequent lack of portability of cloud applications; iii) how to combine the advantages of agile, flexible, and continuous integration, testing, delivery, and deployment, and of mission-critical quality assurance, test, and verification.

The main solution of the aforementioned problems is the automation of De-vOps jobs, in particular test, configuration, deployment, and ops management. This position paper introduces an ongoing research on automating installation, configuration, startup, and operation management of self-adaptive distributed applications on hybrid cloud infrastructures. In particular, it presents an envisaged solution based on three correlated research topics: (i) a declarative, cloud agnostic, and extensible Domain Specific Language (DSL) for structural and behavioral modeling and policy definition for automated deployment and selfmanagement; (ii) a generic and instantiated architecture of an autonomic ops application manager enabling self-configuration, self-healing, and self-protection; (iii) a DSL workbench as a service, equipped with editors, wizards, consoles, and dashboards for deployment and monitoring of self-adaptive cloud applications.

Related work and background

Self-aware management is becoming commonplace to address the scale, growth, and reliability of cloud applications. The authors of [IZM + 17] propose a conceptual framework for analyzing the state-of-the-art and comparing practical characteristics, benefits, and drawbacks of self-awareness approaches used for cloud applications in different domains. A big challenge of self-aware and adaptive distributed systems on cloud is achieving self-protection as well as guaranteeing self-configuration, self-healing, and self-optimization. Aceto et al. [START_REF] Aceto | Cloud monitoring: A survey[END_REF] examine current platforms and services for cloud monitoring pointing out their issues and challenges whereas the authors of [START_REF] Vanitha | Software testing in cloud platform: A survey[END_REF] investigate testing models, recent research works, and commercial tools for cloud testing. However, the main open issues emerging from the analysis of the literature related to the management of elasticity, dependability, and security of cloud applications are mainly about the limitations of flexibility and portability in a multi-cloud environment.

The OASIS TOSCA [START_REF]TOSCA Simple Profile in YAML Version 1.0[END_REF] standard covers the cloud-portable automation of installation, configuration, and startup of conventional cloud applications. TOSCA is a declarative language that let model the distributed application topology independently from the particular target cloud infrastructure. It uses the following concepts: Nodes -nodes represent components of an application or service and their properties; example nodes are computer, network, storage (i.e. infrastructure-oriented), OS, VM, DB, Web Server (i.e. platform-oriented), functional libraries, or modules (i.e. applicative); (ii) Relationships -they represent the logical relations between nodes (e.g. hosted on, connects to), and describe the valid source and target nodes they link together; (iii) Artifacts -they describe installable and executable objects required to instantiate and manage a service; (iv) Service Templates -they group the nodes and relationships that make up a service's topology. In summary, TOSCA DSL allows describing a distributed cloud application at the infrastructure level in a portable way. Current TOSCA implementations target the main cloud provider infrastructures.

Outline of the solution

A DSL for self-adaptive cloud applications. We plan to overcome the limitations sketched in the section above by extending the TOSCA standard and implementations. The planned extensions of the TOSCA standard are about: (i) language traits for the installation, configuration, and setup of security and data protection provisions; (ii) language traits for structural and behavioral modeling of distributed applications, beyond the infrastructure level, at the application/service level; (iii) a policy description language that enables the definition of self-configuration, self-healing, and self-protection policies to be fulfilled at runtime. Security and data protection provisions to be automatically installed, configured and setup with the DSL deal with standard authentication, confidentiality, and integrity. Besides the DSL language traits to efficiently and conveniently express these security requirements, the implementation shall take into account security matters that are particular to cloud deployment. These concerns are about multi-sites communication, different hypervisors (code which manages the virtual machines), different hypervisor provided services that could have various security issues or expose security loopholes, different CPU/memory/storage. The TOSCA standard already provides a rich and flexible language for structural modeling (topology) at the infrastructure level. We plan to enrich the language with traits for structural modeling at the applicative level, by describing a distributed application as a graph of logical components connected by service dependency wires. At deployment time, this structure shall be installed, configured and setup at the applicative level too. This structural model is referenced by the self-configuration, self-healing and self-protection policies. Behavioral modeling shall leverage an existing standardized notation, such as the State Chart XML (SCXML) [W3C14]. SCXML provides a powerful, general-purpose and declarative modeling language to describe the behavior of timed, event-driven, state-based systems. Therefore, in our context, state machines shall describe the external interactions between service components, explicitly showing the states of the conversation of each component with its wired interlocutors. Behavioral modeling will enable runtime checks and adaptation thanks to monitoring. The runtime policy language extension shall enable: (i) non-intrusive logging of events at the infrastructure and the applicative levels; (ii) non-intrusive monitoring (analysis of the logging stream) with slightly delayed evaluation of infrastructure events and distributed application behavior (asynchronous passive testing); (iii) non-intrusive active testing of the deployed application in a concurrent staging environment; (iv) intrusive active testing in the production environment to check application robustness and fault tolerance; (v) runtime installation, configuration and setup of components without service interruption, including new version deployment and version backtracking; (vi) automatic elasticity (scalability up and down); (vii) server failover (self-recovery); (viii) masking of transient network failures; (ix) circuit breaking at the application level; (x) generalized timeout management. An autonomic ops application manager shall implement and enforce these policies at run time.

The autonomic ops application manager. The classical autonomic architecture combines the managed application and an autonomic manager that oversees it. The abstract (platform independent) architecture of our ops manager includes (i) a supervisor, and (ii) a collection of concurrent feedback loops. Each feedback loop monitors and analyses a particular aspect of the managed application and its cloud environment and, if needed, plans and executes an adaptation process, driven by DSL policies, which brings the managed system to a new state. The abstract (platform independent) model of the feedback loop is the original MAPE-K generic architecture [START_REF] Didac | MAPE-K Formal Templates to Rigorously Design Behaviors for Self-Adaptive Systems[END_REF]. The supervisor controls the concurrent feedback loops and handles the interaction with the user, through the monitoring facility of the DSL workbench. There are no interferences between the managed application and the Ops manager, except for those performed by the adaptation processes. Significant concerns for the supervisor are the stability, accuracy, short settling-time, robustness, termination (no deadlock), consistency, scalability, and security of the adaptation processes. Important research questions are (i) the coordination of autonomous ops manager within cross-organizational distributed applications, and (ii) the DSL policy change at run time.

