C Gomez

B Burt-Pichat

F Mallein-Gerin

B Merle

P D Delmas

T M Skerry

L Vico

Luc Malaval

C Chenu

Expression of Semaphorin-3A and its receptors in endochondral ossification -potential role in skeletal development and innervation

Keywords: immunocytochemistry, RT-PCR, bone innervation, bone development, semaphorin-3A

Expression of Semaphorin-3A and its receptors in endochondral ossification: potential role in skeletal development and innervation.

INTRODUCTION

Early work as well as recent immunocytochemical studies have highlighted the rich innervation of bone [START_REF] Calvo | On the development of bone marrow innervation in new-born rats as studied with silver impregnation and electron microscopy[END_REF][START_REF] Serre | Evidence for a dense and intimate innervation of the bone tissue, including glutamate-containing fibers[END_REF]. The expression of neuromediators in skeletal nerve fibres, and of their receptors in osteoblasts and osteoclasts has been established. The presence in bone of cell processes labeled with neural tip markers like synaptophysin has also been documented. Overall, these data provide evidence for the existence of functional nerve fibres within the bone tissue [START_REF] Gronblad | Innervation of human bone periosteum by peptidergic nerves[END_REF][START_REF] Hohmann | Innervation of periosteum and bone by sympathetic vasoactive intestinal peptide-containing nerve fibers[END_REF][START_REF] Bjurholm | Substance P-and CGRPimmunoreactive nerves in bone[END_REF][START_REF] Serre | Evidence for a dense and intimate innervation of the bone tissue, including glutamate-containing fibers[END_REF]. Some of these nerve fibres are in contact with cells lining the trabeculae (osteoblasts and osteoclasts), suggesting a direct control of bone cell function [START_REF] Serre | Evidence for a dense and intimate innervation of the bone tissue, including glutamate-containing fibers[END_REF]. Several studies have indeed shown evidence for a neural control of bone growth. Developmental growth in the rat foot is reduced after surgical denervation [START_REF] Edoff | The developmental skeletal growth in the rat foot is reduced after denervation[END_REF]. The remodelling of bone is also affected by surgical or chemical sensory and sympathetic denervations in rat [START_REF] Sandhu | Effect of surgical sympathectomy on bone remodeling at rat incisor and molar root sockets[END_REF][START_REF] Hill | Effects of neonatal sympathectomy and capsaicin treatment on bone remodeling in rats[END_REF][START_REF] Cherruau | Chemical sympathectomy impairs bone resorption in rats: a role for the sympathetic system on bone metabolism[END_REF]. Furthermore, a leptin-dependent regulation of bone mass by the central nervous system has been demonstrated in mice, which appears to be mediated by sympathetic innervation [START_REF] Ducy | Leptin inhibits bone formation through a hypothalamic relay: a central control of bone mass[END_REF][START_REF] Takeda | Leptin regulates bone formation via the sympathetic nervous system[END_REF]. The role of sympathetic innervation in bone remodelling is also supported by the changes in bone mass observed in mice deficient in genes coding for dopamine β-hydroxylase, dopamine transporter and Y2 receptors [START_REF] Bliziotes | Bone histomorphometric and biomechanical abnormalities in mice homozygous for deletion of the dopamine transporter gene[END_REF][START_REF] Baldock | Hypothalamic Y2 receptors regulate bone formation[END_REF][START_REF] Takeda | Leptin regulates bone formation via the sympathetic nervous system[END_REF].

As early as embryonic day (E)15, nerve fibres appear in the perichondrium of the diaphysis of rat long bones. They become functional at E18-19, when neural tips are formed and neuropeptide expression starts [START_REF] Sisask | Ontogeny of sensory nerves in the developing skeleton[END_REF][START_REF] Sisask | The development of autonomic innervation in bone and joints of the rat[END_REF]. There is little understanding of the molecular mechanisms by which neurons reach their targets in bones. In the central nervous system, many studies described several families of proteins which are involved in wiring and allow axons and dendrites to locate and reach their appropriate targets. The most prominent families of neural signaling molecules are Semaphorins, Netrins, Slits and Ephrins. The Semaphorins, a large family of cell surface and secreted guidance molecules, is one of the best understood. In particular, it has been shown that Semaphorin-3A (Sema-3A) signaling is very important for neuronal targeting in the peripheral nervous system (PNS), as shown for trigeminal and dorsal root ganglia which are involved in bone innervation [START_REF] Kitsukawa | Neuropilin-semaphorin III/D-mediated chemorepulsive signals play a crucial role in peripheral nerve projection in mice[END_REF][START_REF] Taniguchi | Disruption of semaphorin III/D gene causes severe abnormality in peripheral nerve projection[END_REF][START_REF] Ulupinar | Role of semaphorin III in the developing rodent trigeminal system[END_REF]. Sema-3A, a soluble molecule, signals through multimeric receptors always combining at least one of the 3 plexins (Plx) and neuropilin-1 (NP-1). NPs do not appear to have a signaling function.

Plx contribute to both ligand specificity and signal transduction, activating RhoA through poorly known mediators [START_REF] Goshima | Semaphorins as signals for cell repulsion and invasion[END_REF]. Recently the expression of Sema-3A and NP-1 was demonstrated in osteoblast and osteoclast cell lines, and NP-1 was shown to be expressed in bone [START_REF] Togari | Expression of mRNA for axon guidance molecules, such as semaphorin-III, netrins and neurotrophins, in human osteoblasts and osteoclasts[END_REF][START_REF] Harper | Neuropilin-1 expression in osteogenic cells: down-regulation during differentiation of osteoblasts into osteocytes[END_REF]. Furthermore, abnormal bone and cartilage development, with vertebral fusions and partial rib duplication, has been observed after a targeted mutation of Sema-3A in mouse, indicating that this signaling pathway may play a role in bone development [START_REF] Behar | Semaphorin III is needed for normal patterning and growth of nerves, bones and heart[END_REF].

The aim of our study was to investigate the expression of the molecules in the Sema-3A signaling network during skeletal development, focusing on the spatial and temporal relationship with nerve fibre growth.

RESULTS

Multiple bone cell lineages express components of the Semaphorin axon guidance system

Messenger RNA for Sema-3A, NP-1, Plx-A1 and Plx-A2 but not Plx-A3 were detected in osteoblastic cells. They were also present, with the exception of Sema-3A, in differentiated osteoclasts (fig. 1A). Sema-3A, NP-1 and Plx-A1 and -A2 mRNA are expressed at all stages of chondrocyte differentiation, including the final hypertrophic stage, reached under the highest dose of BMP-2 (fig. 1B). Plx-A3 was not detectable in chondrocytes at any stage. As shown on figure 2, NP-1, Plx-A1 and A2 proteins are expressed by both osteoblasts and osteoclasts. Sema-3A expression was only detected in osteoblasts, confirming RT-PCR data.

Only a very faint staining was observed in MC 615 cells for all markers (not shown).

Sema-3A and its receptor chain, NP-1, have the same pattern of expression during endochondral ossification

We observed that Sema-3A and NP-1 displayed nearly identical patterns of expression at all stages of bone development. At E17 Sema-3A (fig. 3C) and NP-1 (not shown) were detected in the outer perichondrium of the diaphysis and metaphysis, and in the pre-hypertrophic and hypertrophic chondrocytes within the future primary ossification center (fig. 3 C, enlargement in D). Staining was also present in resting chondrocytes within the epiphysis, the site of the future secondary ossification center (fig. 3C). At E19 NP-1 (fig. 3F) and Sema-3A (not shown) presented a patchy staining in the periosteum of the diaphysis, restricted to the area lining the primary ossification center. At this stage NP-1, but not Sema-3A, was also expressed in the bone collar (fig. 3F) and in blood vessels in the primary ossification center (Fig. 3G). At E21, staining for Sema-3A (fig. 3H) and NP-1 (not shown) was located in osteoblasts and in the bone marrow and blood vessels, in trabecular bone and only for Sema-3A in the endosteum (fig. 3H). NP-1 (fig. 3J) and Sema-3A (fig. 3K) at P14 were both still located in osteoblasts lining the bone trabeculae but only under the growth plate. Both molecules were also expressed in pre-hypertrophic and hypertrophic chondrocytes of the growth plate (NP-1: Fig. 3L, enlarged in M and M; Sema-3A: data not shown). At P14, Sema-3A (fig. 3O) and NP-1 (fig. 3P) were also detected in large multinucleated (DAPI staining of nuclei in insets) osteoclasts along the trabeculae under the growth plate.

Co-receptors Plexins A1 and A2 are expressed by chondrocytes and osteoblasts

At E17, Plx-A1 was expressed in the perichondrium of the diaphysis and the metaphysis (fig 4A), in hypertrophic chondrocytes at mid-diaphysis and also in the resting chondrocytes in the epiphysis (not shown). Plx-A1 was still present at E19 in hypertrophic chondrocytes (fig 4B), especially those facing the osteogenic perichondrium. It was expressed at lower levels in the periosteum and by bone collar forming osteoblasts (4B, short arrows) and in the primary ossification center (not shown). Plx-A1 was expressed by osteoblasts lining all bone trabeculae at E21 (fig. 4C, arrows) and only those under the growth plate at P14 (fig. 4D, arrows). At the latter stage, it was also expressed by periosteal osteoblasts (fig. 4D, arrowheads), by some osteocytes (fig. 4D, dotted line circles) and in osteoclasts (fig. 4E). At E17, Plx-A2 was expressed in hypertrophic chondrocytes, in resting chondrocytes of the epiphysis and also at lower levels in the perichondrium (fig 4F). At E19, staining extended to the primary ossification center (not shown). At E21, Plx-A2 (Fig. 4G) was present in the periosteum (short arrows) and in some osteoblasts (long arrows) and osteocytes (dotted line circles). At P14, expression of Plx-A2 (fig 4H) was mostly restricted to endosteal (arrows) and periosteal osteoblasts (short arrows) and osteocytes (dotted line circles). At this stage it was also detected in a few osteoblasts and in osteoclasts under the growth plate (fig. 4I). Both markers were also expressed by other cell types in the marrow and the periosteum (4D, H).

Plx-A3 presents a restricted pattern of expression during endochondral ossification

At E17, Plx-A3 was expressed in the perichondrium of the metaphysis (fig. 5A), where some profiles seemed to outline blood vessels (5A, arrows), and by hypertrophic as well as some resting chondrocytes (not shown). Staining at E19 (fig. 5B) was located in discrete profiles within the periosteum at the mid-diaphysis, some of which suggestive of vascular staining (5C, arrow). Plx-A3 was also detected in the bone collar (5D) and in hypertrophic and at lower levels in resting chondrocytes (not shown). No labeling was observed at this stage within the primary ossification center. At E21, Plx-A3 was expressed in the secondary spongiosa as a loose network along bone trabeculae and in endothelial cells (fig. 5E, enlarged in F). It was also present in the periosteum and hypertrophic and some resting chondrocytes (not shown). At P14, Plx-A3 was strongly expressed in endothelial cells of vascular buds in the ossification front under the growth plate (fig. 5G andH), and in capillaries in the metaphysis (fig. J).

The pattern of expression of neural cell markers in developing rat femur parallels the development of endochondral ossification

To correlate the in vivo expression of Sema-3A and its receptor subunits with the appearance of nerve fibres, we analysed the expression of nerve markers during bone development.

