Noël Nguyen

Véronique Delvaux

Role of imitation in the emergence of phonological systems

The issue we address in this review paper is to what extent does mutual adaptation play a role in the emergence and evolution of phonological systems.

Adaptation to the interlocutor has been shown to take many forms and to embrace all the levels of spoken language, from adjustments in vocal intensity to changes in word forms over the course of a conversational exchange, as well as lexical and syntactic alignment across speakers, to name but a few examples. Phonetic convergence, that is, the tendency for two speakers engaged in a conversational exchange to sound more like each other, is one important aspect of between-speaker adaptation. Empirical evidence has recently accumulated that shows that phonetic convergence is a recurrent phenomenon in mature speakers. This phenomenon relies on sensory-motor abilities that infants may already possess at birth. Phonetic convergence affects the way in which both speakers speak after their interaction has ended, and may build up over long periods of time. It may also be a driving mechanism in the acquisition of the phonology and phonetics of a second language.

In this paper, (i) we outline the role of imitation in modern speech and language; (ii) we review the evidence provided by experimental and modelling studies for the potential role of imitation in the emergence and evolution of phonological systems; and (iii) we discuss how the resulting hypotheses could be tested in the framework provided by the multi-agent computational COSMO model.

Introduction

In their target paper for the present Journal of Phonetics Special Issue, Moulin-Frier, Diard, Schwartz, and Bessière (this volume) present an ambitious and far-reaching account of how phonological systems may emerge from a small number of general principles governing interactions between agents in a speech communication task. One key feature of the COSMO model relates to the mechanism that allows agents to converge towards a common set of wordlike forms for referring to external objects. In COSMO, these forms gradually arise through a sequence of pairwise communications that take place between agents assigned as speakers and a nts ss n s l st n rs. E ommun t on nt ls t sp k r's s n t n n obj t n r sults n t justm nt of bot t sp k r's motor prototyp s n l st n r's auditory prototypes associated with that object. Importantly, each instance of communication in COSMO is asymmetrical: the speaker speaks, the listener hears that speaker. Which agent is the speaker and which is the listener is subject to random changes across time. As a result, w l t sp k r's pro u sp p tt rn s n mp ct on the auditory prototypes associated with the object in the listener, there is no reciprocal influence of the listener on the speaker and, more generally, no interaction between the two agents, inasmuch as an interaction entails an influence of both agents on each other. In that respect, COSMO may be seen as a non-interactional account of the emergence of a speech-mediated code.

COS O's ppro

s t l st n p rt n or w t ot r nflu nt l t or t l fr m works such as the one developed by [START_REF] Oudeyer | The self-organization of speech sounds[END_REF] for example. For Oudeyer, it is possible to simulate how a speech-mediated code may form in a society of agents without having to endow these agents with the capacity to interact with each other. According to an assumption made by Oudeyer, agents do not communicate, in the sense of intentionally conveying meaning to one another, and do not necessarily make the distinction between the speech signals they produce and those produced by others. This approach stands in sharp contrast with models of the emergence of phonological systems in which reciprocal influences of agents upon each other pl y ntr l rol . n Bo r's (2000) mo l for x mpl t form t on of p onolo l system is the collective and cumulative by-product of a large number of local, pairwise interactions between agents one of which aims to imitate the speech pattern the other agent has produced.

Adaptation to the interlocutor has been shown to take many forms and to embrace all the levels of spoken language, from adjustments in vocal intensity [START_REF] Natale | Convergence of mean vocal intensity in dyadic communication as a function of social desirability[END_REF] to changes in word forms over the course of a conversational exchange [START_REF] Fowler | Differential shortening of repeated content words produced in various communicative contexts[END_REF], as well as lexical and syntactic alignment across speakers (Garrod & Pickering, 2004), to name but a few examples. Phonetic convergence, that is, the tendency for two speakers engaged in a conversational exchange to sound more like each other, is one important aspect of between-speaker adaptation. Empirical evidence has recently accumulated that shows that phonetic convergence is a recurrent phenomenon in mature speakers (e.g., [START_REF] Babel | Imitation in speech[END_REF]. This phenomenon relies on sensory-motor abilities that infants may already possess at birth [START_REF] Meltzoff | Imitation of facial and manual gestures by human neonates[END_REF]; but see [START_REF] Jones | The development of imitation in infancy[END_REF]. Phonetic convergence affects the way in which both speakers speak after their interaction has ended [START_REF] Pardo | On phonetic convergence during conversational interaction[END_REF], and may build up over long periods of time [START_REF] Harrington | Does the Queen speak the Queen's English?[END_REF][START_REF] Pardo | Phonetic convergence in college roommates[END_REF]. It may also be a driving mechanism in the acquisition of the phonology and phonetics of a second language [START_REF] Lewandowski | Perception-production loop in native-non-native dialogs: Phonetic convergence[END_REF][START_REF] Sancier | Gestural drift in a bilingual speaker of Brazilian Portuguese and English[END_REF].

The issue we propose to address in this paper is to what extent does between-speaker phonetic convergence play a role in the emergence and evolution of phonological systems. In Moulin-Fr r n oll u s' COSMO model, as pointed out above, the focus is placed on reference, i.e. the setting up of a common speech-mediated code for designating external objects, to a much greater extent than on mutual adaptation. Because information passes in one way only (from the speaker to the listener), it appears to us that phonetic convergence can only occur in an indirect and delayed manner (and only in the sensory-motor version of the model), through t n t t t sp k r's pro u sp p tt rn n u s n t l st n r's u tory prototyp s s t s r l t r brou t nto pl y by t l st n r's own sp pro u t on syst m when that listener becomes a speaker. By contrast, in other theoretical frameworks such as de Bo r's (2000) m t t on s onsubst nt l to t way in which speakers are assumed to interact with each other. In the following, we outline the role of imitation in modern speech and language (Section 2), we review the experimental evidence that may exist for a potential role of imitation in the emergence and evolution of phonological systems (Section 3), and we discuss how the resulting hypotheses could be tested in the framework provided by COSMO (Section 4).