The DSL workbench. The DSL workbench (WB) is a Platform as a Service deployed on the cloud. It shall be accessible by the user via i) the Web-based Graphical User Interface, and ii) the WB API, a REST interface. The workbench is composed of three main layers: GUI, Processing, and Storage. The GUI Edit wizard allows: i) easy graphical drafting of the DSL artifacts (models, policies), ii) reverse engineering of existing legacy artifacts into the DSL ones, and iii) straightforward building of the DSL archives (sets of models and policies for an application to be deployed) and self-adaptive application releases (AR). The GUI Deploy wizard lets initiate and supervise the deployment of a AR on the target Limitations of the approach. The main limitations of the proposed approach relate to: i) the fact that the proposed DSL is based on TOSCA and could collide or overlap with other emerging standards; ii) the inability or difficulty to integrate the proposed solution within all the available cloud provider infrastructures (i.e. Microsoft Azure, Google Cloud, etc.); iii) the lack of optimization methods for self-adaptive applications on cloud, except for the automatic scal-ing up and down -the current project focuses on security, dependability, and fault-tolerance, not on self-optimization.

Real-world trials

We plan to try our solution with real-world applications and systems, in particular in the logistics and high-tech industries. In the logistics domain, decision making is distributed. All stakeholders make decisions locally and autonomously, so the most important challenge is to achieve collaborative decision-making in practice. The proposed solution can be adopted in the logistic domain for building a cloud-based logistics information platform, as a general exchange platform, where cloud services are composed to collect, classify, store, analyze, evaluate, publish (release), manage and control relevant information on interorganizational logistics operations, processes, and management.

The proposed solution shall be applied to simplyTestify [sim], which is a geo-distributed, multi-instance and multi-tenant PaaS offering self-provisioning and pay-as-you-go test automation services. Even if simplyTestify core modules have been designed and implemented for cloud portability, the implementation of strong elasticity, dependability, security, and performance requirements is IaaSdependent (the current version of simplyTestify runs on the Amazon Web Services public cloud). The DSL, the autonomic ops application manager, and the workbench shall allow the automatic installation, configuration, and startup of the PaaS and the policy-driven implementation of the mentioned requirements in a hybrid cloud including private and other public clouds, such as Microsoft Azure, Google Cloud, IBM Cloud, etc.

Fig. 1 .

 1 Fig. 1. General Architecture


Lucas Garnier
email: lucas.garnier@univ-rennes1.fr

C Saavedra

Rigoberto Castro -Beltran

José Luis Lucio

Etienne Gaviot

R Castro-Beltrán

B Bêche

Hybrid composed method associating conformal transformation with matrix formulation for computing eigenvalues and eigenvectors in bended optical waveguides

Keywords: Integrated optics, hybrid composed method, mathematic conformal transformation with multilayer formalism methods

come

Accurate knowledge of how light behave within a waveguide with a given radius of curvature becomes more and more important because of the development of integrated photonics [START_REF] Adams | Introduction to optical waveguides[END_REF][START_REF] Tamir | Guided-wave optoelectronics[END_REF][START_REF] Snyder | Optical waveguide theory[END_REF], which include ring micro-resonators, phasars, and other devices fitted with a radius of curvature. Sometimes, it can be original and interesting to modify, adapt, enhance or merge various methods of resolution [START_REF] Bêche | Another way to shape the comprehensive analytical approach describing electromagnetic energy distribution through four-slab-layer structures[END_REF][START_REF] Begou | Marcatili's extended approach: comparison to semi-vectorial methods applied to pedestal waveguide design[END_REF]. This work presents a hybrid numerical calculation method (HNM) to determine the eigenvalues (effective indices) and eigenvectors related to the optical field within a curved waveguide. This method is a hybrid method which uses at first an apt conformal transformation (CT) of the complex plane plus secondly a matrix multilayer formalism (MMF) developed at the origin for real materials. As a 'o' composition law, we can write HNM ≡(MMF o CT). The aim of the conformal transformation is to convert a curved geometry into a virtual straight geometry structure; such a conversion entails a significant effect on the index profile of the structure and modifies its pattern. Various kinds of mathematic conformal transformations can be considered [START_REF] Murray | Variables complexes[END_REF], and used in physics, optoelectronics and hyper-frequency such as: inversion, rotation, homographic or more particular transformations like the Schwarz-Christoffel or Joukovsky ones; the specific one used in this work will be detailed later. In our case, it will be a shortcut and a mathematical trick that makes it possible to dispense with the 'true' Jacobian method and the equation of propagation in polar coordinates. Then, as regards the modified index profile, we use the matrix multilayer formalism first developed by P. Yeh and A. Yariv to describe the behavior of light in stratified structures composed with layers with different indices [START_REF] Yeh | Electromagnetic propagation in periodic stratified media. I. General theory[END_REF][START_REF] Adams | Introduction to optical waveguides[END_REF]. So as to take advantage from this formalism, the modified index profile obtained thanks to the CT is discretized into virtual layers prior to apply the MMF. In our case, the MMF will be used to follow step by step (discretization) the profile created by the CT. This method allows to make the calculation on a personal computer in a very short amount of time, and its results can be compared to that of the commercial software COMSOL-3D [8].