Neurofilament heavy chain (NH) is a general nerve marker, while tyrosine hydroxylase (TH), an enzyme involved in the synthesis of catecholamines such as noradrenalin, is a specific marker for sympathetic nerves fibres. The staining patterns of both markers were identical at all development stages, although TH gave a higher background at late stages. At E17, both markers were expressed only in the perichondrium of the metaphysis (fig. 6A,F). At E19, NH and TH positive fibres were found in the periosteum of the diaphysis along the primary ossification center (fig. 6B,G). At this stage, we also found NH and TH labeling in the bone collar (fig. 6B,G, long arrows). At E21, the nerve fibres were located along the bone trabeculae in the diaphysis (fig. 6C,H, short arrows), in close vicinity to osteoblasts and lining cells. At P14, labeling was weaker in bone and was still present in the periosteum of the diaphysis (NH in fig. 6E, not shown for TH).

DISCUSSION

Three functional receptor complexes have been described for Sema-3A: NP-1/Plx-A1, NP-1/Plx-A2, NP-1/Plx-A3 [START_REF] Nakamura | Molecular basis of semaphorin-mediated axon guidance[END_REF][START_REF] Raper | Semaphorins and their receptors in vertebrates and invertebrates[END_REF]. These complexes also bind other ligands in a competitive fashion: Sema-3B, 3C and 3F, and vascular endothelial growth factor-165 (VEGF 165 ). Target genes and signal transduction pathways for this signaling have not been fully elucidated, but in neural cells are known to involve the reorganisation of the cytoskeleton via Rho GTPases, which enables axonal guidance [START_REF] Castellani | Control of semaphorin signaling[END_REF]. In this study, we show that all components of the Sema-3A signaling system are expressed in mature stages of all 3 bone cell lineages (chondrocytes, osteoblasts and osteoclasts), with the exception of Plx-A3. Although osteoclastic cells from RAW 264.7 cultures do not express Sema-3A, we found labelling in resorbing osteoclasts within the bone tissue. Others reported its expression in osteoclast-like cells from human bone marrow cultures [START_REF] Togari | Expression of mRNA for axon guidance molecules, such as semaphorin-III, netrins and neurotrophins, in human osteoblasts and osteoclasts[END_REF]. This may reflect specificities of immortalised cell lines and/or culture conditions, which could also explain the low levels/absence of Sema-3A signaling proteins in the MC 615 chondrogenic cells, in spite of messenger expression. It was previously reported that MC3T3-E1 osteoblastic cells express NP-1 mRNA and that several human osteoblastic cell lines express Sema-3A mRNA and protein [START_REF] Togari | Expression of mRNA for axon guidance molecules, such as semaphorin-III, netrins and neurotrophins, in human osteoblasts and osteoclasts[END_REF][START_REF] Harper | Neuropilin-1 expression in osteogenic cells: down-regulation during differentiation of osteoblasts into osteocytes[END_REF]. Expression of NP-1 mRNA was also shown in human osteoarthritic cartilage [START_REF] Enomoto | Vascular endothelial growth factor isoforms and their receptors are expressed in human osteoarthritic cartilage[END_REF]. To the best of our knowledge we are the first to show the expression of Plexins-A1 and -A2 in osteoblasts, osteoclasts and chondrocytes, of NP-1 in osteoclasts, and of Sema-3A in chondrocytes.

During bone development, we first observed Sema-3A and all components of its receptor system in pre-hypertrophic and hypertrophic chondrocytes in ossification centers, and also in the facing periosteum, before and after the onset of endochondral ossification and vascular invasion. We found in particular that Plx-A1 is more strongly expressed by the outermost layer of chondrocytes of the mid-diaphysis, in direct contact with the osteogenic perichondrium (fig. 4B). Such "borderline" chondrocytes have been suggested to interact with perichondral osteoblasts for the regulation of ossification [START_REF] Bianco | Bone formation via cartilage models: the "borderline" chondrocyte[END_REF]. Thus, Sema-3A signaling precedes or coincides with the invasion of bone by blood vessels and nerve fibres (Table 2). At later stages, Sema-3A and its receptor components (except Plx-A3) are co-expressed by osteoblasts and osteoclasts, with regional differences. Such coexpression is compatible with an autocrine/paracrine pathway, such as has been shown for instance in breast carcinoma cells [START_REF] Bachelder | Competing autocrine pathways involving alternative neuropilin-1 ligands regulate chemotaxis of carcinoma cells[END_REF]. However, expression of Sema-3A receptors in bone is not restricted to osteoblast and osteoclasts, but is also found in blood vessels and probably nerve processes. Thus, Sema-3A signaling molecules are in a position to modulate the vascularisation of bone, and the innervation of osteoblasts and osteoclasts during bone development and remodelling.

In keeping with a number of recent studies showing evidence that bone modelling and remodelling are controlled by the nervous system [START_REF] Sandhu | Effect of surgical sympathectomy on bone remodeling at rat incisor and molar root sockets[END_REF][START_REF] Hill | Effects of neonatal sympathectomy and capsaicin treatment on bone remodeling in rats[END_REF][START_REF] Edoff | The developmental skeletal growth in the rat foot is reduced after denervation[END_REF][START_REF] Cherruau | Chemical sympathectomy impairs bone resorption in rats: a role for the sympathetic system on bone metabolism[END_REF][START_REF] Takeda | Central control of bone formation[END_REF][START_REF] Takeda | Leptin regulates bone formation via the sympathetic nervous system[END_REF], we previously showed a dense innervation of bone in close contact with osteoblasts and osteoclasts, although no typical synapses were observed [START_REF] Serre | Evidence for a dense and intimate innervation of the bone tissue, including glutamate-containing fibers[END_REF]. We observed here the same labeling pattern and density for NH, a general nerve marker and TH, a specific marker for sympathetic fibres. This suggests that the sympathetic system, which is known to innervate blood vessels and suspected to play a part in the regulation of bone remodelling, is the major one in bone, at least at the stages that we investigated [START_REF] Ducy | Leptin inhibits bone formation through a hypothalamic relay: a central control of bone mass[END_REF][START_REF] Takeda | Leptin regulates bone formation via the sympathetic nervous system[END_REF][START_REF] Levasseur | Sympathetic nervous system as transmitter of mechanical loading in bone[END_REF]. In ossified areas, we found staining for NP-1 not only in osteoblasts and osteoclasts, but also in endothelial cells and within the bone marrow. Such generalised expression did not allow identification of a "network" pattern similar to nerve markers; nonetheless, expression of NP-1 in nerve processes is amply documented in the literature [START_REF] Yu | Dynamic regulation of axon guidance[END_REF]. In contrast, the expression pattern of Plx-A3 in developing bone strikingly parallels the one observed for nerve markers from E17 to E21. It is, however, also localised in the endothelial cells of capillaries, particularly at the ossification front in later stages. Peripheral nerves and blood vessels have similar patterns of development in the forelimb, and Sema-3A may function as a common signal molecule for the establishment of the two systems [START_REF] Bates | Neurovascular congruence results from a shared patterning mechanism that utilizes Semaphorin3A and Neuropilin-1[END_REF]. Our unablility to detect Plx-A3 expression within the primary ossification center at E19 could be explained by a downregulation allowing vascular penetration (see below). Overall, our observations suggest that Plx-A3 is the candidate partner of NP-1 for the regulation of neurovascular patterning by Sema-3A.

Sema-3A has been shown in numerous studies to be a repulsive neural guidance cue, crucial for the normal patterning of nerves fibres in vivo [START_REF] Schwarting | Semaphorin 3A is required for guidance of olfactory axons in mice[END_REF][START_REF] Kawasaki | Requirement of neuropilin 1-mediated Sema3A signals in patterning of the sympathetic nervous system[END_REF]. However, this action can be counteracted by competitive inhibitors. For instance, the competition for NP-1 occurring between Sema-3A and VEGF 165 , which is expressed by human hypertrophic, but not proliferative chondrocytes of the growth plate [START_REF] Petersen | Expression of VEGF121 and VEGF165 in hypertrophic chondrocytes of the human growth plate and epiphyseal cartilage[END_REF], seems to play a key role in mechanisms of innervation and angiogenesis. The effect of VEGF 165 on both peripheral nerves and blood vessels opposes the effect of sema-3A [START_REF] Nakamura | Molecular basis of semaphorin-mediated axon guidance[END_REF]. VEGF 165 enhances endothelial cell survival and mobility and increases axonal outgrowth and neural survival in vitro [START_REF] Castellani | Control of semaphorin signaling[END_REF][START_REF] Petersen | Expression of VEGF121 and VEGF165 in hypertrophic chondrocytes of the human growth plate and epiphyseal cartilage[END_REF][START_REF] Enomoto | Vascular endothelial growth factor isoforms and their receptors are expressed in human osteoarthritic cartilage[END_REF]. In contrast, Sema-3A inhibits endothelial cells mobility and survival, as well as axonal outgrowth [START_REF] Deckers | Expression of vascular endothelial growth factors and their receptors during osteoblast differentiation[END_REF][START_REF] Castellani | Control of semaphorin signaling[END_REF][START_REF] Bachelder | Competing autocrine pathways involving alternative neuropilin-1 ligands regulate chemotaxis of carcinoma cells[END_REF][START_REF] Serini | Class 3 semaphorins control vascular morphogenesis by inhibiting integrin function[END_REF]. An hypothesis suggested by our observations on early bone development is that the Sema-3A/Plx-A3/NP-1 pathway would inhibit neurovascularisation of the cartilage anlage early in skeletal development. This block could be removed later and locally by the down regulation of key receptor chains (e.g. Plx-A3), the elimination of Sema-3A producing cells (e.g. hypertrophic chondrocytes, cf. POC at E19, Table 2) or by the expression of competitors of Sema-3A (such as VEGF 165 ), for instance by perichondral cells and hypertrophic chondrocytes. Inhibition of Sema3A signaling would then allow blood vessel and nerve fibre invasion of the diaphysis, and thus the onset of endochondral ossification together with the start of local neural and/or vascular regulation of bone formation. Later, during the ossification process, Sema-3A may provide a repulsive guidance allowing the growing nerve fibres to be directed to their targets in the bone tissue. However, other guidance sytems are certainly involved in bone neurovascular development and are likely to interfere with the Sema-3A pathway.

In conclusion, we show that the Sema-3A signaling network appears to be well positioned to regulate the development of bone innervation, and perhaps other aspects of bone development and remodelling.

MATERIALS AND METHODS

In vitro models

We analysed the expression of the Sema-3A signaling molecules in in vitro models of osteoblasts, osteoclasts and chondrocytes. In mature (day 15) rat calvaria (RC) cell cultures, osteoprogenitors form bone colonies expressing high levels of osteoblast markers, osteocalcin and bone sialoprotein [START_REF] Malaval | Kinetics of osteoprogenitor proliferation and osteoblast differentiation in vitro[END_REF](Fig. 1A,2B). As an osteoclast model, we used the RAW 264.7 cell line, which, when treated with the cytokine RANKL, fully differentiate in multinucleated osteoclast-like cells expressing calcitonin receptor and tartrate resistant acid phosphatase [START_REF] Hsu | Tumor necrosis factor receptor family member RANK mediates osteoclast differentiation and activation induced by osteoprotegerin ligand[END_REF](Fig 1A ,2D). As a chondrocyte model we used the immortalised mouse cell line MC 615, which can differentiate into hypertrophic chondrocytes when treated with BMP-2, as documented by the concomitant expression of Collagen Xα1 and osteocalcin (Fig 1B) [START_REF] Valcourt | Functions of transforming growth factor-beta family type I receptors and Smad proteins in the hypertrophic maturation and osteoblastic differentiation of chondrocytes[END_REF].