Role of imitation in speech and language

In human beings, vocal imitation is a behavior that manifests itself over the lifespan. While some empirical studies have reported mimicry of sounds as early as 2-6 months of age (e.g. [START_REF] Kuhl | Infant vocalizations in response to speech: vocal imitation and developmental change[END_REF][START_REF] Kokkinaki | Basic aspects of vocal imitation in infant-parent interaction during the first 6 months[END_REF][START_REF] Gratier | Imitation and repetition of prosodic contour in vocal interaction at 3 months[END_REF], a phenomenon that could be facilitated by audio-visual congruence in the model (Legerstee, 1990), other data suggest that the ability to imitate does not fully develop until the second year of life [START_REF] Jones | The development of imitation in infancy[END_REF]. Imitation involves a variety of perceptuo-motor, cognitive and social skills which makes it one of the "building blocks from which spoken language develops in typical development" (Charman, 2006:106). In particular, early vocal imitation has been found to positively correlate with later lexical development (e.g. Masur & Eichorst, 2002). The acquisition of L2 phonetics and phonology is also considered to be partly grounded on the ability to reproduce foreign speech sounds, so that individual differences in "speech imitation ability" [START_REF] Reiterer | Are you a good mimic? Neuro-acoustic signatures for speech imitation ability[END_REF] or "phonetic compliance" [START_REF] Delvaux | Phonetic compliance: a proof-ofconcept study[END_REF] may result in behavioral foreign accent differences in late L2 learners. Elderly speakers also show some ability to reproduce unfamiliar speech sounds [START_REF] Delvaux | Capacit ppr nt ss p on qu t troubl s u l n olo r br l n So k[END_REF].

Regardless of its role in language development and second language acquisition, imitation of speech sounds is typically observed in mature speakers, i.e. in speakers who have reached the full mastery of their native language. Empirical evidences of phonetic convergence have accumulated over the last decade, whether in laboratory settings exposing a speaker to another individual's speech productions [START_REF] Namy | Gender differences in vocal accommodation: The role of perception[END_REF][START_REF] Goldinger | Episodic memory in printed word naming[END_REF][START_REF] Shockley | Imitation in shadowing words[END_REF][START_REF] Delvaux | The influence of ambient speech on adult speech productions through unintentional imitation[END_REF][START_REF] Nielsen | Specificity and abstractness of VOT imitation[END_REF][START_REF] Babel | Dialect convergence and divergence in New Zealand English[END_REF][START_REF] Babel | Evidence for phonetic and social selectivity in spontaneous phonetic imitation[END_REF][START_REF] Babel | The Role of Fundamental Frequency in Phonetic Accommodation[END_REF]Gentilucci & Bernadis, 2007;[START_REF] Honorof | Articulatory events are imitated under rapid shadowing[END_REF][START_REF] Miller | Is Speech Alignment to Talkers or Tasks? Attention[END_REF][START_REF] Mitterer | The link between perception and production is phonological and abstract: evidence from the shadowing task[END_REF][START_REF] Yu | Phonetic Imitation from an Individual-Difference P rsp t v : Subj t v Att tu P rson l ty n ''Aut st '' Tr ts[END_REF][START_REF] Dufour | How much imitation is there in a shadowing task[END_REF][START_REF] Lelong | Study of the phenomenon of phonetic convergence thanks to speech dominoes[END_REF]Lelong, 2012;[START_REF] Nguyen | Does imitation facilitate word recognition in a non-native regional accent?[END_REF][START_REF] Sato | Converging toward a common speech code: imitative and perceptuo-motor recalibration processes in speech production[END_REF], or in actual conversational interactions [START_REF] Pardo | On phonetic convergence during conversational interaction[END_REF][START_REF] Pardo | Conversational role influences speech imitation[END_REF][START_REF] Kim | Phonetic convergence in spontaneous conversations as a function of interlocutor language distance[END_REF][START_REF] Aubanel | Variation phonologique régionale en interaction conversationnelle Unpublished doctoral dissertation[END_REF] but the exact role of phonetic convergence in speech and language remains an open question. Still, phonetic convergence may inform us on how speech sounds are dealt with, i.e. how they are structurally organized, cognitively processed and socially used.

First, phonetic convergence resides in the active exploitation of an effective sensory-motor link in processing speech sounds. Adapting to the interlocutor's speech initially requires the speaker to be able to make a cross-modal correspondence between the sounds he has just perceived (in the auditory domain) and the sounds he is about to produce (using motor commands), independently of the utterances they encode. As recalled by Moulin-Frier and colleagues (section 1.2), recent findings on the existence of a mirror system in humans from which Broca's area may have evolved (Arbib, 2005a) are valuable in the search for a sensory-motor association system. Note, however, that while sound imitation can not be achieved in the absence of such a "parity" mechanism, the reverse is not true: the existence of a sensory-motor link does not imply per se that it will be exploited to support the imitation of the interlocutor's vocal productions. After all, mirror neurons have been first discovered in macaque monkeys, a species with poor imitation skills [START_REF] Kopp | Imitation in embodied communicationfrom monkey mirror neurons to artificial humans[END_REF]; but see [START_REF] Kumashiro | Natural imitation induced by joint attention in Japanese monkeys[END_REF], and their potential role in supporting action understanding (through mental simulation) and imitation is still under debate [START_REF] Hickok | The role of mirror neurons in speech perception and action word semantics[END_REF].