Let us consider a slab waveguide of width L (Fig. 1) on which will be applied, firstly a conformal transformation (CT) of the complex plane (Fig. 2a) and secondly a matrix multilayer formalism (MMF, Fig. 2b). By way of the conformal transformation, one can go from a z = x + iy = ρe iθ complex plane to a pattern transformed as: w = u + iv = f(z). The goal of this transformation is to take advantage of calculations for a non-bended effective waveguide. To achieve this goal, the apt transformation is w = R ext ln(z/R ext ) with R ext the external radius of curvature of the guide. The following relations can thus be found (Fig. 2a):

ρ = √x 2 + y 2 = R ext ln ( z R ext ) and θ = Arctan ( y x ) = v R ext (1)
In the z-space, the wave equation governing the propagation of light in the waveguide is [Δ x,y + k 0 2 n 2 (x, y)]. ϕ(x, y) = 0, with ϕ ≡E or H that represent respectively the electric or magnetic field associated to the wave. By carrying out basic calculations based on partial differentials as

∂ ∂( x y ) = ∂u ∂( x y ) ∂ ∂u + ∂v ∂( x y ) ∂ ∂v
plus the second order, it will occur

∂ 2 u ∂( x y ) 2 = R ext (x 2 +y 2 )-2R ext ( x y ) 2 (x 2 +y 2 ) 2 and ∂ 2 v ∂( x y ) 2 = (±) 2xyR ext (x 2 +y 2 ) 2 .
Then, a new Laplacian can be expressed by such a way with Δ x,y = Δ u,v exp(-2u/ R ext ). So the wave equation becomes:

[Δ u,v + k 0 2 . n 2 (u)exp(2u/R ext )]. ϕ(u, v) = 0 (2)
This equation is similar to the previous one but with a correction term on the index that becomes n(u). exp(u/R ext ). Thanks to the CT, it is possible to compute the modes in a curved waveguide by doing the calculation on a straight waveguide fitted with a modified index profile. This new profile will be sliced into virtual layers in order to carry out the multilayer matrix formalism. To each virtual layer i with its index n i , this formalism associates a matrix M i that can take both different forms according to the oscillating or evanescent natures of the fields [START_REF] Adams | Introduction to optical waveguides[END_REF]:

M i = ( cos(q i l i ) j q i sin(q i l i ) jq i sin(q i l i ) cos(q i l i ) ) , for n i > n eff (3)
and,

M i = ( cosh(ϑ i l i ) j ϑ i sinh(ϑ i l i ) jϑ i sinh(ϑ i l i ) cosh(ϑ i l i ) ) , for n i < n eff (4)
with, j 2 = -1 the imaginary number, q i = √k 0 2 n i 2 -β 2 , ϑ i = jq i = √β 2 -k 0 2 n i 2 , l i the length of the layer, n i the index of the i th layer, k 0 =2/ 0 the wave-vector,  0 the wavelength, and β = k 0 n eff the effective propagation constant that represent the eigenvalue of the system exactly as the effective index n eff . The matrix describing the whole structure is the product of each one of the local matrices:

M ̅ = ∏ M i N i=1
= ( m 11 m 12 m 21 m 22 ) with N the total number of layers (Fig. 2b).

One may consider now the eigenvalue equations of such a structure given by (TE/TM modes):

{
with, n p =n(-2u) and n r =n(u=0)=n clad into the TC index profile (4 th diagram in Fig. 2a).

Considering the previous equation ( 5) theory of multilayer structures, both terms n p and n r are respectively the indices of the substrate and superstrate. In the case of virtual layers on an index profile modified by a CT, as the index continuously varies, there is no such thing as constant substrate or superstrate, so the choice of the values of n p and n r has an influence on the calculation of eigenvalues per definition (see n p onto the 4 th diagram Fig. 2a). In this computing method, the chosen values are the lowest of each side of the cladding (that is n p on the left side area and n r =n clad on the right side area): then it depends on the lower bound chosen for the coordinate -2u that has been taken equal to -2R ext ln(R ext R int ⁄ ) so that each region (the left side, the core, and the right side) has the same length and the same number of i-iterations concerning the MMF. As an example, the computation of the TE mode eigenvalue equation for different values of R ext , L the width of the guide, and (n core ; n cladd ) yields relevant results summarized in Table 1 plus the comparison with COMSOL-3D computing.

It is clear that the results provided by the hybrid numerical method (HNM) present the same global kind of variation in function of the different opto-geometric parameters (R, L, n i ,  0 ) of the structure, providing then a most valuable approximation of actual values. For the majority of the cases (see Table 1), precision can be observed weaker for L = 2µm than for L = 1µm. Indeed, as the three regions (cladding on the left, core, and cladding on the right) feature a same length (4 th diagram, Fig. 2a), the value of the lower bound n p =n(-2u) is defined by the effect of the CT and then the value of R ext . The value n p (and also (n r -n p ) due to the fixed value of n r =n clad =n(u=0) after the CT) for respectively a width L=2µm will be below a value of n p for L=1µm. It is also remarkable that the greater R ext is, the more precise the results are. This can be explained by the fact that, the smaller R ext is, the more enforced will be the CT effect on the virtual index directly proportional to the factor e (u/R ext ) , which will increase the value(n r -n p ) = (n clad -n(-2u)). Indeed, if we consider the specific asymptotic case R ext → +∞, it will lead to e (u/R ext ) → 1 and n p =n clad =n r representing no effect of CT. We can notice that the value n p , defined by the effect of CT and also the distance L according to Fig. 2a, is squared in the impact calculation of p into [START_REF] Begou | Marcatili's extended approach: comparison to semi-vectorial methods applied to pedestal waveguide design[END_REF].