In situ model

To analyse the spatial expression of sema-3A and NP-1 during bone development, we performed immunocytochemical studies on developing rat femurs at days E17 and E19 (beginning of endochondral ossification), at E21 (birth) and 14 days after birth (P14, end of the endochondral ossification). At E17 the bone is still a cartilage model in the center of which (mid-diaphysis) some chondrocytes become hypertrophic. It is limited by a layer of undifferentiated osteogenic cells, the perichondrium (termed periosteum at later stages). At E19 the bone collar has formed, and vascular invasion and endochondral ossification has begun in the mid-diaphysis (primary ossification center). The growth plates are formed soon after E19, separating the epiphysis from the metaphysis. In fully developed bone, trabecular bone tissue is restricted to the metaphysis and the epiphysis; the midshaft (diaphysis)

Cell culture

Culture reagents were from Sigma (Saint-Quentin Fallavier, France) unless otherwise indicated. 21 days pregnant female Wistar rats were killed by cervical dislocation, in accordance with the local Ethical Committee recommendations, and the fetuses were collected through C-section. Fetal rat calvaria (RC) cells were enzymatically isolated by sequential digestion with collagenase as described previously (Bellows et al., 1986;[START_REF] Malaval | Kinetics of osteoprogenitor proliferation and osteoblast differentiation in vitro[END_REF]; populations obtained from digests II through V were kept and plated separately.

After 24 h, cells were trypsinised (0.01% trypsin in citrate saline), counted, populations II-V were pooled, and cells were plated at 3 x 10 3 cells/cm 2 in 35 mm dishes for RNA isolation and on 1cm 2 glass coverslips for immunostaining. RC cells were grown in α-MEM medium containing antibiotics, 10% FBS, 50μg/ml ascorbic acid, and 10 mM sodium βglycerophosphate. The medium was changed every 2-3 days. All dishes were incubated for 15 days at 37°C in a humidified atmosphere of 95%air/5% CO 2 . The RAW 264.7 monocytic cell line was cultured in 25cm 2 flasks or on glass coverslips, in DMEM medium supplemented with 10% fetal calf serum, 1% penicillin/streptomycin, 1% glutamate and 0.5% fongizone. The cultures were kept for 6 days to differentiate into osteoclasts in whole medium described above supplemented with 30ng/ml Rank Ligand (kindly provided by Amgen, Thousands Oaks, CA, USA). The mouse chondrogenic MC 615 cell line has been characterised previously [START_REF] Mallein-Gerin | Expression of simian virus 40 large T (tumor) oncogene in mouse chondrocytes induces cell proliferation without loss of the differentiated phenotype[END_REF][START_REF] Valcourt | Functions of transforming growth factor-beta family type I receptors and Smad proteins in the hypertrophic maturation and osteoblastic differentiation of chondrocytes[END_REF]. MC 615 cells were maintained in 1:1 high glucose DMEM/Ham's F-12 containing 10% FBS and supplemented with 2mM L-glutamine, 100 units/ml penicillin, 100µg/ml streptomycin (all product from in Vitrogen). For differentiation into hypertrophic chondrocytes, MC 615 cells were grown for 3 days in flasks or on glass coverslips as above, except that FBS was reduced to 1% and BMP-2 (0-100ng/ml) was added to the medium. Recombinant human BMP-2 was produced and purified by Genetics Institute, Inc. (Cambridge, MA, USA). The culture medium supplemented with BMP-2 was replaced each day. At the end of the culture time, all cell types were either processed for RNA isolation or fixed and permeabilised for immunostaining.

RNA isolation and RT-PCR

Cells were scraped in lysis buffer and total RNA was isolated using the Rneasy R minikit (Qiagen SA, Courtaboeuf, France). 2μg of total RNA were reverse-transcribed using Omniscript reverse transcriptase from Qiagen SA (Courtaboeuf, France) and oligo (dT) 12-18 primers (Amersham Pharmacia Biotech, Orsay, France). Polymerase Chain Reaction (PCR) was performed using 2μl (equivalent to 0.2µg of RNA) of each RT reaction as template with the HotStarTaq TM DNA polymerase kit (Qiagen SA). PCR primers were designed from published DNA sequences (Table I). Efficiency and specificity of amplification with the designed primer pairs were controlled using mRNA isolated from mouse or rat brain.

Tissue processing

The fetuses were collected as described in the Cell Culture section. The femurs of fetuses and born rats were rapidly excised and cleaned of skin and muscles, then cut longitudinally at the diaphysis level (for E18 and older fetuses) and immediately fixed in 4% paraformaldehyde in 0.1M phosphate buffer, pH 7.3 at 4°C for 24h [START_REF] Mckee | Effects of fixation and demineralization on the retention of bone phosphoprotein and other matrix components as evaluated by biochemical analyses and quantitative immunocytochemistry[END_REF]. The samples were decalcified in 4% di-sodium ethylenediamine-tetraacetic acid (EDTA) for 1 up to 5 weeks before dehydratation through a graded ethanol series and embedded in paraffin. Sections (5μm) were cut on a Leica RM2145 microtome (Leica instruments, Heidelberg, Germany) and processed for immunocytochemistry.

Immunocytochemistry

Cultures on glass coverslips were fixed in 3.7% formaldehyde, permeabilised in -20°C methanol and stored in PBS. Cytoenzymology for TRAP activity in RAW cells was performed with a commercial kit (#386-A, Sigma). Sections from decalcified samples were laid on silanised slides and dried overnight at 37°C. After two baths of methylcyclohexane and rehydration, the sections were treated, along with coverslips, for 15min with 100mM glycine and 50mM ammonium chloride in Tris buffer, pH 7.6, in order to saturate free aldehydic groups. Endogenous peroxidase activity was blocked by incubation for 15min with 1% sodium azide and 1.5% H 2 O 2 in 50% methanol. Sections and cultures were then incubated overnight at 4°C with primary antibodies diluted in Tris Buffer Saline (TBS: Tris-HCl 50mM pH: 7.6, 0.9% NaCl, 0.01% BSA) containing 1% normal rat serum to saturate non-specific sites. All goat polyclonal antibodies against Sema-3A and its receptor components were purchased from Santa-Cruz Biotechnology (Santa-Cruz, CA, USA). On sections, the anti rat-Sema-3A (sc1148) and anti human-NP-1 (sc7240) were used at 2μg/ml, the anti human-Plx-A1 (sc10139) at 0.4µg/ml, the anti human-Plx-A2 (sc10144) at 4µg/ml and the anti human-Plx-A3 (sc10134) at 0.8µg/ml. The rabbit anti rat-Tyrosine Hydroxylase (TH) (AB151), used at 1/5000, and the rabbit anti rat-Neurofilament Heavy Chain (NH) (AB1991), used at 1/2000, were purchased from Chemicon (Temecula, CA, USA). The production and use of the rabbit antiserum directed against bone sialoprotein (BSP) have been described previously [START_REF] Malaval | Kinetics of osteoprogenitor proliferation and osteoblast differentiation in vitro[END_REF] and references therein). Controls were incubated with non-immune goat or rabbit serum. For goat primary antibodies on sections, we used a second step incubation with a rabbit anti-goat IgG antibody (Dako, Copenhagen, Denmark) for 30 min. Both goat and rabbit antibody-antigen complexes were detected with a rabbit Dako EnVision + K4002 kit (Dako). For coverslips, the incubation time of primary antibodies was reduced to 30 minutes at room temperature, and the signal was enhanced by the use of a biotin labeled anti-goat secondary antibody with an additional incubation step with streptavidin-peroxidase. Labeling was revealed with 3-3' diaminobenzidine tetrahydrochloride (DAB) (Sigma) in Tris buffer containing 0.01% H 2 O 2 . All washings were done with TBS containing 0.02% Triton X100. Some labeled sections were counter-stained either with toluidine blue or with the DNA labeling compound, 4', 6 diamidino-2phenylindole (DAPI, Sigma); some unlabeled sections at P14 were stained with Goldner's trichrome for histological visualisation. Sections were dehydrated, mounted in Xam (Gurr-BDH Laboratory, Poole, UK) and then observed on a Leica DMRB microscope (Leica, Paris, France). Coverslips were mounted in Fluoprep (Biomérieux, Marcy l'Etoile, France). A: E17, arrows: staining in the periosteum (X600); B: E19, diaphysis along the primary ossification centre (X200), selected areas enlarged in C (periosteum) and D (cortical bone); C and D: arrows: labeled profiles (X1000); E: E21, diaphysis (X100), selected area enlarged in F (X600); G: P14, ossification front, arrows: vascular buds (X800), enlarged in H (X1200); I: Non-immune control at E21; J: Trabecular bone in the metaphysis at P14; stars: lumen of labeled capillaries (X1000). All sections were counter-stained with toluidin blue. staining in the bone collar (X800). C: NH at E21, diaphysis (X800). D: Non-immune control at E21 (X600). E: NH at P14, periosteum of the diaphysis; arrows: staining (X800). F: TH at E17, metaphysis; arrows: labeling in the periosteum (X1200). G: TH at E19, diaphysis along the primary ossification center; long arrows: staining in the periosteum, short arrows: staining in the bone collar (X400). H: TH at E21, diaphysis (X1200). I: non-immune control at E21 (X1200). All sections were counter-stained with toluidin blue.

LEGENDS OF FIGURES

  contains only bone marrow. At E21 vascular invasion of the epiphysis and development of the secondary ossification center has begun, and is well established at P14. Histologically, osteoblasts are observed lining the outer cortical bone, as the basal cell layer of the periosteum (fig. 3A, short arrows). They also line the inner cortical bone (endosteum) and are present as cuboidal cells along bone trabeculae in the metaphysis (fig. 3B, short arrows). Osteoclasts are larger, multinucleated cells, usually in direct contact with the bone matrix (Fig 3A, B, long arrows).

Figure 1 :

 1 Figure 1: Expression of RNA for Sema-3A signaling molecules by osteoblasts, osteoclasts

Figure 2 :

 2 Figure 2: Immunodetection of Sema-3A signaling molecules in osteoblasts and

Figure 3 :

 3 Figure 3: Immunodetection of sema-3A and NP-1 during endochondral development of

Figure 4 :

 4 Figure 4: Immunodetection of Plx-A1 and Plx-A2 during endochondral development of

Figure 5 :

 5 Figure 5: Immunodetection of Plx-A3 during endochondral development of rat femur.

Figure 6 :

 6 Figure 6: Immunodetection of NH and TH during endochondral development of rat

Table 1 : Primers sequences and PCR conditions.

 1

	Target	Origin Primer sequences (5' to 3')	Accession Number	Product length (bp)	Number of cycles	Annealing temperature (ºC)
	Sema-3A Rat	GTGTTCCTTGG CCATATGCTC CAGCAGTTGAGCCAATGT CAG	NM_017310 472	35	48
	NP-1	Mouse	CTGGTGAGCCCTGTGGTCTATTCC ATATCATCCACAGCAATCCCACCG	AF 018957	272	35	55
	Plx-A1	Mouse	TCACTCACCTGGTGGTTCA CACAAAGCCAAACATGTCAGC	D 86948	505	35	50
	Plx-A2	Mouse	CATGTCTGAGAGACAGGTCACCA CCAGGTGAGATGCAGATGACTTG	D 86949	360	35	55
	Plx-A3	Mouse	GCTGTTGATGGCAAGTCTGA GAAGGAGCACTGACAAAGC	NM_008883 198	40	52
	ColXa1 Mouse	GCAACTAAGGGCCTCAATGG GAGCCACTAGGAATCCTGAG	NM_009925 596	35	57
	OCN	Mouse	CAAGTCCCACACAGCAGCTT AAAGCCGAGCTGCCAGAGTT	X 04142	371	30	58
	OCN	Rat	AGGACCCTCTCTCTGCTCAC AACGGTGGTGCCATAGATGC	X04141	294	25	60
	BSP	Rat	ATGGAGATGGCGATAGTTCG TGAAACCCGTTCAGAAGGAC	NM_012587 504	30	60
	CTR	Mouse	GTCTTGCAACTACTTCTGGATGC AAGAAGAAGTTGACCACCAGAGC	U18542	255	40	55
	TRAP	Mouse	CTCTCTGACCACCTGTGCTTCCTC GAACCTCTTGTCGCTGGCATCGTG	NM_007388 292	26	55
	GAPDH Mouse	ATCACTGCCACCCAGAAGAC ATGAGGTCCACCACCCTGTT	AK015422	443	25	57

Table 2 -Expression of Sema-3A signaling molecules during endochondral ossification.