In COSMO, sensory-motor agents do not imitate each other, although they are attuned to their environment in that they update their motor and/or auditory prototypes following each deictic game. Phonetic convergence effects indicate that mature human speakers do not only use their sensory-motor representations to ensure speech perception, but that they actively exploit them to drive their own productions (towards their interlocutor's) during a specific conversational interaction. Besides the need for these mental representations to be rich and flexible in structure, it is not yet fully understood how the routine imitative process affects functional aspects of speech-related representations, i.e. how they are built in, stored, retrieved, and updated (for a review, see [START_REF] Nielsen | Specificity and abstractness of VOT imitation[END_REF].

Second, phonetic convergence is only an aspect of the phenomenon of adaptation to the interlocutor that has been extensively documented over the years, in particular in the sociolinguistics literature in relation with the concepts of interpersonal accommodation [START_REF] Giles | Accommodation Theory: Communication, Context, and Consequence[END_REF][START_REF] Gallois | Communication Accommodation Theory: A look Back and a Look Ahead[END_REF][START_REF] Trudgill | Colonial Dialect Contact in the History of European Languages: On the Irrelevance of Identity to New-Dialect Formation[END_REF] and style-shifting behaviour (e.g. Eckert & Rickford, 2001). Well-known occurrences of adaptation to the interlocutor include infant-directed speech, clear speech and foreign-directed speech [START_REF] Uther | Do you speak E-N-G-L-I-S-H? A comparison of foreigner-and infant-directed speech[END_REF][START_REF] Beckford-Wassink | Intraspeaker variability in vowel production: An investigation of motherese, hyperspeech, and Lombard speech in Jamaican speakers[END_REF][START_REF] Smiljanic | Speaking and hearing clearly: Talker and listener factors in speaking style changes[END_REF].

Convergence between interlocutors concerns postures, mannerisms, facial expressions (the "chameleon effect": Chartrand & Bargh, 1999), as well as linguistic form at virtually all levels of linguistic hierarchy along a variety of parameters: prosody (pause frequency, pause duration [START_REF] Gregory | Conversation partner mutual adaptation as demonstrated by Fourier series analysis[END_REF]; overall intensity [START_REF] Natale | Convergence of mean vocal intensity in dyadic communication as a function of social desirability[END_REF]; fundamental frequency [START_REF] Gregory | Analysis of fundamental frequency reveals covariation in interview p rtn rs' sp[END_REF]; speaking rate [START_REF] Webb | Interview synchrony: An investigation of two speech rate measures in an automated standardized interview[END_REF]), lexicon (lexical choice: [START_REF] Brennan | Conceptual pacts and lexical choice in conversation[END_REF], syntax (syntactic structure: Pickering & Garrod, 2004;[START_REF] Branigan | Syntactic coordination in dialogue[END_REF][START_REF] Branigan | Syntactic alignment and participant role in dialogue[END_REF], and, of course, phonetics and phonology (vowel formants, vowel duration, VOT, etc.). Moreover, phonetic convergence arises in conversational interactions although there is no explicit instruction to imitate (e.g. [START_REF] Aubanel | Variation phonologique régionale en interaction conversationnelle Unpublished doctoral dissertation[END_REF][START_REF] Pardo | On phonetic convergence during conversational interaction[END_REF], as well as in minimally interactional experimental settings involving exposure to another individual's speech [START_REF] Delvaux | The influence of ambient speech on adult speech productions through unintentional imitation[END_REF][START_REF] Nielsen | Specificity and abstractness of VOT imitation[END_REF][START_REF] Yu | Phonetic Imitation from an Individual-Difference P rsp t v : Subj t v Att tu P rson l ty n ''Aut st '' Tr ts[END_REF], including shadowed speech [START_REF] Namy | Gender differences in vocal accommodation: The role of perception[END_REF][START_REF] Goldinger | Echoes of echoes? An episodic theory of lexical access[END_REF][START_REF] Shockley | Imitation in shadowing words[END_REF][START_REF] Babel | Dialect convergence and divergence in New Zealand English[END_REF][START_REF] Babel | Evidence for phonetic and social selectivity in spontaneous phonetic imitation[END_REF][START_REF] Babel | The Role of Fundamental Frequency in Phonetic Accommodation[END_REF][START_REF] Honorof | Articulatory events are imitated under rapid shadowing[END_REF][START_REF] Miller | Is Speech Alignment to Talkers or Tasks? Attention[END_REF][START_REF] Mitterer | The link between perception and production is phonological and abstract: evidence from the shadowing task[END_REF]. Besides, shadowers have been shown to align to a mo l's spok n wor s w t r t os wor s r pr s nt u tor ly or v su lly . . n l preading task [START_REF] Gentilucci | Imitation during phoneme production[END_REF][START_REF] Miller | Alignment to visual speech information[END_REF][START_REF] Sanchez | Visual influences on alignment to voice onset time[END_REF]. Altogether, empirical evidence thus points out to a widespread behavior which relies on processes that may be consubstantial to human spoken interaction, or at least to the cognitive processing of speech whenever the perception of another's speech accompanies one's own productions.