Concerning now the right area, as depicted on Fig. 2a, the index is increased on such area of the waveguide in the w-plane, which corresponds to the external side of the guide in the z-space, so we can expect the light to be deported outwards, in the same way that one could figure about centrifugal acceleration (Fig. 3). The mechanism to shape successively the amplitude of the discretized optical mode by classical MMF is defined by the procedure: Field initial=1 = M ̅ . Field final=N . The values of Δx peak effectively highlight such a phenomenon obviously, and, as may be expected, the greater R ext is considered, the smaller Δx peak is, varying down to zero as R ext tends to infinity, which yields the previous asymptotic case of a straight waveguide. As regards index contrast, we can also point out the fact that for (n core ; n cladd ) = (2; 1.5), Δx peak is greater than for (n core ; n cladd ) = (2; 1) or (n core ; n cladd ) = (2.5; 1.5). Indeed, as the difference between n core and n cladd is reduced, the mode becomes less guided, so the more it is deported outwards. To conclude, all the previous classical physical behaviors in propagation are fortunately found in the results of such hybrid numerical method HNM = (MMF o CT) computing without requiring a tremendous power of calculation. This hybrid method allows valuable approximation of the characteristics of such structures. of the Z-plane to the W-plane. The Z-plane corresponds to the real configuration where the waveguide is curved, on which the operation w=R ext .ln(z/R ext ) is applied prior to further calculations on a waveguide in the W-plane that is devoid of any radius of curvature, but is fitted with a modified index profile. b) Multilayer matrix formalism (MMF) that allow to slice the index profile into virtual layers so as to determine the eigenvalues (effective propagation constant or effective index) and the eigenvectors (optical field) in such a structure. As an example, this table represents the eigenvalues of the TE 0 -modes for different width (L) of the guide and for different indices regarding the cladding (n clad ) and the core (n core ) for an optical excitation of wavelength λ 0 =1µm. These values are compared to those given by COMSOL-3D software. The Δx peak represents the shift between the positions of the maxima of intensity of the light and the center of the guide (where the maximum is when considering a straight guide).

Figure caption : Figure 1 :

 :1 Figure caption:

Figure 2 :

 2 Figure 2: Explanations about both elements of the hybrid method. a) conformal transformation (CT)

Figure 3 :

 3 Figure 3: Representation of the optical field in a non-curved waveguide (dashed line) and in a waveguide with a 10μm radius of curvature (continuous line). The term Δx peak represents the shift of position of the maximum regarding intensity between the two cases.

  Fig 1:

  Fig 2:

Table caption :

 caption


Introduction

Let us consider land rent as a monetary surplus generated by a producti ve activity. Land is allocated to the activity which yields the highest rent, as a result of a bidding process. This principle which THÜNEN (1826) and later [START_REF] Lösch | Die räumliche Ordnung der Wirtschaft, Iena, Gustav Fischer. English trans[END_REF] applied to agricultural location, leads to the famous theory of concentric circles which is still at the heart of present day urban structure theories. These theories concentrate on uti lity and demand rather than on the conditions of production.

In a different approach, the aim of this paper is to generalize some aspects of the allocation of land to productive activities. For diffe rent activities, we shall define different vectors of technical coeffi cients, showing different degrees of intensity in the use of land, labor and capital. Then, we shall try to determine how price-, cost-and dis tribution parameters influence the relative locations of these activi ties.

This work is related to papers by [START_REF] Scott | Land Use and Commodity Production[END_REF]SCOTT ( , 1979SCOTT ( , 1980) ) and by [START_REF] Barnes | Technical Choice and Reswitching in Space Economies[END_REF]. We shall use rent equations which are formally con nected with SRAFFA's equations of production [START_REF] Sraffa | Production of Commodities by Means of Commodities[END_REF]), yet we shall not concentrate on distribution-price relations, as SRAFFA's fol lowers do (for example, ABRAHAM-FROIS and[START_REF]Théorie de la valeur, des prix et de l'accumulation[END_REF][START_REF]The Urban Land Nexus and the State[END_REF], but shall seek merely to highlight spatial patterns. The interrelated endo genous variables of neo-Ricardian economics become here exogenous pa rameters. On the contrary, distances and locations, which are simply ignored by the latter school, now become endogenous variables. More precisely, we need the following assumptions.

Space is like a homogeneous THUNEN's plain with a central market where all the product is sold and/or distributed. Land is privately owned and the landowner and the land user are different.

Production is described by fixed coefficients for the use of land, labor, capital and transport. There is only circulating capital. Produc tion of one commodity can be achieved by different techniques, or dif ferent combinations of factors, but only one technique is used at any given place.

The value of the product is distributed in the form of wages, profits and land rent, relations between costs, production and distribution being viewed in the classical framework of advance economics [START_REF] Schumpeter | History of Economic Analysis[END_REF]. At the beginning of the production period, producers must advance capital outlay, including the cost of transporting this capital from the central market where it is bought. At the end of the production period the producer must pay the cost of transporting his product to the market. It will be assumed that the product is sold.

After the capital has been reconstituted, the net income from the sale is distributed, i.e. wages, profits and rents are paid.

Land rent is both differential and residual. It is paid after the insti tutional wage rate and the normal profit rate have been deducted. Land rent also plays an active role in determining locations according to THUNEN's principle.

Differential rents stem from differences in locations, i.e. from diffe rences in transport costs to the center, and from differences in the techniques employed. The analysis uses comparative statics. Every com modity produced by a given technique generates a rent depending on the technical coefficients, the market prices, the distribution variables and the cost of transport to the center. The equilibrium allocation of land is characterized by spatial variables, namely the limits of rings corresponding to the different commodities or techniques. The values of these variables are obtained by solving a system of rent equations, from which we can see how location depends on technical price and distribution parameters.

While section 2 presents the structure of two very simple models, sec tion 3 gives some elements of analysis, with theoretical exercices illustrated by imaginary numerical examples.

Two models of land allocation

These models have been kept as simple as possible. The first one consi ders only one commodity produced by two alternative techniques. The second one deals with two commodities, each one being produced in this case by only one technique.

One commodity , two techniques

Only one commodity -say corn-is produced and used as capital in produc tion and as consumer commodity. Wages, profits and rents are paid in terms of this commodity. There is no price involved, because only corn is circulating. The commodity can be produced by one of two fixed tech niques, identified by the superscript k(k = 1,2). One unit of land prok k duces x units of commodity by technique k (x is the inverse of the quantity of land used per unit of commodity). The production of one unit k k of commodity uses a units of capital (the commodity itself) and 1 units of labor, w is the wage rate and r is the profit rate, both of which are uniform. The cost of transport per unit of distance and per unit of com modity is the transport rate, x . The land rent, p, depends on the tech nique, k, and on the distance to the center, 6 ; we shall call this rent Pk (6).