 2

	Stage Site	Sema-3A/NP-1 Plx-A1/Plx-A2	Plx-A3	Nerve markers
		Periosteum	+	++/+	+	Along the metaphysis
	E17
		HC	++	++	+	-
		Periosteum	++	+	+	Ubiquitous
	E19
		POC	NP-1	+	-	-
		Periosteum	++	+	+	+
	E21	Cortical bone	+	+	+	+
		Metaphysis	++	+	+	++
		Periosteum	+	+	+	+
	P14	Cortical bone	-	Osteoblasts Osteocytes	+	+
		Metaphysis	Osteoblasts and osteoclasts under the growth plate	Osteoblasts (few for A2) and osteoclasts under the growth plate	Endothelial cells under the growth plate	Loose network in the marrow
	+: Expressed, -: Not detected. HC: Hypertrophic chondrocytes; POC: Primary ossification center.

ACKNOWLEDGEMENTS

This work was supported by the "Institut National de la Santé et de la Recherche Médicale" (INSERM), Paris, France, and by an "Emergence" grant funding (to CC and LM) and a postgraduate fellowship (CG) from the Rhône-Alpes Region, France. The authors thank Irma Machuca and Frédéric Saltel (UMR5161) for advice and help with the RAW 264.7 cell line.


Serge Guillaume
email: serge.guillaume@cemagref.fr

Brigitte Charnomordic

Knowledge discovery for control purposes in food industry databases

Keywords: Fuzzy Logic, Genetic Algorithm, Machine Learning, Knowledge Discovery, Process Control, Food Industry, Cheesemaking

Knowledge discovery for control purposes in food industry databases

Introduction

Food Industry involves many strongly non linear multivariable complex processes.

Control of processes by classical methods is based on model use. But, for complex and imperfectly understood processes, phenomena are not sufficiently well known to be mathematically characterized, hence the importance of empirical knowledge of a human operator. To act, the operator relies above all on his know-how, in conjunction with the interpretation of physical measurements given by sensors and partly through subjective appraisal of sensory criteria.

The class of problems we are interested in involves a great number of variables. Expert knowledge does exist, but it does not cover all situations. Let us take as an example the case of a traditional cheesemaking process, such as the making of Comt cheese. The process can be viewed as a series of sequential subprocesses, complex and multivariable themselves. To reach a high quality product, the human operator must be fully aware of the characteristics of subprocesses and he must integrate them into the whole process.

He acquires deep knowledge on the influence of a particular control variable over the final product, as well as about the impact of some state variables, at a given step. What remains very difficult for him to appreciate concerns the interactions between variables which belong to different subprocesses. Nevertheless, these interaction factors are presumably important.

In an adaptive tracking perspective, for target following through and to correct drifting off, this process, as many similar food industry processes, needs a decision support tool.

Over the past several years, fuzzy control based on expert knowledge has proven a successful method of designing controllers [START_REF] Mamdani | Twenty years of fuzzy control: Experiences gained and lessons learnt[END_REF], [START_REF] Perrot | Evaluation temps rel de la couleur de biscuits. comparaison entre deux approches : floue et baysienne[END_REF]. A similar approach was tried out for Comt [START_REF] Charnomordic | Knowledge-based fuzzy control in food industry : A decision support tool for cheesemaking[END_REF]. Expert knowledge introduced in the fuzzy inference system was not sufficient to give a satisfactory predictive quality, the lack of knowledge about variable interaction being strongly suspected for the poor predictive performance.

Even for non automated processes, process data are usually available. For quality control purposes field data are collected on a regular basis. Process databases constitute a potential huge source of information still to be made the most of.

Process data describe the product being processed. They contain implicit control decisions and their consequences on the product. In other words they hold expert knowledge. Knowledge discovery based on process data can help to build a realistic decision support tool. Extracted knowledge must be interpretable by experts to allow a better understanding of the process, and a profitable cooperation between human knowledge and computer learning.

Quality criteria for a knowledge discovery model include and go beyond quality criteria for a blind learning model. Common learning criteria must be respected including quality of fit for the learning database and use of validation data for robustness. Complementary criteria concern discovered knowledge. The validity range, the novelty as compared to what is already known by experts, the interpretability level are all important and not easily quantified.

Our method will use a fuzzy inference system framework and data from a database.

Training procedures have been applied to fuzzy inference systems for tuning fuzzy set parameters or learning the rule base. Training techniques were often inspired by optimization methods used in other fields. Neural networks learning procedures combined with fuzzy control led to a new family of fuzzy controllers, called neuro-fuzzy networks [START_REF] Jang | Structure determination in fuzzy modeling: a fuzzy cart approach[END_REF]. Stochastic algorithms, for instance genetic algorithms, were used for automatically generating a rule base from training data. Although these approaches seem to be interesting, they have been applied to academic problems only, such as pole-cart balancing [START_REF] Chiang | A self-learning fuzzy logic controller using genetic algorithms with reinforcements[END_REF], [START_REF] Leitch | New techniques for genetic development of fuzzy controllers[END_REF]. These involve a few variables. Data bases from food processes are likely to be different. Numerous variables are collected, data are not especially designed for our use : data are noisy, some values are missing, the experiments do not cover the input space.

When there are more than a few input variables, the number of generated rules dramatically increases with the number of variables. This is known as "the combinatorial curse of dimensionality", and it is a serious restriction to the applicability of automatic generation techniques, for real world problems, such as the ones we are interested in.

The objective of this paper is to generate a set of interpretable fuzzy rules from a training data base. Training can be seen as a validation procedure based on available data: potential rules contain knowledge which has to be confirmed or infirmed by data. Each rule of the final fuzzy rule set describes a trajectory in the variable space. A procedure will be discussed for rule and variable reduction, to avoid a too high dimensionality and to insure consistency. This procedure is not yet completely automatic: experts are asked to confirm variable removal. Practical aspects will be discussed in a case study, using data from a cheesemaking process.

System Description

Fuzzy Logic for learning: why and how

The problem of control can be viewed as one of decision making, i.e. given the observation of the process state to decide which action to take. Decision making can be achieved by a fuzzy controller. A fuzzy controller is a rule based system whose inference engine benefits from the fuzzy logic capability to handle human reasoning. The fuzzy controller architecture includes three principal components: a fuzzification procedure, an inference engine and a defuzzification step. The fuzzification procedure converts the values of the fuzzy controller input variables into linguistic values, corresponding to fuzzy sets. The fuzzy inference engine relies on a set of fuzzy rules to infer a fuzzy output. A fuzzy rule is divided into two parts : the antecedent part and the consequent part. Suppose that we have the two following rules :

rule1 : if X is A 1 and Y is B 1 then Z is C 1 rule2 : if X is A 2 and Y is B 2 then Z is C 2
At time t, the process state is given by the (x t , y t ) vector, represented by linguistic variables X and Y. Given the membership function for each linguistic label (A 1 , A 2 , B 1 , B 2 ), membership degrees µ A i (x t ) and µ B i (y t ) can be computed for each rule. Combination of conditions in the antecedent part is done using a fuzzy logic AND operator, denoted as ∧, most often chosen as a product or minimum. The matching strength of Rule i , also called rule weight, is called w i and defined as follows:

w i = µ A i (x t ) ∧ µ B i (y t ) (1)
The consequent part z i of Rule i is calculated by applying a fuzzy implication mechanism. The fuzzy output is obtained by aggregating Rule 1 and Rule 2 consequent parts. The simplest defuzzification procedure is done by computing a weighted average. If we generalize to a set of n rules, the fuzzy controller output is

z = i w i .z i i w i (2)
To be more useful in modeling process drifts, the output can be expressed in terms of a fixed target value moderated by a fuzzy inferred output, which comes to

O = T + i w i .z i i w i (3)
We will use the fuzzy controller as a knowledge learning framework. Available data are considered as an implicit recording of human decisions and process state all through the process. A rule base will be built that best reflects the data used for learning and the hidden knowledge they contain. The generated rules are checked for consistency and interpretability. A rule consistency criterion will be applied to avoid internal contradictions. For interpretability sake, the rule base must be kept to a reasonable size. Variable and rule reduction will be done to point out major trends and to make the rule base easier to be interpreted by experts.

It should be emphasized that the performance of the fuzzy rule base is highly dependent on the database quality. Ideally the database should represent an extensive coverage of all kinds of various situations.

Practical Implementation

The fuzzy controller is used as a fuzzy inference system: instead of a control action, its output is a context dependent target value.

Its input variables are state or action variables. For each variable, fuzzy sets need to be defined. Two possibilities arise for defining the fuzzy sets. Partitioning can be self-generated by observing the distribution of values in the database, optimized by using stochastic techniques such as genetic algorithms, or it can be established by taking advice from experts. We decided on the last method for a better readability, linguistic labels chosen by experts being more meaningful.

Once the partitioning decided upon, a first set of rules is generated by considering all possible combinations of input fuzzy variables.

Learning consists of running a genetic algorithm to optimize the consequent parts of the fuzzy rules in order to design the rule base best suited to the database. The best fit is considered in terms of performance as well as rule consistency and interpretability. Genetic algorithms offer two main advantages in our case. First a global optimization technique is needed because of the complete lack of knowledge about each rule consequent part. Moreover the same computational frame could be reused for automatically partitioning the input space of some variables.

The method used must be applicable to real world databases, which are often plagued by a non neglectable number of missing values. A strategy for dealing with missing values without loosing too much information is needed. It is essential not to increase dimensionality, thus we do not define a specific linguistic label for missing values. Missing values are mostly due to manual information collecting rather than sensor faults in a specific range. In order not to eliminate possibly important combinations in rule definition, missing values are considered as equally distributed over the existing fuzzy sets. Indeed each of the linguistic labels is meaningful, because of the partitioning being done by experts. This assumption would be questionable in the case of automatic partitioning.

The set of rules will be built accordingly. During the optimization step, missing values will be replaced by randomly picked values in the corresponding variable range, and then fuzzified. Of course, the accuracy of the results is dependent on the ratio of missing values. A reasonable limit seems to keep this ratio less than 20%.

Genetic Algorithms

Since they have been proposed by Goldberg [START_REF] Goldberg | Genetic Algorithm in Search, Optimization and Machine Learning[END_REF], Genetic Algorithms (GA) have been widely used in optimization and machine learning. GA are part of stochastic techniques. These are particularly useful when the number of possibly solutions of a problem is so large that it is impossible to test each of them. Their objective is twofold : explore the solution space using a random way to find better and better solutions and, optimize these by finding the local optima, that means the best solutions in their neighborhood.

The solution found is not guaranteed to be the best one, but the aim of the iterative process is to reduce the set of solutions to the global optima.

GA use biologically inspired concepts.

A set of solutions, called a population, is alive all along the process. Each solution, called a chromosome, is made of a set of genes which encode the problem.

1 0 0 1 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 0 1 1 Rule 1 Rule 2 Rule 4 Rule 5 1 Figure 2: Chromosome encoding
In our n-rule problem, a chromosome is made of n substrings, each substring representing the encoded consequent part of a rule. All substrings have the same length. Figure 2 illustrates a 5-rule, 5-bit substring chromosome.

Each step of the iterative process is called a generation.

At the end of a step, each chromosome is evaluated through an objective function. Those which have the best performances according to this function are allowed to breed. This is done by two kinds of operators : crossover and mutation.