The extensive evidence on linguistic convergence should not, however, obscure the fact that convergence effects are typically subtle ("perfect" imitation never occurs, actually the size of the effects is typically small) and variable across social and situational factors. It is not a purely automatic process. In particular, phonetic convergence has been shown to be mediated by (i) linguistic, system-internal, factors (e.g. American English speakers exhibited deferred, postexposure, imitation of extended VOT but not of reduced VOT in word-initial voiceless stops, presumably because reducing VOT could hold the risk of confusion with voiced stops [START_REF] Nielsen | Specificity and abstractness of VOT imitation[END_REF]; see also Olmstead et al., 2013, on the effect of native language on patterns of VOT imitation); (ii) speaker-specific, situation-independent factors which are related with personality ("openness" [START_REF] Yu | Phonetic Imitation from an Individual-Difference P rsp t v : Subj t v Att tu P rson l ty n ''Aut st '' Tr ts[END_REF], one of the Big 5 personality traits [START_REF] Costa | Revised NEO Personality Inventory (NEO-PI-R) and NEO Five-Factor Inventory (NEO-FFI) manual[END_REF]), cognitive functioning (in particular: the ability to allocate attention resources, more than their availability as measured by working memory capacity; [START_REF] Yu | Phonetic Imitation from an Individual-Difference P rsp t v : Subj t v Att tu P rson l ty n ''Aut st '' Tr ts[END_REF], and attitude (towards race, nationality and dialect; [START_REF] Babel | Dialect convergence and divergence in New Zealand English[END_REF][START_REF] Babel | Evidence for phonetic and social selectivity in spontaneous phonetic imitation[END_REF]; (iii) situation-specific factors, such as attitude towards the talker [START_REF] Yu | Phonetic Imitation from an Individual-Difference P rsp t v : Subj t v Att tu P rson l ty n ''Aut st '' Tr ts[END_REF][START_REF] Pardo | Phonetic convergence in college roommates[END_REF], gender of the pair of talkers and talker's role in conversation [START_REF] Pardo | On phonetic convergence during conversational interaction[END_REF][START_REF] Pardo | Conversational role influences speech imitation[END_REF] and the degree of social context embedded in the situation [START_REF] Babel | Evidence for phonetic and social selectivity in spontaneous phonetic imitation[END_REF].

Altogether, the reviewed evidence suggests that phonetic convergence in conversational interactions is grounded on a low-level cognitive process involving a strong sensory-motor association, hence its pervasiveness in minimally interactive designs. However, phonetic convergence does not result from a purely reflexive process, since it is selective (and as such, requires selective attention to the fine-grained phonetic details of the imitated sounds and some kind of higher-level matching process between production and perception: [START_REF] Babel | Imitation in speech[END_REF][START_REF] Babel | Evidence for phonetic and social selectivity in spontaneous phonetic imitation[END_REF]see Studdert-Kennedy (2000a) for a view of imitation as a "purely formal process") and is typically modulated by a variety of psychological and social factors which are partly under the control of the talker.

Actually, communication accommodation theory first developed in the 1970s by Giles and colleagues [START_REF] Giles | Accent mobility: A model and some data[END_REF]Giles et al., 2001;[START_REF] Gallois | Communication Accommodation Theory: A look Back and a Look Ahead[END_REF] claims that linguistic convergence and its opposite, divergence, are strategically used by speakers engaged in spoken interactions in order to respectively minimize or maximize the social distance with their interlocutors, so as to reinforce their own social identity. Whether driven by social identity matters and available for conscious manipulation [START_REF] Eckert | Style and social meaning[END_REF], or largely automatic and "deterministic" [START_REF] Trudgill | New-dialect formation. The inevitability of colonial Englishes[END_REF][START_REF] Trudgill | Colonial Dialect Contact in the History of European Languages: On the Irrelevance of Identity to New-Dialect Formation[END_REF], linguistic accommodation is typically considered in sociolinguistics as one of the mechanisms potentially playing an influential role in channelling linguistic variation towards dialect formation, and ultimately language change. That is, local, short-term accommodation in repeated conversational interactions could lead, at the individual level, to long-term accommodation, and at the community level to the propagation and diffusion of innovative variants (Auer & Hiskens, 2005), especially if they are associated with perceived prestige [START_REF] Lev-Ari | An experimental study of the role of social factors in language change: The case of loanword adaptations[END_REF], thus leading to sound change. Depending on the structure of the social network disseminating the innovations, accommodation could account for both dialect levelling and dialect formation (e.g. along highly inner-connected but loosely outerconnected subparts of the network).

Two types of empirical studies support this scenario. First, convergence effects elicited in the laboratory have been shown to persist several days after exposure [START_REF] Goldinger | Episodic memory in printed word naming[END_REF], while long-term accent changes can usually be retraced to extensive contact with the accommodated dialect (Harrington, 2006;[START_REF] Munro | Canadians in Alabama: A perceptual study of dialect acquisition in adults[END_REF][START_REF] Evans | Plasticity in vowel perception and production: A study of accent change in young adults[END_REF], laying the foundation for a link between short-term and long-term accommodation within individuals. Second, recent sociolinguistic studies have used computer simulations on a social structure and game theory to test scenarios of language change [START_REF] Ke | Language change and social networks[END_REF][START_REF] Fagyal | Centers, peripheries, and popularity: The emergence of norms in simulated networks of linguistic influence[END_REF][START_REF] Mühlenbernd | Signaling and Simulations in Sociolinguistics[END_REF]. Fagyal and colleagues [START_REF] Fagyal | Centers, peripheries, and popularity: The emergence of norms in simulated networks of linguistic influence[END_REF] have shown that in scale-free networks the process of an innovative variant's spread, stabilization, and gradual replacement by a new one can be modelled providing that (i) the network includes both strongly influential, high-connected agents who support the propagation of new variants, as well as peripheral, low-connected agents whose conservative variants may constitute a reservoir for further innovations, and (ii) agents select which variants to adopt in their one-to-one interactions based on their interlocutor's social status, defined as the agent's number of outgoing connections.