In a one-commodity-economy, there is no exchange, and the center cannot be a market. It retains its attractive power if we assume that it is the sole distribution place, meaning that the payment of wages, profits and rents are made at that place. When one unit of commodity is produced, the quantity a^ is kept as capital and the quantity 1 -a* = t^ has to k k be transported to the distribution center ; we have 0 < t < 1. t is a coefficient of production. Let us distinguish the two techniques in as suming that technique one is more land-intensive (or that it makes the land less productive), and that it is less labor-and capital-intensive than technique two. This means that :

1 . 2 x < x ,
(1 ) a < a2 , hence t^ > t2 l1 < l2

Finally, we have to specify the spatial variables. For a given technique, the land rent is a decreasing function of distance. So, we can define the distance 6^ such that (6^) = 0.

Since each rent function is linear, there exists a unique value of 6 such that p1(S) = p2(S).

-1 2 -6 is the boundary between the two technique rings.6 , 6 and 6 determine the relative circular locations of the techniques, as shown on figure 1.

Figure 1. The spatial variables

According to the definition, the expression of the rent per unit of land, when technique k is used, is :

/n\ k k k,. \ ,k k ,k k . . oN (2) p (6) = x -a x (1 + r) -1 x w -t x t6 , (k : 1,2)
where the negative terms measure the amounts of commodity used to recons titute capital (akxk), to pay profit (akxkr), wages (lkxkw) and transport costs (tkxkt6). k k k At distance 6 , we have p (6 ) = 0, and, from (2) :

k k k/. v ,k k .k k rk " , x -axU+rj-lxw-txxo =0 (.k = 1,2)
, or more simply :

(3) 1 -ak(l + r) -lkw -tkx6k = 0 (k = 1,2). 1 2 This system can be solved in 6 , 6 . Then we can write the rent functions in terms of these values :

(4) p k(6) = tkxkT(Sk -6).

According to the definition of 6, we have :

t1x1x(S1 -Z) = t2x2x(62 -S) , thus (5)
t1x1(61 -6) = t2x2(S2 -6), which we can solve in 6.

From (3), we find immediately :

( 6 ) S k = - , ( k = 1 , 2 ) . t T k
For given techniques, the precise value of 6 depends on the values taken by r, w and t .

From ( 5) and (6), we have :

J 1 ,1 -a1(1 + r) -l1w -fn ,2 2 ,1 -a2 (1 + r) -l2w t X ( . ------i-----------6) = t X (--------o----------to) , t 1 T t T
which, after some arrangements, gives :

2/. 2v 1,, 1, , 1 1 2 2. n 1 1 ,2 2n ■ 5 -x(1-a)-x(1-a)+(ax -a x ) r + ( 1 x -lx)w (J2 2 .1 1v t(t X -t X ) k k
Remembering that t = 1 -a , and putting k k k a x = a , the quantity of capital per unit of land, using technique k, and the quanl nique k.

k k k 1 x = X , the quantity of labor per unit of land, using tech-Then we obtain :

7 t2x2 -t V + (a1 -a 2)r + (X1 -X 2 )w (7) 0 = ------------------------------------> W f 2 2 f1 T(t X -t X ) k k
where, from the inequalities (1) and the definitions of a and X , (8) x1 < x2, a1 < a2, X1 < X2 .

6 depends on the values taken by r, w and t , for a given technique.

Two commodities, no technical alternative

In our second model, we have two commodities respectively identified by the subscript i (i = 1, 2), each of them being produced by only one technique. These two commodities have to be distinguished by their con ditions of production. But technical coefficients in physical terms are not directly comparable for two different commodities when units are not. For this reason, we define the unit of commodity i as the quantity of that commodity produced on one unit of land, so that = x^ = 1. We assume that the production of one unit of commodity i uses labor in quantity 1^, and the other commodity -as intermediate input, i.e. as ca pital-in quantity a^ . Note that 1^ and a^ could also be defined as quantities used per unit of land ; a^ could also measure the indirect use of land, i.e. the quantity of land used to produce the input needed for the production of one unit of commodity i, i.e. for one unit of land devoted to commodity i. We assume that one of the production processes is more labor-intensive, while the other one is more capital-intensive, so that :

(9)

1^ < 12 and &2 < .

Commodities 1 and 2 are evaluated by their delivery prices p^ and p^ to the central market. Since we are only interested here in relative prices, we can assume that p^ = 1, so that

The other preceding notations hold.

The rent function is now expressed in value terms. For one unit of land devoted to commodity 1, the value of the product is p. The capital advance increased by the corresponding profit is a^ (1 + r) ;

the expenditure on labor is 1^ w ; the transport cost is t6 for the output and a^xfi for the input. This transport cost is also an advance which generates a profit a^xSr. It is easy to state the same items for the production of commodity 2. Then we have :

(10)

P1 (<5) = p -a^ (1 + r) -1 ^ -[1 + a^(1 + r)]xS, P2(6) = 1 -pa2(1 + r) -l2w -[1 + a2(1 + r)]x6.
If C6^) = P2(^2) = 0, we have the new system :

( 11) p -a^d + r) -l^w -[1 + a^(1 + r)]x6/ j = 0 1 -pa2 (1 + r) -l2w -[1 + a2 (1 + r)]x62 = 0 ,
which is solved in 6^ and 62 .

The rent functions become :

(12) p,j(<S) = [1 + a^(1 + r)]x(6,j -6) P 2 (6) = [1 + a2(1 + r)]x(«2 -6)
From the definition of 6 :

(13) [1 + a^l + r)]t(61 -5 ) = [1 + a2(1 + r)]x(6 -6 )
which gives the value of 6.

From (11), we find directly :

p -a (1 + r) -1 w = 1 1 (14) t [1 + a,|(1 + r)] 1 -pa2(1 + r) -l2w x [1 + a2(1 + r)]
By substituting ( 14) in ( 13) and solving in 6, we obtain :

p -1 + (a p -a.) (1 + r) + (1 -1 )w (15) J = ---------2-----1 --------------2----1 _ T(ai -a2) (1 + r)

The spatial patterns of production

From the preceding models we have derived expressions of the spatial va riables in terms of price, distribution and transport cost parameters.