The sexual crossover operator produces two children from two parents chosen among the best candidates, a mutation operator is applied to bring disorder.

In the framework of our application, GA are used to optimize the consequent part of the rules, their antecedent being fixed. The chromosome is made of a set of encoded numerical values, each one being associated to a rule. A value is encoded as a binary string. The chromosome may thus be considered as a sequence of substrings.

The crossover operator we use is called 'OnePointCrossover' : the crossover point is randomly chosen within the chromosome length. The chromosome of the first child is made by combining the part of the chromosome of the first parent located before the crossover point and the part of the chromosome of the second parent located after this point. The chromosome of the second child is its counter part.

This operator seems to be suitable for substring handling : at most one substring is modified. The mutation is done by inverting some genes, randomly chosen according to a given probability.

Figure 3 shows the effect of these operators on the substrings located near the crossover point for a 5-bit encoding. The bit inside the grey box has been changed by the mutation operator.

Each rule consequent part is allowed to vary within two limits Low and High. If l is the length of the binary string, the consequent part associated to a rule is :

Learning is based on an objective function. The data base is a set of multidimensional data points, including input and output variables. The objective function will assess the chromosome performance by comparing the observed output with the fuzzy system output. The fuzzy system output is given by equation ( 2), where z i is the consequent part of rule i. The best chromosomes are those which minimize the square root of the squared sum of errors, also called RSSE, given in the following equation:

RSSE = n (P redicted output -Observed output) 2 (5)
where n is a given number of data points either randomly chosen in the base or taken in a sequential order.

The complete architecture of the controller is given in figure 4.

3 Space reduction to make inferred knowledge easy to understand

An exhaustive system would be quite complex. Such a system presents some drawbacks :

• The size of the system : the larger it is, the more difficult the finding of an optimal solution is. Let us consider a process involving 12 input variables, each of them described using 3 membership functions. Then the number of rules in the system is 3 12 , more than 500000.

Let us suppose than the consequent part of a rule may take 32 values, then the number of combinations is 32 500000 .

• The robustness : a complete system is likely to be over dimensioned. Extra variables bring more noise than information. Training in these conditions is less robust, as it is likely to learn noise instead of the underlying process represented by the training set.

• The knowledge : instead of getting the best fit we are rather interested in finding out major trends. Our aim is knowledge discovery and not the design of an accurate predictive system. This knowledge may be used, in cooperation with the experts one, to control the process.

The proposed methodology aims to reduce dimension in both spaces : the variable space (number of inputs) and the rule space.

The selection is done through 2 iterative procedures.

Rule selection

Considering a given set of input variables and the corresponding partitioning, all the possible rules are implemented.

Let us give some basic definitions :

• activity : a data point activates a rule when the corresponding rule weight is non zero.

• active data point : a data point is active when it activates at least one rule of the rule base.

• active rule : a rule is active when it is activated by at least one data point.

Obviously, the only rules which may be validated by the procedure are the active ones. The rule selection is done according to 3 criteria :

1. Influence :

The first criterion is the influence of the rule in the data base. For a given data point the rule weight is the product of all input membership degrees, as shown in Equation 1. For a given rule, the influence is measured by its cumulated weight in the data base. Only the most influent rules are kept. Let us note that this selection step does not involve GA.

Consequent part level :

GA associate a consequent part C i to each rule R i in the system. Remember that C i represents a gap to be added to a base target value (as in equation 3). GA being random based techniques, they are non deterministic : successive runs may lead to different results. We choose to repeat r times the GA with the same configuration to improve reliability. At the end of the repetitions, we have, for each rule R i , r values for C i : C i 1 , C i 2 , . . . , C ir . The final value is taken as the median of the C i j , j = 1 . . . r values. Thus a value about zero might result from a combination of negative and positive values, and cannot be trusted.

We can distinguish between two kinds of rules :

• neutral rules : a rule R i is considered as neutral when median(C i j ) < C th , j = 1 . . . r, where C th is a given threshold. Neutral rules contain standard knowledge.

• sensitive effect rules are rules whose consequent part final value is above a given threshold. They interest us most as they are likely to explain deviations from the standard product being processed.

In our approach, we choose to remove neutral rules.

Consistency :

The last criterion takes into account the rule consistency. It is expected for rules activated by the same group of data points to have a similar behavior. The link between two rules i and j is defined as follows :

Link i,j = N i,j max(N i , N j )
where N i , N j , are cardinals of the sets E i , E j of data points which activate respectively rules i and j; N i,j is the cardinal of E i ∩ E j .

When two rules are strongly linked, Link i,j greater than a given threshold, their consequent parts either have the same sign or are inconsistent. Inconsistent rules are eliminated before next iteration.

It is important to note that as the number of rules is reduced, the number of active data points decreases.

Variable selection

Once the rule selection is done, the selected rules can be analyzed in order to assess the relative importance of the input variables.

The variable selection step is necessary. Databases are constituted of field data, and recorded variables do not necessary present a real interest for control purposes. The goal of this procedure is to keep, as a data point descriptor, the variables which are useful for control, and is at the heart of knowledge discovery.

The variable selection is based on the following axiom, which requires a rule distance function:

IF two rules are close AND IF they are weakly linked AND IF their consequent parts have opposite signs THEN the inputs which differ are of major importance for the output, within the context described by the other inputs.

Similarly, if the rules are close and weakly linked, and if their consequent parts have the same sign, then the inputs which differ do not have a noticeable effect on the output, according to the whole context.

The distance between two rules can be defined as the number of variables whose linguistic label values differ in the rule descriptor. A rule descriptor is defined as a sequence of integers, and only describes the antecedent part of the rule. Each integer corresponds to the linguistic label value for the corresponding input variable. Let us consider the following example, where we have four input variables.

Rule 1 : if V ar 1 is A 3 and V ar 2 is B 2 and V ar 3 is C 2 and V ar 4 is D 3 Rule 2 : if V ar 1 is A 1 and V ar 2 is B 4 and V ar 3 is C 2 and V ar 4 is D 2
Rule 1 and Rule 2 have the following descriptors: 3223 and 1422. They differ in three digits: first, second and fourth digits. The distance between rule 1 and rule 2 is considered as equal to three.

Generalizing to a n-input system, the distance between any two rules is given by n i=1

(1)

{m i = l i } (6)
where m i and l i are respective linguistic label values taken by variable i in Rule 1 and Rule 2, and n is the number of input variables. Variables should be removed which have not a noticeable effect on the output, whatever the context. The variable removal is currently achieved with the help of experts to prevent errors in the elimination of variables. Indeed, some rules in the system may be activated by a very small numbers of data points. Upon removal, a new reduced more significant system can be built. The complete reduction procedure for rules and variables, shown in figure 5, is reiterated.

4 Application to a cooked and pressed cheesemaking process (Comt)

Cheesemaking for cooked and pressed type cheese such as Comt runs over at least 4 months until 6 to 12 months including ripening. The first process stage that produces young cheese takes about 20 hours. This stage is divided into several steps: milk maturation, coagulation, draining in vat-cookingstirring, and pressing, as shown in figure 6. Each step is characterized by a set of variables, described in table 1. All variables correspond to numerical measurements, except those marked with a bullet (•) in the second column. These variables are routinely measured and available in technical report sheets. At the end of pressing, some analytical tests may be applied to selected young cheeses, in order to assess properties of a cheese set. The most common tests are physico-chemical analyses (pH, moisture content, fat content and lactate ratio being the most common ones). The present paper is restricted to generating the rules which mark the influence of process variables onto moisture content (in young cheese). Moisture content, abbreviated as MC, is the fuzzy system output variable. Partitioning the input space was done with the help of experts. The database included 172 data points, with a number of missing values distributed among all variables. The method described in section 2 was used to generate a first rule base, including all possible 11 input variables. The variable selection procedure was applied to reduce the number of relevant variables and to build a second rule base through reiteration of the rule selection procedure. The third rule base was built again in a similar way. Table 2 gives a summary of the results. For each rule base, the number of variables, the percentage of missing values are given in the first two lines. The total number of possible rules corresponding to the input space partitioning is indicated. Data points available in the database only activate some of these rules, called active rules. The fuzzy system output is computed from a base value, chosen as 55, which is modified by all active rules (see equation 4). Each rule is allowed to influence the base value within the range [-5,+5].

Input variable/

Rule Let us give some comments about the first rule base. Two indices give a measurement of the system performance: the total RSSE, defined as in equation 5, and the standard error of calibration SEC, given by

SEC = RSSE (n)
where n is the number of active data points.

Considering the high dimension of the initial rule base (500 000 possible rules), the performance of the method appears quite satisfactory. In the fuzzy system, each rule is allowed a given influence on the output ([-5,+5]). Thus a n-rule system output might range between [-5*n, +5*n] about the base output, which, in our case, is far from reality: moisture content is a percentage and only varies from 0 to 100. No validity control is done, convergence is achieved by the optimization algorithm. Even so, the maximum error (3.2) and the total RSSE (8.5) resulting from the generated rule set are very reasonable. The relative error given by SEC is 1.0, which is close to the known accuracy of moisture content measurements (0.6). The variables designated for removal at the end of first step are given in column 1 of table 2. In concert with experts, that choice was confirmed. The first three variables to be removed have in common to be poorly measured for different reasons. RpH suffers from a high inaccuracy in measurements due to the small observed variability. RT is not of much use in practice and has many missing values. The grain size GS could certainly be very informative, but is currently attributed on subjective criteria with no established common reference.

The second system built shows the best performance of the three: all error markers decrease. The third system is characterized by a small set of four variables. Variable selection procedure shows that, although available variables all contribute to the process state, moisture content is essentially determined by the cooking stage.

Even though error markers increase in value, they remain reasonable. This last system was useful to bring out some new elements of knowledge about the process.

Discussion

At each step, the procedure highlighted some variables, as being more of interest for process control. The most valuable elements of knowledge discovered were:

• statement 1: Use High maximum cooking temperature (MT) to raise moisture content.

• statement 2: Use LowActivity starters to raise moisture content, if MT is not High.

• statement 3: Stirring Duration (SD) and Cooking Duration (CD) are complementary.

From the first system a well known fact in process characteristics came out. The maximum cooking temperature MT is a dominant feature, and tends to shadow other processing factors. Our variable selection procedure also identified three variables (RT, RpH and DT) as of little importance. After discussion experts confirmed the dominant influence of MT, as well as the secondary role of the three variables listed above for the output being studied (Moisture Content). They also suggested GS (curd grain size) removal, due to a lack of established reference values. More was expected concerning starters influence. Starters influence may be hidden by MT effects, but should come out somehow. Experts then realized that the initial fuzzy partitioning of starters space was not well adapted and could be improved. A new partitioning was suggested.

After removal of four variables, the phenomenon described in the second statement came out clearly. Rule Base 2 shows that starters behavior is context dependent. It may influence moisture content if and only if Maximum Temperature is not High. A high temperature slows down starters activity. The starters variable results from a multidimensional analysis of various characteristics. The new partitioning was done by giving more weight to some of them. Results show that they are more important for modifying moisture content that the ones used in the previous partitioning.

The third statement is still under study. It somehow reflects the diversity of know-how. Depending on the manufacturing unit, cheesemakers seem preferably to use either one or the other of CD and SD to control moisture content.

Both rule and variable selection may be improved. The main drawback with rule selection is that neutral rules are eliminated by the consequent part level criterion, as explained in section 3.1. However they contain the knowledge to get a standard product, and are activated by numerous data points. Once the reduction is done, it would be interesting to expand the selected rule set by reintroducing some of the neutral ones according to new criteria yet to be developed.