To sum up, imitation of speech sounds is a pervasive, multifaceted phenomenon in speech and language, which has been claimed to play a role in language development, second language acquisition, conversational interactions and language variation and change. It is not clear however, whether all the phenomena that may be regrouped under this label of "imitation of speech sounds" are actually different aspects of a single, unified process. In particular, phonetic convergence effects might result from automatic attunement to ambient speech via the constant, real-time updating of sensory-motor representations, as well as from controlled, deliberate imitation of one's interlocutor for social purposes. The former could be considered as one of the learning mechanisms supporting language development, that would remain in mature speakers as an automatic process of sensorimotor recalibration primarily facilitating speech perception through comparison between one's own productions and external speech inputs provided by others [START_REF] Garnier | Neural correlates of phonetic convergence and imitation of speech[END_REF]. The latter could be described as a complex cognitive ability acquired through practice and learning, that would involve coordinating action and perception based on cognitive processes such as "conscious maintenance and recall of past auditory or vocal episodes, selective attention to subcomponents of experienced and produced sounds, identification of specific goals of reproducing certain acoustic features, and awareness of possible benefits that can be attained through successful sound reproduction" (Mercado et al., 2014:10).

Role of imitation in the emergence of phonological systems

In a COSMO deictic game, the listener does not respond to the speaker. An important consequence of this is that reciprocal adaptation of the speaker and listener to each other does not occur. In particular, phonetic convergence, or imitation, of the two agents towards each other, is not deemed to significantly contribute to the building up of a phonological system. As underlined above, there is in that respect a marked difference between COSMO and other theoretical frameworks which, on the contrary, make imitation a driving force in the emergence of language and its phonological component.

The discovery of the mirror system in area F5 of the premotor cortex in the macaque's brain, and the role this system has been established to play in both the execution and recognition of grasping actions, has given rise to the idea that imitation, taken as a general sensory-motor ability, is one of the factors that, in the course of evolution, made our brains "ready for language" [START_REF] Arbib | The mirror system, imitation, and the evolution of language[END_REF][START_REF] Arbib | The mirror system hypothesis: How did protolanguage evolve[END_REF][START_REF] Rizzolatti | Language within our grasp[END_REF]. Relying on the fact that area F5 in the monkey brain is regarded as an homologue of Broca's area in the human brain, Arbib has proposed an evolutionary scenario in which human language arose from a primitive system of communication based on manual gesture. Although the phonological component of language is not its primary focus, two key features of this theoretical framework are particularly relevant to the issues discussed here. First, the mirror system is said to be at the origin of parity, defined as the fact that "what counts for the speaker must (approximately) count for the hearer" [START_REF] Arbib | The mirror system hypothesis: How did protolanguage evolve[END_REF]), a property central to human language and one of the basic requirements for speech communication to be successful in Moulin-Frier and colleagues' COSMO model. Second, in Arb b's proposed framework, imitation is not a behavior that merely mirrors that observed in another individual. Rather, it involves mapping the action to be imitated onto a repertoire of already available motor schemas that may then be recomposed, fractionated and/or tuned in a novel way. In other words, while it allows sets of actions to be passed on from one individual to another through observation, imitation is also conducive to the emergence of behavioral patterns that deviate from those they are based upon, and can therefore be a factor of innovation and change.

That imitation contributes to accounting for the building up of phonological systems is a central tenet of Studdert-Kennedy's approach to language (e.g., Studdert-Kennedy, 2000b, 2005). According to Studdert-Kennedy, human language developed from an early form of facial and vocal imitation in Homo erectus, with a mirror system akin to that found in the monkey as possible neural substrate. Like Arbib, Studdert-Kennedy holds that imitation is much more than mere behavioral echo, as the perceived action is assumed to undergo parsing into a set of elementary components that are then reassembled. It is because of the conjunction of these two major phenomena, the propensity shown by humans to imitate each other, and the decomposition/recomposition process through which imitation is performed, that language came to be endowed with the form it has today. In the course of its phylogenetic trajectory, the vocal apparatus has evolved towards differentiation into a set of functionally (semi-)independent articulatory organs, the lips, tongue tip, body and root, velum, and larynx. In Studdert-Kennedy's view, imitation operates through the mediation of this discrete and finite set of independently controllable organs, and this is what caused discreteness and combinatoriality to emerge in speech patterns. Speech development in children follows a course that to a certain extent parallels this phylogenetic scenario, and leads children to gradually confer a discrete, segmental structure to the quasi-continuous speech flow directed to them as they endeavor to imitate that speech flow. As in Arbib's account, imitation is for Studdert-Kennedy a selective mechanism, which subjects a perceived action to conversion into a set of component gestures, and may generate novel patterns in the process.

Articulatory Phonology has also assumed that gestural structures may emerge as the byproduct of imitation between speakers. For example, [START_REF] Browman | Competing constraints on intergestural coordination and self-organization of phonological structures[END_REF] conducted a computational experiment in which speakers were expected to conform to an "accommodation condition", which entailed their wanting to act like each other. As these simulated interactions in dyads of speakers developed, phase relationships between gestures arose that were both stable and shared across speakers, by virtue of what Browman and Goldstein regard as a selforganization process. In a follow-up to this work, [START_REF] Goldstein | Emergence of discrete gestures[END_REF] see also [START_REF] Goldstein | Emergence of discrete gestures[END_REF][START_REF] Goldstein | Articulatory phonology: A phonology for public language use[END_REF][START_REF] Goldstein | The role of vocal tract gestural action units in understanding the evolution of phonology. Action to language via the mirror neuron system[END_REF] asked whether the establishment of gestural contrasts associated with a single vocal organ, can also be modeled as resulting from interactions between speakers in an imitation task. The results showed that, subject to certain conditions (see [START_REF] Goldstein | Emergence of discrete gestures[END_REF][START_REF] Goldstein | Emergence of discrete gestures[END_REF]Fowler, 2003, for detail), mutual imitation allowed computational agents to converge towards a partitioning of a gestural continuum into a number of discrete intervals.