These results can be used in two ways.

First, we can specify in terms of the parameters above the conditions for concentric locations. This is a generalization of the Löschian con ditions [START_REF] Lösch | Die räumliche Ordnung der Wirtschaft, Iena, Gustav Fischer. English trans[END_REF]. Secondly, we can show how the spatial pattern can change with the values of the parameters (comparative statics).

The conditions for concentric locations

For which possible combinations of values of the parameters are the two activities actually present, the one at the center, and the other one at the periphery (the spatial pattern we term concentric locations) ?

The concentric locations of the two techniques

When both techniques are used, their spatial order depends on the order of the slopes of the corresponding rent curves. If technique 2 is used at the center and technique 1 at the periphery, the slope of the second curve must be steeper than the slope of the first one (see figure 1).

Then, from (4) :

i-2 2 x J 1 (16) t x > t x ,
which means that the amount of commodity to be transported per unit of land is greater with the second technique than with the first one. This is a necessary condition. If the spatial order is reversed so is this condition.

Suppose now that condition ( 16) is realized. The sufficient conditions for concentric locations must be stated. We have two sets of simulta neous conditions.

1 . Technique 2 is actually used at the center. This is true if and only if (17) p2(0) > p1(0) , and ( 18)

P (0) > 0 .
k k The first inequality gives, from (2) and the definitions of a and X

2 2/" N , 2 . 1 1 , . v , 1 x -a (1 + r) -X w > x -a (1 + r) -X w, or ( 1 2n ,A , 2 , 2 1 1 2 N n (a -a )r + (X -X )w + x -x + a -a > 0 , with 2 1 1 2 2 1 1 1 2 2 x -x + a -a = x -x + a x -a x ^ 2N 2 1,1 , 2 2 .11 = (1 -a )x -(1 -a )x = t x -t a
Thus, we have :

(19) / -iJ 2n ,,1 -> 2 .
.2 2 .11.. r (a -a )r + (X -X )w + t x -t x > 0, or f^Cr, w) > 0

The second inequality, (18), is equivalent to x2 -a2(1 + r) -X2w > 0 , and, if x2 > 0, (20) J 2 2 2 1-ar-lw+1-a >0, or, in other terms ( 19) and ( 20) define the set of vectors (r, w) such that technique 2 is actually used at the center.

2. Technique 1 is actually used at the periphery. This means that (see figure 1) :

(21) 6^ > 62 , and

(22) 61 > 0 .

From ( 6) and ( 21), we have :

1 -a1(1 + r) -l1w N 1 -a2(1 + r) -l2w -----------------> ---------------- ^ thus t T t T I-2 4-2 h-, \ «-2!1 x J J v .1.2 t -t a (1 + r) -t 1 w > t -t a (1 + r) -t 1 w, and 2 f2 /12,1 ,1,2s .1 .2 ,.1 2 ,2 1. . n ( t a -t a )r -(t 1 -t 1 )w -t + t +(ta -t a ) > 0 , f1 2 * -2 1 t* 1 % 2 /a 2, 1 2 1 TU ta -ta = ( 1 -a ) a -(1-a)a = a -a. Then, ( 2 1n
,2,1, .2 .1 2 1 . _ u (a -a )r + (t 1 -t 1 )w + t -t + a -a > 0 , where 2 1 2 1 ? ?

1 1 t -t + a -a =(1-a)+a -(1 -a ) -a =0; then 2 1 1 2 2 1
(a -a )r + (t 1 -t 1 )w > 0, or, in other terms

(23) f (r, w) > 0 .
Note that, from (1), a2 -a1 >0, and t1l2 -t2l1 > 0, so that ( 23) is always satisfied for positive values of r and w.

(22) is equivalent to :

1 -a^(1 + r) -l^w K n ! -------z---------> □, or simply t T (24)
1 1 1 -ar-lw+1-a >0, or in other terms f2 (r, w) > 0, where which defines the set of vectors (r, w) such that technique 1 is actual ly used at the periphery.

To sum up, when the technical conditions imply inequality (16) -the second rent curve is steeper than the first one-the concentric loca tions are obtained if and only if the vector (r, w) belongs to the set defined by inequations (19), ( 20) and (24) for positive values of r and w. Thus we have related spatial patterns to distribution parameters.

Since the aforementioned conditions have no obvious meaning, we shall now propose a numerical example using the data of table 1. (19 e) f^r, w) = -8r -w + 2 > 0, thus r < -0.125 w + 0.25 .

k a i.k -, k t = 1 -a lk T k X k k k a = a x ,k ,k k A = 1 X
(20 e) f (r, w) = -12r -3w + 18 > 0, thus r < -0.25 w + 1.5 .

(24 e) ^2 ^» w) = -0.2r -0.1w + 0.8 > 0, thus r < -0.5w + 4 .

In figure 2, the set of vectors (r, w) corresponding to concentric loca tions is represented by the small shaded triangle CW^ . Outside this triangle, but inside the triangle R 0W7 , condition (19) does not hold, 1 2 which means that p (0) > p (0). Condition (24) holds, indicating that technique 1 is used everywhere. In the real world, r will probably be small, and we can consider that r < 1. (21') leads to

(23') (a1 -a2)r + ( t V -t V ) w > 0,
which is never realized since the coefficients of r and w are both nega tive. As it is impossible to produce anywhere using technique 2, it is therefore impossible to obtain concentric locations. Technique 1 is em ployed only if inequality (24) holds.

The second model

As for the first model, let us first compare the slopes of the rent cur ves, namely, from ( 12), [1 +8^1 + r)]x and [1 + a2 (1 + r)]x. Since al ^ 32' sl°Pe is in any case the steepest one. So, in any concentric pattern the first commodity will be more central than the second one. The necessary and sufficient conditions are derived as in the first model.