Variable selection is based upon rule distance. Rule distance definition given in equation 6 can be improved. One way would be to consider than an elementary distance, for a given factor, between 0 ant 2 is double than the elementary distance between 0 and 1, or between 1 and 2. This is particularly true if the factors 0, 1 and 2 correspond respectively to the linguistic concepts of SMALL, MEDIUM and LARGE. Another complementary way would introduce a weight on variables, which could be adjusted during the variable selection step.

The objective of this work was knowledge discovery in food industry process data bases. A self learning fuzzy inference system selects among all the possible rules those which are best suited to the data. A variable selection step eliminates variables which do not present a real interest for control purposes. These two iterative procedures lead to a small set of selected rules and variables.

The method has been applied to a cheesemaking process. The discovered knowledge has been confronted with the expert knowledge. The discovered knowledge partly confirms the expert one. It also brings out new questions, especially by highlighting context dependent effects of some variables. Later on, the rule base will be introduced into the process control loop to build the fuzzy inference engine. On line learning should be used to validate and improve it.

The rule selection procedure eliminates neutral rules. Nevertheless, these are important for the process, they contain the recipe to make a standard product. Further work will have to develop new criteria to keep neutral rules.

A

  Figure 1: A fuzzy controller

  Figure 3: Genetic operators

  Figure 4: The fuzzy controller

Figure 5 :

 5 Figure 5: Selection of rules and variables

Figure 6 :

 6 Figure 6: Summary of Cheesemaking

6

  Author-produced version of the article published in FuzzySets and Systems, 2001, N°122(3), p.487-497.

	The original publication is available at http://www.sciencedirect.com
	Doi: 10.1016/S0165-0114(00)00094-4
	Observed output	Difference
		Genetic Algorithm
	Fuzzy Inference System
	Input

Table 1 :

 1 Available variables in database

	Step	qualitative variable abbreviated as
	Maturation of milk
	Milk (in vat) type	•	Milk
	Starters combination	•	Starter
	Coagulation
	Renneting temperature		RT
	Renneting pH		RpH
	Cutting time to clotting time ratio		CCR
	Curd grain size	•	GS
	Draining/Cooking/Stirring
	Maximum temperature		MT
	Cooking Duration		CD
	Stirring Duration		SD
	Draining Temperature		DT
	Pressing
	Dornic acidity (of serum)		DA
	Output variable
	Moisture Content		MC

Table 2 :

 2 Summary of the results for the three rule bases

	Base

Author-produced version of the article published in Fuzzy Sets and Systems, 2001, N°122(3), p.487-497. The original publication is available at http://www.sciencedirect.com Doi: 10.1016/S0165-0114(00)00094-4

Acknowledgments

The authors would like to thank Antoine Brodier and Yannick Renard, from Comit Technique du Comt, for giving us access to valuable expert advice and technical data sheets; Yolande Nol and Mickal Glaudel, from SRTAL INRA, for helping us in understanding milk and cheesemaking technology.

Author-produced version of the article published in Fuzzy Sets and Systems, 2001, N°122(3)


Josu Doncel
email: josu.doncel@inria.fr

Nicolas Gast
email: nicolas.gast@inria.fr

Bruno Gaujal
email: bruno.gaujal@inria.fr

Are mean-field games the limits of finite stochastic games? *

INTRODUCTION

Mean-field games model the rational behavior of an infinite number of indistinguishable players in interaction [START_REF] Lasry | Mean field games[END_REF]. An important assumption of mean-field games is that, as the number of player is infinite, the decisions of an individual player do not affect the dynamics of the mass. Each player plays against the mass. A mean-field equilibrium corresponds to the case when the optimal decisions of a player coincide with the decisions of the mass. This leads to a simpler computation of the equilibrium.

It has been shown in [START_REF] Gomes | Discrete time, finite state space mean field games[END_REF][START_REF] Guéant | Mean field games and applications[END_REF][START_REF] Tembine | Mean field asymptotics of markov decision evolutionary games and teams[END_REF] that for some games with a finite number of players, the Nash equilibria converge to mean-field equilibria as the number of players tends to infinity. In fact, many authors argue that mean-field games are a good approximation of symmetric stochastic games with a large number of players, the rationale behind this being that the impact of one player becomes negligible when the number of players goes to infinity.

In this paper, we question this assertion. We show that, in general, this convergence does not hold. In fact, the "tit for tat" principle allows one to define many equilibria in repeated or stochastic games with N players. However, in mean-field games, the deviation of a single player is not visible by the population and therefore the "tit for tat" principle cannot be applied. The conclusion is that, even if N -player games have many equilibria with a good social cost, this may not be the case for the limit game.

STOCHASTIC GAMES WITH IDENTI-CAL PLAYERS

We consider a class of stochastic games, as introduced by Shapley [START_REF] Shapley | Stochastic games[END_REF] for zero sum games and generalized by Fink [START_REF] Fink | Equilibrium in a stochastic n-person game[END_REF] to N players. Players are identical and anonymous, in the sense that the dynamics, the costs and the action policies only depend on the population distribution, as detailed below.

The finite state space of each player is S = {1, . . . , S} and its finite action set is A = {1, . . . , A}. The players' states at time t are denoted by X(t) = (X1(t), . . . , Xn(t), . . . , XN (t)), with Xn(t) ∈ S. The state of the players evolve in continuous time: each player n takes actions (denoted An(t) ∈ A) at instants distributed according to a Poisson process, independently of the others. The superposition of all these N activation processes forms a Poisson process whose intensity is proportional to N , the number of players. If we only observe the system at these activation instants, and using uniformization, the system can be seen as a discrete time model where players take actions at discrete times TN = {i/N } i∈N .

The population distribution at time t is denoted by M(t) ∈ P(S), where P(S) is the set of probability measures on S. As the set S is finite, M(t) is a vector with |S| components and for all s ∈ S, Ms(t) being the fraction of players in state s at time t:

Ms(t) = 1 N N n=1 1 {Xn(t)=s} .
We assume that players interact according to a mean field model, namely, the Markovian evolution of X(t) at time t ∈ TN can be written as

P Xn(t + 1 N ) = j Xn(t) = i, M(t) = m, An(t) = a = 1 N Pij(a, m). (1)
We consider that each player chooses its own stationary mixed strategies. Such a strategy is a measurable function π : S ×P(S) → P(A), that associates to each state i ∈ S and each population distribution, a probability measure πi(m) on the set of possible actions -P(S) and P(A) are the sets of probability measures over S and A (as A is finite, P(A) is the simplex). We denote by πi,a(m) the probability that, under m, a player in state i takes action a, At time t ∈ TN , the player n suffers an instantaneous cost c Xn(t),An(t) (M(t)), function of her state Xn(t), the action that she takes An(t) and the population distribution M(t). The objective of player n is to choose a strategy π n from some set of admissible strategies Π, in order to minimize her expected discounted payoff, knowing the strategies of the others. The discount factor is denoted by β. Given a strategy π n ∈ Π used by player n and a strategy π ∈ Π used by all the others, we denote by V (π n , π) the expected discounted payoff of player n:

V N (π n , π) = E   t∈T N e -βt c Xn(t),An(t) (M(t)) An has d.b. π n A n has d.b. π (n = n)   .
An equilibrium for this game is a strategy π such that a player has no another admissible strategy that leads to a higher payoff. This notion depends naturally on the set of admissible strategies.

Definition 1 (Nash Equilibrium). For a given set of strategies Π, a strategy π ∈ Π is called a symmetric Nash equilibrium in Π for the N player game if, for any strategy

π n ∈ Π, V N (π, π) ≤ V N (π n , π).
The existence of a stationary Nash equilibria is proven in [START_REF] Fink | Equilibrium in a stochastic n-person game[END_REF] when cia(m) and Pij(a, m) are continuous in m.

To make the connection with the mean field game model presented below, let us remark that when all players use a stationary strategy π, the evolution of the population distribution satisfies

1 N E Mj(t + 1 N ) -Mj(t) M(t) = i∈S a∈A Mi(t)Qij(a, M(t))πi,a(M(t)), (2)
where the rate matrix Q(a, M(t)) := P (a, M(t)) -Id.

MEAN-FIELD GAME MODEL

We consider an infinite homogeneous population of players. We denote by m π (t) ∈ P(S) the population distribution at time t. As the state space is finite, m π (t) is a vector whose ith component, m π i (t), is the proportion of players in state i at time t. We assume that the initial condition m π (0) = m0 is fixed. For t ≥ 0, the population distribution evolves over time according to the following differential equation: for j ∈ S

ṁπ j (t) = i∈S a∈A m π i (t)Qij(a, m π (t))πi,a(m π (t)). (3)
The rationale behind this differential equation is Equation ( 2): the players in state i that take action a ∈ A move to state j at rate Qij(a, m π (t)).

We now concentrate on a particular player, that we call Player 0. Player 0 chooses her own strategy π 0 . We denote by x(t) the probability distribution of Player 0 when she applies strategy π 0 and the population uses strategy π. For a given state i ∈ S, xi(t) denotes the probability for Player 0 to be in state i at time t. The distribution x evolves over time according to the following equation: for j ∈ S

ẋj(t) = i∈S a∈A xi(t)Qij(a, m π (t))π n i,a (t). ( 4
)
If Player 0 is in state i and takes an action a, it suffers from the instantaneous cost ci,a(m π (t)), introduced in Section 2. Given a population strategy π and the strategy of Player 0 π n , we define the discounted cost of Player 0 as

V (π n , π) = ∞ 0 i∈S a∈A xi(t)ci,a(m π (t))π n i,a (m π (t))e -βt dt,
where β is the discount factor. Player 0 chooses the strategy that minimizes her expected cost, which depends as well on the strategy π. When Player 0 does so, we say it does the best-response to the mass strategy π.

BR(π) = arg min

π n V (π n , π).
Definition 2 (Mean-Field Equilibrium). A strategy is a (symmetric) mean-field equilibrium if it is a fixed point for the best-response function, that is,

π M F E ∈ BR(π M F E ).
Applying the Kakutani fixed point theorem for infinite dimension spaces over a modified version of the correspondence BR, one can show the existence of a mean-field equilibrium for these mean-field games under very mild continuity assumptions. The proof is available in [START_REF] Doncel | Mean-Field Games with Explicit Interactions[END_REF].

Theorem 1 (Existence of equilibrium, [START_REF] Doncel | Mean-Field Games with Explicit Interactions[END_REF]). Assume that Qij(a, m) and cia(m) are continuous in m. Then, there always exists a mean-field equilibrium.

(NON-)CONVERGENCE

Mean-field games as a natural limit

Using classical arguments from mean field theory, one can show that when the strategy of the players is given, the population distribution for the N player game described in Equation (2) converges to the population distribution of the mean field limit (3). More precisely, in an N player-game, the influence of a single player on the mass m is of order 1/N . In fact, it is shown in [START_REF] Tembine | Mean field asymptotics of markov decision evolutionary games and teams[END_REF] that if the cost functions ci,a(m), the transition kernels Qij(a, m) and the policy πi(m) are continuous in m then the dynamics of the population converges to the solution of the differential equation ( 3) and the evolution of one player converges to solution of (4).

Furthermore, it is shown in [START_REF] Doncel | Mean-Field Games with Explicit Interactions[END_REF] that under such continuity conditions (i) if π be a mean-field equilibrium, then there exists N0 such that for all N ≥ N0, π is a ε-equilibrium of the N player game; and (ii) if (π N )N is a sequence of Lipschitz-continuous strategies such that π N is a equilibrium for the N player game, then, any sub-sequence of the sequence (π N )N has a sub-sequence that converges weakly to a mean-field equilibrium. This is a more or less direct consequence of Theorem 3.3.2 in [START_REF] Tembine | Mean field asymptotics of markov decision evolutionary games and teams[END_REF].