As recalled above, imitation between speakers is a central mechanism in de Boer's (2000) computational model of the emergence of vowel systems. The model is made up of a population of agents that are each equipped with an articulatory synthesizer, a perceptual device that allows distances between vowels to be computed in the formant space, and a memory in which both the articulatory and acoustic properties associated with vowel prototypes are stored. These agents engage in a sequence of pairwise interactions that are referred to as imitation games and which, in essence, consists for one of two agents (the imitator) to imitate the vowel produced by the other agent (the initiator). Quite importantly, imitation is only approximate, in the sense that it is achieved through the filter of the vowel prototypes already available to the imitator. More specifically, the perceived vowel sound is mapped onto the closest imitator's vowel prototype, and is reproduced by the imitator as this vowel prototype. The initiator's and imitator's repertoires of vowel prototypes are then updated in a way which depends on whether the game is found successful or unsuccessful by both agents. Thus, whether in [START_REF] Browman | Competing constraints on intergestural coordination and self-organization of phonological structures[END_REF], [START_REF] Goldstein | Emergence of discrete gestures[END_REF], or de [START_REF] De Boer | Self-organization in vowel systems[END_REF] modelling enterprise, the computational agents' goal is to sound more like each other. In COSMO's deictic games, the agents' goal is of a quite different nature: to use the same word forms when referring to the same objects. Agents interacting with each other may end up sounding more alike but only as an indirect and delayed consequence of this primary goal. The word form employed by Agent A as speaker to designate a particular object has an impact on the auditory prototype associated with that object in Agent B as listener. This may affect the way in which Agent B will in turn refer to the object when this agent is called on to assume the role of speaker and that object presents itself again. However, before this particular conjunction of events happens, and as the sequence of dyadic interactions unfolds, Agent B may be exposed as listener to a potentially high number of word forms produced by other speakers in relationship with that same object. Thus, Agent A's produced word form can influence Agent B's own produced form only by percolation and through what may be a long series of dyadic interactions.

However, the imitation-based models, COSMO, and other models of the emergence of phonological systems such as [START_REF] Oudeyer | The self-organization of speech sounds[END_REF] have one important commonality: they all attribute an essential role to the links between perception and action. Imitation obviously requires that speakers be able to match perceived speech forms to those they in turn produce. In their target paper, Moulin-Frier and colleagues convincingly demonstrate that the sensorymotor version of COSMO does a better job of simulating how phonological systems may emerge than both the motor-only and auditory-only versions. Thus, in our view, the difference between imitation-based models and sensory-motor models such as COSMO centers on how these models answer the following question: in the emergence of phonological systems, is it necessary to assume that adaptation between partners of an interaction is reciprocal and, more specifically, that each partner overtly or covertly endeavors to mimic the speech patterns produced by the other partner?

An important contribution towards answering this question can be found in the experimental work that, in parallel with the development of multi-agent computational models, has been recently conducted in an attempt to reproduce in the laboratory the conditions that may have presided over the emergence of language (Scott-Phillips & Kirby, 2010). Most of this work has been concerned with the emergence of graphical communication systems, and there are yet very few experimental studies on how acoustic communication systems may have formed. In a recent study, however, de Boer and Verhoef (2012, see also [START_REF] Verhoef | Cultural evolution of combinatorial structure in ongoing artificial speech learning experiments[END_REF] examined how a repertoire of vowel-like sounds gradually took shape as groups of participants were successively asked to reproduce these sounds and transmit them to the next group, along what was made to look like a generational chain. The vowel sounds were associated with as many visual forms constructed by systematic combinations of different shapes and colors, and were originally synthesized from random patterns in the F1-F2 formant space. Within each generational group, participants learned to reproduce the vowel sound they heard in association with a particular visual form, by drawing a trajectory on a computer screen, which was then converted into a time series of F1 and F2 frequency values themselves passed on to a speech synthesizer. The authors' hypothesis was that, as the correspondences between vowel sounds and visual forms were transmitted across generational groups, a combinatorial structure would gradually emerge within the vowel sounds, on par with increased learnability. The results did not conform to these predictions, possibly because the learning task was too difficult (see Verhoef et al., 2011, for detailed discussion).

In yet a more recent work, [START_REF] Verhoef | Emergence of combinatorial structure and economy through iterated learning with continuous acoustic signals[END_REF] used another task which, instead of vowel sounds reproduced through a graphical medium, involved learning an artificial whistle language. [START_REF] Verhoef | Emergence of combinatorial structure and economy through iterated learning with continuous acoustic signals[END_REF] had four parallel chains of ten participants memorize and reproduce twelve whistle sounds of a variety of forms (as freely produced and recorded by different people prior to the experiment), using a slide whistle. Directly relevant to the present paper is the fact that the whistle sounds did not refer to anything, and that participants were simply instructed to imitate these sounds as best they could. The experiment showed that, in the course of being transmitted from one participant to the following one along each chain, the whistle sounds came to be broken down into smaller components that were then reused in combination with each other. Quantitative analyses revealed that the sounds became both increasingly structured and increasingly learnable. According to [START_REF] Verhoef | Emergence of combinatorial structure and economy through iterated learning with continuous acoustic signals[END_REF], the cognitive and sensory-motor constraints associated with the transmission task caused a combinatorial structure to form because it made the sounds easier to memorize and reproduce. Thus, this study demonstrates that in an experimental setting where each individual is asked to learn through imitation a repertoire of sounds produced by another individual, along a linear transmission chain, combinatorial properties emerge in that repertoire of sounds that resemble those found in phonological systems.