For the first commodity to be produced at the center, we must have

(25) (26)
p ^O ) > P2 (0), and

P1CQ) > o

Combining ( 25) and ( 10) gives ( 27) p _ a<](1 + r) -l.jW > 1 -pa2(1 + r) -l2w, and thus :

(1 + a2)p -a^r + (12 -l ^w + a2pr -1 -a1 > 0, or g (p, r, w) > 0 I

From ( 14) and ( 26), we obtain (28)

p -a r -1 w -a > 0, or gJJ(p, r, w) > 0 ( 27) and ( 28) define the set of vectors (p, r, w) corresponding to the production of the first commodity at the center.

For the second commodity to be produced at the periphery, we must have

(29) (30)
ô2 > 61 , and ô2 > 0

(29) is equivalent to 1 -pa2(1 + r) -1 w p -a ^l + r) -1 ^ T [1 + a (1 + r)] t [1 + a ^l + r)]
After multiplying the first term by [1 + ^ (1 + r)] and the second one by [1 + a^ (1 + r)], rearranging and symplifying, we obtain condition (31) 92^P> r> w ) > with 92 = + 2a^ + a^a^)p + (2a^ + 2a,ja2)r + (1^ -^ + a^l-j -a 1 )w 2 2 -(2a2 + 2a1a2 )pr + ^l, ,a ^^w r + a ^r -a1a2pr + 1 + 2a

From ( 14) and.(30), we have : i -a2P -¿2W " a2Pr + 1 > 0» or> in other terms g'2 (p, r, w) > 0 .

(31) and ( 32) define the set of vectors (p, r, w) such that the second commodity is produced at the periphery. Thus ( 27), ( 28), ( 31) and ( 32)

together determine the set of vectors (p, r, w) corresponding to concen tric locations. These conditions define a three-dimensional set and the equations of the bounds are very complicated. We will specify technical data (see table 2) and successively fix one of the three parameters so as to show two-dimensional sections of the concentric locations set. g'^ = p -0.1 w -0.44 > 0 ^ p > 0.1 w + 0.44 .

= -1.5 p -0.17 w + 1.98 > 0 => p > -0.11 w + 1.31 .

g'2 = -0.22 p -0.2 w + 1 > 0 => p < -0.91 w + 4.55.

Then figure 3 shows what happens for different vectors (p, w). Only in the shaded zone do the concentric locations hold. Outside this area, only one of the two commodities can be produced ; however this is impos sible since each of the commodities needs the other as input. If we fix w = 1, the conditions become : g = 1.2 p -0.4 r + 0.2 pr -1.3 > 0 => p > 4 r + 13 . 2 r + 12 g'-j = P -0-4 r -0.5 > 0 ^ p > 0.4 r + 0.5 . g2 = -1.48 p + 0.9 r -0.56 pr + 0.08 r2 -0.08 pr2 + 1.74 > 0 , 1.74 + 0.9 r + 0.08 r2 = > p < -----------------.

1.48 + 0.56 r + 0.08 r g' = -0.2 p -0.2 pr + 0.8 > 0 => p < ---.

+ r

The ad hoc set is shaded in figure 4 (with r varying between 0 and 1). Finally, let us fix p = 1.10 . Then we have = 0.18 r + 0.1 w + 0.08 > 0 î w > 1.B r + 0.8 .

g'^ = -0.4 r -0.1 w + 0.7 > 0 => w < -0.4 r + 7 .

n -T / , n n: n nz: n nno 2 . n ^ . 0.25 + 0.34 r -0.008 g7 = 0.34 r -0.16 w -0.06 wr -0.008 r > 0 => w < - ---------------------L 0.16 + 0.06 r g'2 = -0.22 r -0.2 w + 0.78 > 0 => w < -1.1 r + 3.9 .

These conditions define the shaded set in figure 5 (with r varying bet ween 0 and 1).

JL

The changes of locations

Suppose that the preceding conditions for concentric locations are ful filled. How does the spatial pattern of activities change in response to variations in the parameters p, r, w and t ? Once again, the two models are examined.

The first model

In the case of inequality (16), i.e. when the second rent curve is stee per than the first one, the boundaries of the rings corresponding to the A __ two techniques are determined by 6 and 6, as defined by ( 6) and ( 7),and as shown on figure 6. It must be remembered that ,_,x -j -t2x2 -t.1x1 + (a1 -a2)r + (X1 -X2 )w ( 7)

6 = -------------------------------- /. 2 2 ,1 1v T (t X -t X )
The problem of the changes in locations can be studied theoretically by 1 means of the derivatives of 6 and 6, and numerically with an example ^ _ and graphs of 6 and 6 • The example is based on the data already used for model 1 (see table 1).

The effect of variations in the rate of profit

a T 1 n 2
We have :

-= -a9 Z---T~T~ 3r T(t x -t x ) j l 3 r t1T < 0
The inner ring, where technique 2 is used, becomes smaller, which means that there is a substitution of techniques on the land located near the _ 1 initial boundary 6. 6 is decreasing but we cannot ascertain whether the outer ring becomes smaller or larger. The direction in which this area

'j __ __
varies is given by the sign of the derivative of ( 6) -( 6)2 for 6 > 0.

The general case leads to very complicated formulae, and consequently this problem shall be treated only for the numerical example.

Let us fix w = 1 (which is compatible with concentric locations). We have -40r + 5 61 = -2.5r + 8.75 The effect of variations in the wage rate This case is analogous to the preceding one. In the example, let us fix r = 0.10, so that : 6 = -5w + 6 , and S1 = -1.6w + 9.33 .

_

-) Figure 8 shows how 6 and 6 vary with a change in w. When w increases the surface S9 decreases. When w varies from 0 to 1.2 (field of concentrie locations), the study of the derivative of (6 )2 -( 6)2 shows that the area begins to increase and then decreases. But the ratio of the S-, two areas, -g-, appears to be increasing :

f l -n(61)2 -n(6)2 s2 nC6)2
-1, of which the derivative is always posi tive. Thus, even if decreases, it will do so less than S 2, with the result that the change is relatively beneficial to technique 1. Again, an increase in the wage rate leads to a relative spatial expansion of the more labor-saving technique.

We can extend the preceding analysis to the transport rate. Here the re sult will be slightly different.