Counter-example in the general case

The positive result presented in the previous section uses an important unnatural assumption: the strategy of a player should be continuous in m. In the definition of a Nash equilibrium, there is no a priori reason to restrict to strategies that are continuous in m, even when the cost and transition kernel are continuous in m. In this section, we show that removing this assumption leads to a very different result.

To construct such an example, let is consider a symmetrical version of the prisoner's dilemma. The state space of a player is S = {C, D} (that stand for Cooperate and Defect, respectively) and the action set is A = S. Let mC (resp. mD) be the proportion of players in state C (resp. D). The population distribution is m = (mC , mD). The instantaneous cost of a player depends on her state i and on the mass m:

ci,a(m) = mC + 3mD if i = C 2mD if i = D
This cost corresponds to the expected cost of a player that is matched with another player at random and suffers a cost according to the following cost matrix:

C D C 1, 1 3, 0 D 0, 3 2, 2 (5)
The state of a player coincides with her current action. When a player chooses action a, the next state becomes a with probability one. This gives the following transition matrices:

P (C, m) = 1 0 1 0 and P (D, m) = 0 1 0 1 .
A player that has no impact on the actions of the other players should play D: indeed, as indicated by the cost matrix (5), its cost is smaller when she plays D. This leads to the following result: Lemma 1. There exists a unique mean-field equilibrium π ∞ that consists in always playing D.

Proof. We consider that Player 0 has state vector x and the mean-field is m. Taking into account that

xC (t) + xD(t) = mC (t) + mD(t) = 1, her expected cost is ∞ 0 [xC (t)mC (t) + 3xC (t)mD(t) + 2xD(t)mD(t)]e -βt dt = ∞ 0 [xC (t) + 2mD(t)]e -βt dt.
This cost is minimal when xC (t) is minimal. This shows that the best response of player 0 is to always play D.

A similar proof shows that π ∞ is also a Nash equilibrium for the corresponding stochastic game with N players. However it is not the only one. Let us define the following stationary strategy:

π N (m) = D if mC < 1 C if mC = 1.
This strategy can be rephrased as "play C as long as everyone else is playing C. Play D as soon as another player deviates and play D". This strategy is called a reward and punishment strategy: as long as the others are cooperating, I reward them by also cooperating myself. If one person defects, playing D will punish everyone, including the defector, who therefore has no incentive to defect. This strategy is an equilibrium for the N player game:

Lemma 2. For β < 1 and N large, π N is a Nash equilibrium of the N -player stochastic game.

Proof. Assume that all players, except player 0, play the strategy π and let us compute the best response of player 0.

If at time 0, mC < 1, then the best response of player 0 is to play D. On the other hand, if mC = 1 and if player 0 becomes active, then if she uses π, she will suffer a cost

1 N ∞ i=0 e -βi/N = exp(-βt)dt+O(1/N ) = 1/β + O(1/N).
If player 0 deviates from π at this time step and chooses action D, all players will also deviate after the next time step. This implies that mD(t) = 1 -exp(-t) and that the player 0 will suffer a cost equal to ∞ 0 (xC (t) + 2 -2e -t )e -βt dt + O(1/N ) ≥ 2/(β(β + 1)) + O(1/N ) when N is large. This shows that when β < 1, player 0 has no incentive to deviate from the strategy π so that, π is a Nash equilibrium.

The Nash-equilibrium π N leads to a totally different situation from the mean-field equilibrium. In the first case, when all players start in C, they always play C whereas in the second case they will all play D. This shows that for this game, there exists a sequence of Nash-equilibria that do not converge to a mean-field equilibrium (although the cost and the transition kernel are both Lipschitz-continuous). π N is a social optimum while π ∞ has the worst possible social cost.

In mean-field games, a single player have a negligible impact in the population distribution so that her individual actions are not visible by the population. This is the reason why the population can not punish a single player that deviates from the desired pattern. In other words, in the limit game the "tit for tat" principle cannot be applied.

CONCLUSION

In this paper, we consider the convergence of finite stochastic games to mean-field games when the number of players tends to infinity. When we restrict to strategies that are continuous in m, every Nash equilibria converge to a meanfield equilibrium. However, when removing this restriction, convergence to mean-field games does not hold in general.

When the number of players is finite, it is possible to define many equilibria by using the "tit for tat" principle. Indeed, players can cooperate to punish a player that does not follow a desired pattern. These equilibria are linked to the Folk theorem for repeated games, that states that, when a one-shot game is repeated infinitely often, for each achievable cost that is not worse than the Nash equilibrium of the one-shot game, there exists a "tit for tat" equilibrium that achieves this payoff. When the number of players is infinite, the deviation of a single player is not visible by the population and, therefore, the equilibria that are based on the "tit for tat" principle do not scale at the mean-field limit. This is all the more damaging because these equilibria have very good social costs: mean-field games fail to describe the best equilibria.

* This work is partially supported by the EU project QUAN-TICOL, 600708.


Abdeljalil Saghe
email: saghe007@gmail.com

Nonstandard analysis : The field of Omicran-reals and its applications

Keywords: Nonstadard analysis, Omicran-reals, hyppereals, Abraham Robinson, infinitesimal

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

Introduction

This paper is organized as follows, we restart in the first section of preliminary results by some notions for define the field of Omicran-reals [START_REF] Saghe | The field of Omicran-reals A new approach to nonstandard analysis[END_REF], in the next section we define the exact limit, and we show that the limit of a sequence (u n ) is an approximation of the exact limit of (u n ), this important result leads to define the projection of an Omicran-real x onto R, and we can verify that the limit of (u n ) is the projection of the exact limit of (u n ) onto the set of the real numbers. In the section 3, we present the exact derivate with its applications. In the fifth section, we give the relations which calculates the logarithm, and we verify that ln(x) ≈ x δ -1 δ , which implies that ln(x) = lim α→0

x α -1 α , from the above example, we deduce the possibility to obtain an exact equality by using an approximate relationship. In the sixth section we give a simple method for calculating the length of an arc AB of a curve C f .

Preliminary results

Definition 1 Let u be a map from ]0, 1] to R, such that :

• There exists a map u is defined on D ′ (0, 1), and holomorphic in a neighborhood of 0.

• There exists ε > 0, such that ∀x ∈]0, ε[ we have a : u(x) = u(x).

The map u is called a metalic map, and u is a metalic extension of u.

Notations :

We note :

• ∆ 1 = { u, u is a metalic map }. • A 2 = { 1 u , u ∈ ∆ 1 ∀x ∈]0, 1] u(x) = 0 and lim u( 1 n ) = 0 }. • ∆ 2 = {v / v :]0, 1] -→ R , there exists 1/u ∈ A 2 and ε > 0 such that v /]0,ε] = ( 1 u ) /]0,ε] }. • ∆ = ∆ 1 ∪ ∆ 2 , (∆, +, .
) is a unitary ring.

• Let I 0 the set of maps defined in ]0, 1] and zero on ]0, ε] ( for 0 < ε ≤ 1 ). The set I 0 is a maximal ideal of ∆, then ∆/I 0 is a field.

• We define ∆( 1 n ) = { h( 1 n ) n≥1 , h ∈ ∆ }.
• Let ∼ the equivalence relation on ∆( 1 n ) defined by :

g( 1 n ) n≥1 ∼ h( 1 n ) n≥1 ⇔ ∃n 0 / ∀n ≥ n 0 , h( 1 n ) = g( 1 n
).

• The equivalence class of (f

( 1 n )) n≥1 modulo ∼ is noted (f ( 1 n )) n≥1 . • Let ∆( 1 n ) the set of the equivalence classes (f ( 1 n )) n≥1 .
• Let ∆ 0 = ∆/I 0 , and g the equivalence class of g.

If it has not ambiguity, we replace g by g.

• The field of Omicran-reals is defined as follows [START_REF] Saghe | The field of Omicran-reals A new approach to nonstandard analysis[END_REF] :

O = ∆ 0 (δ) = { g(δ), g ∈ ∆ 0 },
where δ is an indeterminate.

Remark 1

• We define on O the following total order :

g(δ) ≤ h(δ) ⇔ g( 1 n ) ≤ h( 1 n ) f rom a certain rank n 0 .
• The field of Omicran-reals is an extension of R, in addition, the total order relation defined on O extends the usual order relation ≤.

• The number δ is infinitesimal [START_REF] Saghe | The field of Omicran-reals A new approach to nonstandard analysis[END_REF].

The exact limit

Proposition 1 The map ϕ defined as :

ϕ : (∆( 1 n ), +, .) -→ (O, +, .) (g( 1 n )) n≥1 -→ g(δ) is isomorphism.
If we want to define a new concept more precise than the limit that allows to give the value taken by the sequence (f ( 1 n )) n≥1 on ad infinitum, then this concept (called exact limit) is dependent on the values taken by (f ( 1 n )) n≥1 from a certain rank n 0 . Intuitively, the equivalence class (f ( 1 n )) n≥1 is a only concept can give these values independently n 0 . On other hand, if f is an element of ∆, then we can identify the equivalence class (f ( 1 n )) n≥1 by f (δ) from the above proposition, so, we deduce that we can define the new concept as follow :

Definition 2 Let f ∈ ∆. The Omicran-real ϕ(f ( 1 n ) n≥1 ) = f (δ) is called the exact limit of the sequence (f ( 1 n )) n≥1 . We note : lim exact f ( 1 n ) = f (δ).
Remark 2 We remark that lim exact = ϕ • s, where s is a canonical surjection defined as :

s : (∆( 1 n ), +, .) -→ (∆( 1 n ), +, .) (f ( 1 n )) n≥1 -→ (f ( 1 n )) n≥1 . Example 1 • lim exact 1 n = δ. • lim exact sin( 1 n ) = sin(δ).
• lim

exact 1 n + 1 = δ δ + 1 .
• lim exact sin(2πn) = 0, the writing sin( 2π δ ) doesn't make sense, because the function z -→ sin( 2π z ) is not element of ∆.

• We can verify that there does not exist an element f ∈ ∆, such that f ( 1 n ) = (-1) n from a certain rank, then we can't define the exact limit lim exact (-1) n . Definition 3 (The projection) Let f a metalic function, and

x ∈ O such that x = f (δ). If we find an element x * ∈ R such that | x -x * | ≤ | x -y |, ∀y ∈ R, then the real x * is called the projection of x onto R. Remark 3 The distance from x to R is d R (x) = inf y∈R | x -y |=| x -x * |.

Example 2

• δ * = 0.

• ( 1 δ 2 + 1 ) * = 1.
Theorem 1 Let f a metalic function, and

x ∈ O / x = f (δ).
The projection x * of x onto R exist and unique, in addition :

x * = lim n→+∞ f ( 1 n ).
Proof.

Let x 0 = lim f ( 1 n ). Then : ∀ε > 0 ∃n ≥ n 0 , | f ( 1 n ) -x 0 |≤ ε. ⇔ for every n ≥ n 0 , we have : -ε ≤ f ( 1 n ) -x 0 ≤ ε. ⇔ lim exact f ( 1 n ) -x 0 ≤ ε and -ε ≤ lim exact f ( 1 n ) -x 0 . ⇔ f (δ) -x 0 ≤ ε and -ε ≤ f (δ) -x 0 . ⇔ | f (δ) -x 0 |≤ ε (1) Next, we can show that : ∀y ∈ R on a : | f (δ) -x 0 | ≤ | f (δ) -y |. If ∃y ∈ R and | f (δ) -y | ≤ | f (δ) -x 0 |≤ ε ( ∀ε ∈ R + ) Then, we deduce that | y -x 0 | ≤ 2ε ( ∀ε ∈ R + ) Then y = x 0 = x * .
Finally, we deduce the existence and the unicity of x * ∈ R such that :

∀y ∈ R , | f (δ) -x * | ≤ | f (δ) -y | .
In addition x * = lim f ( 1 n ). Theorem 2 Let f the metalic map, and x = f (δ). Then : ∀ε > 0 real, we have :

| x -x * |≤ ε
x * is a unique element of R which verify this property.