The linear transmission chain paradigm employed by [START_REF] Verhoef | Emergence of combinatorial structure and economy through iterated learning with continuous acoustic signals[END_REF] makes it possible to experimentally pinpoint the role of learning by imitation in the structuration of a communication system as information is transmitted in a one-way fashion, from the sender to the receiver, with no information passing back from the receiver to the sender. Other studies (e.g., [START_REF] Galantucci | An experimental study of the emergence of human communication systems[END_REF][START_REF] Garrod | Can iterated learning explain the emergence of graphical symbols?[END_REF] have resorted to experimental paradigms in which participants engage in pairwise interactions. These studies focus on graphical communication systems, and whether their findings can be extended to acoustic communication systems remains to be empirically determined. What they show, however, is that pairwise interactions have a significant influence on how a communication system takes shape. Thus, [START_REF] Garrod | Can iterated learning explain the emergence of graphical symbols?[END_REF] showed that pairwise interactions, in a Pictionary drawing task, led to the development of a simpler, more abstract, and more easily identifiable set of graphical signs compared with the one obtained at the outset of a one-way, linear transmission chain (referred to as a diffusion chain). Similarity of the drawings also increased to a larger extent within interacting pairs than between members of the diffusion chain. These results suggest that the feedback signals that participants were allowed to exchange in pairwise interactions made it easier for them to converge towards a set of common signs. This is consistent with the finding that verbal information is more accurately transmitted along a linear transmission chain when participants in adjacent positions along that chain are permitted to interact with each other than in the absence of such interactions [START_REF] Tan | Cultural transmission in the laboratory: agent interaction improves the intergenerational transfer of information[END_REF].

It is of course difficult to reproduce the conditions conducive to the emergence of a communication system in a laboratory setting, in particular because this usually involves calling on adult human participants whose already acquired language knowledge can influence their behavior. While this caveat must be borne in mind, the experimental evidence now available suggests that both learning by imitation and pairwise interactions have an important role to play in the process that allow human participants to jointly build up a repertoire of acoustic or graphical forms endowed with combinatorial properties.

Discussion

The work we have reviewed in Section 2 strongly suggests that imitation between individuals is pervasive in spoken language, at every stage of life from infants to mature speakers. It is believed to play an important role in both speech development and L2 acquisition. Phonetic convergence from one speaker towards another speaker's voice has been found to occur to a limited yet systematic extent in both non-interactive and interactive settings. In our view, the evidence now available suggests that imitation in speech may in fact be constituted by the combination of a low-level component with a high-level one. The low-level component involves sensory-motor integration processes that allow speakers to establish a correspondence between the speech signal they are exposed to and their own repertoire of speech motor commands. It is triggered in an automatic manner and represents one instance of the links that are more generally formed between perception and action. The high-level component operates under the control of a variety of linguistic factors (phonetic convergence can occur inasmuch as phonological contrasts are preserved), individual, situational and social factors. At this high level, imitation is a strategy deliberately employed by speakers to modulate the position they occupy with respect to each other in a social space, as their interaction unfolds. It is one of the ways that speakers have to indicate to each other to what extent they are like each other, and to set up a conversational common ground. It may be hypothesized that the strategic, sociallydriven component of imitation develops during ontogenesis on top of the sensory-motor component as children acquire the skills that are required for them to engage in social interactions.

As we have turned to the potential role of imitation in the emergence of phonological systems (Section 3), we have found that there is compelling evidence for the fact that imitation, far from being a mere behavioral echo, is selective, compositional, and may lead to innovation and change. It involves the decomposition of the perceived speech signal into smaller units that are then recombined with each other by the listener's own speech production system. Both multiagent computational models, such as those developed by [START_REF] Browman | Competing constraints on intergestural coordination and self-organization of phonological structures[END_REF], [START_REF] Goldstein | Emergence of discrete gestures[END_REF][START_REF] De Boer | Self-organization in vowel systems[END_REF], and experimental studies suggest that with such characteristics, imitation may provide a strong contribution to the building up of a phonological system.

In an in-depth theoretical and methodological discussion, [START_REF] Oudeyer | The self-organization of speech sounds[END_REF] advocates what we would refer to as a minimalist approach to the emergence of language, which causes him to attribute no prior communicative or social skills to the agents in his computational model. In Oudeyer's view, his model is in that respect sharply different from one like de [START_REF] De Boer | Self-organization in vowel systems[END_REF], in which agents aim to imitate each other and are also assumed to jointly establish whether imitation has been successful or not through non-verbal feedback. Because this presupposes that agents already have basic communicative and social skills, Oudeyer claims that de Boer's model offers an account of the cultural evolution of language rather than of the emergence of language. It may be debated, however, whether the use of non-verbal feedback or that of preestablished communicative knowledge necessarily imply that agents already share a full-fledged linguistic system. In this regard, note that, in COSMO, it is assumed that each agent has an internal model of the entire communication situation, although COSMO presents itself as a model of the emergence rather than the evolution of phonological systems. What Oudeyer demonstrates, in quite an elegant and parsimonious manner, is that it is possible, for a society of artificial agents, through a self-organization process, to develop a stable and shared repertoire of speech sounds, even though this repertoire is entirely devoid of referential or social function. Whether this is how language has actually formed is, however, an empirical question. In our own view, a model of the emergence of phonological systems does not necessarily have to be based on the assumption that agents start with a minimal set of cognitive, communicative and social skills. In what follows, we focus on the difference between COSMO and imitationbased models.