The effect of variations in the transport rate .

suits in the two rings decreasing at the same rate, despite the fact that one technique is more transport-using than the other one. It is easy to verify that the derivative of (6^)2 -(6)2 is always nega tive, meaning that S1 varies in the same direction as S^. Clearly, the 2 derivative of -1 is zero. An increase in the transport rate re- and profit rate are sufficiently low, and it can spread towards the periphery when these rates decrease.

The second model

Again, the conditions of concentric locations are supposed to hold, with the first commodity at the center and the second at the periphery. The boundaries of the corresponding rings are 6 and 62 (see figure 10). As for the two-techniques-model, we have to study how the parameters influence 6 and Remember that :

p -1 + (a2p -a ^ (1 + r) + (12 -l ^w xia^ -a2) (1 + r) 1 -a p(1 + r) -1 w (14) S2 = ----1---------- t [1 + a2(1 + r)]
We use the same example as in section (3.1.2.).

The effect of variations in the relative price From ( 15) and ( 9), we have :

1 + a (1 + r) 36 2 > 0 , and 3p -a )(1 + r) 3<5? -a (1 + r) - = ----------------< 0 , 3 p t [1 + a9(1 + r)]
which imply an increase in the inner ring area and a decrease in the outer ring area. This means that the commodity whose relative price ri ses expands in space (commodity 1).

With our data, if we fix r = 0.1, w = 1 and t = 0.1 : 6 = 55.45 p -60.9 and $2 = -1.8p + 6.55 cor-P -< 0 , ] and we observe a spatial expansion of the labor-saving commodity when the wage rate increases.

In the example, if we fix p = 1.1., r = 0.1 and T = 0.1 : 6 = 4.54w -4.45 and ^ = ~ + 6.21 .

We can see these functions in figure 12, where the interval corresponds to the concentric locations. Figure 13 shows the different possibilities of variations of 6 and 62 depending on the values of p and w.

)w]x(a1~a2 ) Thus the effect of the profit rate is undetermined. The expected effect, namely the expansion of the capital-saving production to the detriment of the other one with a rise in the rate of profit, is garanteed only for the subset of values of (p, w) corresponding to zone © in figure 13. We can have an unexpected effect which can be explained by the ap plication of the profit rate not only to the expenditure on direct inputs, but also to the expenditure on transport. If we return to the example and fix p = 1.1., w = 1 and T =0.1, we have :

T s L = J L , and 69 = ^-l i 2 r 1 + r 2 12 + 2r

Both derivatives are negative. The functions are represented in figure 14, where the interval corresponds to the concentric locations of commodities. tive : an increase of the transport rate creates difficulties for both commodities, and the areas of the two rings diminish. The derivative of -1 is zero, so that the two areas diminish at the same rate when t increases (see figure 15).

Figure 2 .

 2 Figure 2. Conditions for concentric locations. First model ,

Figure 3 .

 3 Figure 3. The conditions for concentric locations of commodities, with a fixed r

Figure 4 .

 4 Figure 4. The conditions for concentric locations of commodities, with a fixed w

Figure

  Figure 5. The co nditions for concentric locations of commodities, with a fixed p

Figure

  Figure 6. The technique-rings when condition (16) is realized

Figure1

  Figure 7. The boundaries of the technique-rings as functions of r

Figure 8 .

 8 Figure 8. The boundaries of the technique-rings as functions of w

Figure 9 .

 9 Figure 9. The boundaries of the technique-rings as functions of T

Figure 10 .

 10 Figure 10. The commodity-rings with concentric locations

Figure

  Figure 11. The boundaries of the comnodity-rings as functions of p

Figure 12 .

 12 Figure 12. The boundaries of the comnodity-rings as functions of w

Figure

  Figure 13. Variations of 6 and as functions of r, depending on p and w

Figure

  Figure 14. The boundaries of the commodity-rings as functions of r

Figure 15 .

 15 Figure 15. The boundaries of the comnodity-rings as functions of t

Table 1 . Data for the first model

 1

16) holds. Conditions (19), (20) and (24) become

	Technique 1	0.2	0.8	0.1	0.1	20	4	2
	Technique 2	0.4	0.6	0.1	0.1	30	12	3
	Condition (

respective ly :

Table 2 . Data for the second model

 2

		a . l	1. 1	T
	Commodity 1	0.2	0.1	0.1
	Commodity 2	0.4	0.2	0.1

  than the first technique (condition 16). If it is land-saving, but capital-and labor-using, it can appear if and only if the wage rate

).

Moreover, let us imagine that we start with technique 1 alone, and that a new technique, 2, becomes available. Then this second technique can appear at the center, if and only if it is more transport-using per unit of land

A special case : Suppose that we add a ^ = 0 to the set of our assump tions. Then the second commodity is produced only with labor and land, but it is used to produce the first one. We keep the relation 1 ^ > 1^.

Thus the second commodity can be produced even if the first one cannot. This can be interpreted as a primitive agrarian economy without capital, analogous to that of SMITH (1776) and [START_REF]Principles of Political Economy and Taxation[END_REF]. We can use the preceding analysis to state conditions of appearance and spatial diffu sion of a new process using the agricultural commodity to produce a luxury commodity. These conditions relate only to conditions of produc tion, to prices and distribution. So we could "spatialize" a Ricardian i l economy.

Conclusions

We have examined some features of productive land use, namely the con ditions for concentric locations of techniques or of commodities. We have emphasized the role of conditions of production, through technical coefficients, and the role of relative prices, of wage rate and profit rate, in the framework of advance economics.

We have generalized the Loschian conditions of circle formation and we have stated the effect of price and distribution on the pattern of THÜNEN's rings. In some ca ses, the results confirm what common sense would expect : we see a rela tive spatial expansion of the more factor-saving activity in response to the rising cost of this factor. But this simple general result takes several forms. Moreover, in the case of the profit rate, our expecta tions are not confirmed : indeed here we can even see the opposite ef fect. This shows the usefulness of this kind of analysis. Nevertheless, all that we have done is very limited. We have neglected the demand side and a more complete model would be very helpful if we are to understand the land allocation process more fully.