Proof.

For every n ≥ n 0 we have :

| f ( 1 n ) -x * | ≤ ε, then ∀n ≥ n 0 : x * -ε ≤ f ( 1 n ) ≤ x * + ε, which implies : x * -ε ≤ f (δ) ≤ x * + ε.
We deduce that :

| x -x * |≤ ε.
To show the unicity of x * , we assume there exists an other element y ∈ R such that : | x -y |≤ ε.

Then : | x * -y |≤ 2ε, finally we get y = x * .

Theorem 3 (of the exact limit)

If f ∈ ∆ 1 , then the real lim f ( 1 n ) is the projection of lim exact f ( 1 n
) onto R, so we get :

( lim exact f ( 1 n )) * = lim f ( 1 n ).

The exact derivative

Definition 4 f be a function that is differentiable at the point x 0 ∈ R.

If the function h -→ f (x0+h)-f (x0)
h is metalic, then the exact limit of (

f (x0+ 1 n )-f (x0) 1 n
) exist. We note :

f (x 0 ) = lim exact f (x 0 + 1 n ) -f (x 0 ) 1 n = f (x 0 + δ) -f (x 0 ) δ
The Omicran-real f (x 0 ) is called the exact derivative of the function f at a point x 0 .

Example 3 Consider the following function defined as:

f : x -→ x 2 .
The exact derivative of f is defined by : f (x 0 ) = 2x 0 + δ.

Theorem 4 Let f be a function that is differentiable at the point x 0 ∈ R.

If the function h -→ f (x0+h)-f (x0)
h is metalic, then :

( f (x 0 )) * = f ′ (x 0 )
Proof.

We can apply the theorem of the exact limit.

Example 4 For f : x -→ x 2 , the exact derivative at x 0 is f (x 0 ) = 2x 0 + δ, and the derivative at x 0 is f ′ (x 0 ) = 2x 0 . We verify easily that :

(2x 0 + δ) * = 2x 0 .
Lemma 1 Let f be a metalic function such that for every integer k ∈ N, the function t -→ f (x 0 + kt) is metalic, then :

f (x 0 + N δ) = f (x 0 ) + δ( f (x 0 ) + f (x 0 + δ) + f (x 0 + 2δ) + ... + f (x 0 + (N -1)δ).
Proof.

From the definition of f , we get : Summing these equalities, and we find the desired result .

f (x 0 + δ) = f (x 0 ) + δ f (x 0 ) f (x 0 + 2δ) = f (x 0 + δ) + δ f (x 0 +
Application 1 (Calculate the sum of Σk n )

• For n = 1, if f (x) = x 2 , then f (x) = 2x + δ.
From the above proposition in the case of x 0 = 0, we find :

N 2 δ 2 = (δ( f (0) + f (δ) + f (2δ) + ... + f ((N -1)δ)).
Which implies that :

N 2 δ 2 = δ.( N -1 k=0 2kδ + δ). Then : N 2 = N -1 k=0 (2k + 1) = 2 N -1 k=0 k + N ⇒ N 2 -N 2 = N -1 k=0 k.
• In the case of n = 2, we choose f (x) = x 3 , then f (x) = 3x 2 + 3xδ + δ 2 , by using the above proposition for x 0 = 0, we find :

N 3 δ 3 = δ.( f (0) + f (δ) + f (2δ) + .... + f ((N -1)δ)) = δ. N -1 k=0 (3k 2 δ 2 + 3kδ.δ + δ 2 ) = δ 3 .( N -1 k=0 3k 2 + 3k + 1).
Then :

N 3 = 3 N -1 k=0 k 2 + 3 N -1 k=0 k + N .
We deduce :

N -1 k=0 k 2 = N 3 -N -3 N -1 k=0 k 3 .
Finally, we get :

N -1 k=0 k 2 = N (N -1)(2N -1)

6

. Similarly, we can calculate N -1 k=0 k 3 , N -1 k=0 k 4 , .....

(

The Riemann sum) Let f , g be two metalic functions, such that :

f (δ) = g(δ), then f ( 1 n ) = g( 1 n
) from a certain rank. Consider the function defined as follows :

f n (x) = f (x + 1 n ) -f (x) 1 n .
From the above lemma, we deduce that there exists a natural number n 0 such that ∀n ≥ n 0 , we have :

f (x 0 + N n ) = f (x 0 ) + 1 n (f n (x 0 ) + f n (x 0 + 1 n ) + f n (x 0 + 2 n ) + ... + f n (x 0 + (N -1) n ). If f is differentiable on R, then f n (a + k n ) = f ′ (a + k n ) + o( 1 n ). Then f (a + N n ) = f (a) + 1 n . N -1 k=0 f n (a + k n ) = f (a) + 1 n . N -1 k=0 (f ′ (a + k n ) + o( 1 n )).
We assume that b > a, we can choose N = E[(b -a)n], then we get :

f (a + N n ) -f (a) = 1 n . N -1 k=0 f ′ (a + k n ) + 1 n N -1 k=0 o( 1 n ) = 1 n . N -1 k=0 f ′ (a + k n ) + N n o( 1 n ). Since N = E[(b -a)n], then b -a -1 n < N n ≤ b -a. We deduce lim f (a + N n ) = f (b)
, we pass to the limit and we find :

f (b) -f (a) = lim 1 n . E[(b-a)n]-1 k=0 f ′ (a + k n ).
For b = 1, and a = 0, we get :

f (1) -f (0) = 1 0 f ′ (t)dt = lim 1 n n-1 k=0 f ′ ( k n ).

The logarithmic function

We know that : ∀x ∈ R, lim

n→+∞ (1 + x n ) n = e x .
Let x be a real number, the function f : z -→ (1 + xz)

1 z = e 1 z ln(1+zx) is a holomorphic function on D(0, ε) -{0}. In addition, ln(1 + zx) = zx -z 2 x 2 2 + o(z 2 x 2 ) ( for | z |<< 1)
, we deduce that : lim z→0 ln(1+zx) z = x, and lim z→0 f (z) = e x , then the function f can be extended to a holomorphic function on a neighborhood of 0, which implies that lim exact

(1 + x n ) n
exist and we have :

lim exact (1 + x n ) n = (1 + xδ) 1 δ , and ((1 + xδ) 1 δ ) * = e x .
From the above results we deduce that e x is an infinitesimal approximate value of (1 + δx)

: ξ -1 (x) = x δ -1 δ (if it exists !!!).
We can justify the existence of this function, for that, we consider the real number x ∈ R * + . The map defined as g : z -→ x z -1 z = e z ln(x) -1 z is a holomorphic function on D(0, ε) -{0}, and lim z→0 g(z) = ln(x), then 0 is an artificial singularity of g, we deduce that there exists the exact limit of the sequence (n(x x δ -1 δ . We define the the original logarithm by :

ln o : x -→ x δ -1 δ .
The function ξ is called the function of the the original exponential, we note :

ξ(x) = exp o (x) = (1 + δx) 1 δ ,
and we deduce that :

(ln (x)) * = ln(x). Then : ln(x) = lim α→0

x α -1 α .

• Application 1. From the above results, we can show the following equality :

ln(x) = lim α→0 x α -x -α 2α . Remark 4 ln(x) ln(y) = lim α→0 x α -1 y α -1 = lim α→0 x α -x -α y α -y -α .
• Application 2. By using the above results we can show the following theorem.

Theorem 5 For every 0 < x and x = 1, we have :

x -1 ln(x) = lim n→+∞ 1 n n-1 k=0 x k n .
Proof. We have :

(x 1 n -1)( n-1 k=0 x k n ) = x -1.
Then :

n(x 1 n -1)( 1 n n-1 k=0 x k n ) = x -1.
Since lim α→0

x α -1 α = lim n(x 1 n -1) = ln(x), then :

x -1 ln(x) = lim • Application 3. Consider P the set of the prime numbers. We define the prime-counting function at real values of x by : π(x) = #{p ≤ x : p ∈ P}.

Theorem 6 (Hadamard and de la Vallée Poussin). As x → +∞, π(x) ∼ x ln(x) .

. By using the above theorem, and the formula ln(x) ≈ x δ -1 δ , we can show the following theorem :

Theorem 7 As x → +∞, π(x) ∼ 1 x 1 x -1 . .
Proof. We approach ln(x) by x δ -1 δ , and δ by 1

x , and we deduce that an approximate value of x ln(x) is 1

x 1 x -1
.

Next, we can verify that x ln(x) ∼ 1

x 1 x -1
, in fact : As x → +∞, we have :

  δ) ............................................................. .................................................................. ........................................................................ f (x 0 + N δ) = f (x 0 + (N -1)δ) + δ f (x 0 + (N -1)δ).

1 δ

 1 the function ξ(x) = (1 + δx) , if this function x -→ ξ(x) has an inverse function, then the calcul gives us the function ξ -1 which is defined as

1 n - 1 )

 11 ) n≥1 , and his value is equal to lim exact

= 1 , then : 1 x 1 x - 1 ∼ 1 x - 1 .

 11111 x ln(x) . Finally, we obtain :π(x) ∼1x

Application 4. Let (p n ) the sequence of a prime numbers, we have : Theorem 8

p n ∼ n ln(n), while n → +∞ By using the above theorem and the approximations ln(n) ≈ n δ -1 δ and δ ≃ 1 n , we get the following theorem :

Theorem 9 p n ∼ n 2 ( n √ n -1), while n → +∞ Proof. We can verify that lim n( n √ n-1)

6 The geometry of the points Geometrically, we can represent the infinitesimal numbers as follows :

If f is a metalic function, and f (δ) is an infinitesimal number, the function f is metalic then the sequence (f ( 1 n )) keeps a constant signe from a certain rank, we can assume that the above sequence is positive from a certain rank n 0 , and we know that :

then we can represent f (δ) by the the family of segments (I n ) n≥n0 , we can justify this notation by this equality :

Definition 5 Let f and g be two metalic fonctions. We assume that the number f (δ) -g(δ) is infinitesimal and f (δ) < g(δ).

• If x A and x B are respectively the abscissa of the points A and B, we define the distance between A and B by :

The length of a curve C f

We define the length of an elementary geometric point by :

where x A = g(δ), and g is a metalic function. Let f be a function such that : x -→ f (x * A + x) is metalic, and f the metalic extension of f on a neighborhood of x * A , consider A(x A , f (x A )) and A ′ (x A + δ, f (x A + δ)) two ordered pairs of E 2 , let φ be a function defined as :

On the other hand, lim z→0 g(z) = x * A , and the map θ

z is holomorphic on D(0, ε) -{0}, and we get :

Then the lim z→0 θ(z) exist, and we have :

We deduce that lim z→0 φ(z) = 0, and φ is continuously extendable over 0, then the function φ is holomorphically extendable over 0, which justify the existence of the exact limit of the sequence (φ( 1 n )), and we have

which is infinitesimal. We define the of the segment [A, A ′ [ by :

We note :

If f is a metalic function defined on the interval [0, 1], Let A(0, f (0)) and B(1, f (1)) two points of the plane which define with f the arc AB, if the exact limit of lim

) 2 exist, we note :

We define the exact lenght of the arc AB by :

The lenght of the arc AB is the real denoted by l * ( AB) and defined by :

where

.

On other hand, since f is a metalic function, then it can be extended to a function twice differentiable at 0. We assume that the function is twice differentiable on ]0, 1[. Then :

where 1 ≤ k ≤ n -1, and

We have :

Next, we deduce that :

We can verify that β n,k > 1, and we have :

2n .

:

By using the Riemann sum, we deduce the length of the arc AB :