One major characteristic of COSMO is the emphasis put on deixis and reference. Note that, in the current version of COSMO, there are no actual objects in the agents' virtual world and which agents would have the capacity to detect, recognize and jointly designate. In fact, objects do not seem to have an independent reality, outside the speech code itself, and look functionally identical to abstract labels for sets of motor and auditory patterns. However, this is due to the way in which COSMO has been implemented so far, rather than a feature of COSMO itself, whose future extension towards a yet more comprehensive model that will include genuine deictic and reference mechanisms can easily be envisioned. In imitation-based models, by contrast, it is in our view the social dimension of inter-individual interactions that is dominant. Whether in [START_REF] Browman | Competing constraints on intergestural coordination and self-organization of phonological structures[END_REF], [START_REF] Goldstein | Emergence of discrete gestures[END_REF], or de Boer's (2000) model, agents seek to resemble each other to a greater extent by sounding more like each other. It is unclear, however, upon which grounds one should consider in an a priori manner that either the referential or social dimension of language is predominant. We suggest that computational modeling should aim to investigate what the respective contribution of either dimension may be to the emergence of phonology, as opposed to taking this contribution or absence of contribution as a given.

COSMO provides a computational model that allows studying the emergence of phonological systems as a self-organization process arising from local deictic games between agents. In particular, the architecture of the computational model, which draws from a full internalization of the communication situation, allows for an explicit testing of competing hypotheses about speech-related representations in line with motor vs. auditory vs. sensory-motor theories of speech communication. In our opinion, a valuable extension of the work carried out by Moulin-Frier and collaborators would involve using the COSMO framework to study the potential role of imitation in the emergence of phonological systems. To our knowledge, no computational framework has yet explicitly assessed how varying the rules of the interaction games played by agents may affect the sound systems emerging from these games.

As stated above, phonetic convergence effects in modern speech may be considered as resulting from a variety of cognitive abilities and associated behaviors ranging from loose attunement to ambient speech in minimally interactive settings to deliberate imitation allowing for reciprocal adaptation between communicating partners. As a consequence, testing for the potential role of imitation in the emergence of phonological systems may be best achieved by considering several steps on a "minimally-to-fully-imitative" continuum. A first step on this continuum could be implemented by varying the modalities of the procedure for representations' updating which is currently embedded in COSMO. For example, one could introduce a coupled "recency-decay" function allowing for recently experienced realizations to be favoured in the directly following games while older, for long inactive, realizations would gradually see their probability to be used again tend to zero.

A second step would consist in simulating a minimal interaction in that, for each naming game of a given object o i ,, there would be a two-stage deictic game (keeping C=1 throughout the game): (i) agent A is the speaker; agent B is the listener; both agents update their knowledge respective to their experience; (ii) agent B is the speaker; agent A is the listener; both agents update their knowledge respective to their experience. In other words, agents listen to each other and adapt their auditory prototypes accordingly, but their behavior are still fully supervised by the sharedattention mechanism. A third step would introduce imitation of the interlocutor along with actual perception behavior. At stage 1, the listener (agent B) uses its auditory prototypes to perceive the speaker's production (i.e. to associate it with an object o i ), and in return (stage 2), it performs a speech production based on its closest motor prototype for that object o i , then updates its motor knowledge accordingly. Agent A then updates its own auditory knowledge only if its inference of the object o i based on the reproduction from agent B (at stage 2) matches the object agent A has itself initially named (at stage 1). These rules of interaction may be compared to the simulation used by Goldstein and colleagues [START_REF] Goldstein | Emergence of discrete gestures[END_REF][START_REF] Goldstein | Articulatory phonology: A phonology for public language use[END_REF], to the extent that agents update their "knowledge" based on an internal assessment of the success of the interaction game. A fourth step would include feedback (in line with the simulations carried out by [START_REF] De Boer | Self-organization in vowel systems[END_REF], so that agent A would pass on the result of its assessment of the imitation to agent B (C=1 vs. C=0), which would only then (i.e., at stage 2) update its own motor representations accordingly. Moreover, in our opinion, valuable insight on the role of imitation in shaping phonological systems could be gained if its effects were assessed in heterogeneous populations of agents. For example, one may carry out simulations in which, due to inter-individual variation, some agents are more prone to convergence than others, and some are even prone to divergence. In the third-step game outlined above, it would mean that agent B would perform the production task by selecting any of its motor prototypes associated with the object o i , or even by selecting the prototype which is the furthest away from (vs. the closest to) the initial production of agent A. The proportion of 'divergent' vs. 'indifferent' vs. 'convergent' agents could be a variable manipulated in the simulations.

On a more structural level, a comparison between minimally-to-fully imitative interaction types may be even more insightful if simulations were carried out on a population of agents exhibiting horizontal as well as vertical structuring. By 'horizontal structuring', we refer here to networks of variously interconnected agents aimed at simulating social structure. In these social networks, the probability of occurrence of specific one-to-one interactions is modulated by the structure of the network itself, and both high-connected and low-connected agents may play an influential role on the specifics of the emerging shared system [START_REF] Fagyal | Centers, peripheries, and popularity: The emergence of norms in simulated networks of linguistic influence[END_REF]. As to 'vertical structuring', it could be implemented by gradually introducing newly-initialized agents ('newborns') as well as removing mature agents ('deceased') as simulations accumulate, so as to ensure generational replacement in the long run.

Admittedly, both horizontal and vertical structuring of the agents' population may be more related to the propagation and transmission of the specifics of a shared repertoire of sounds than to its proper building, i.e. it may enlighten us on the cultural evolution, rather than on the emergence, of phonological systems. However, since culture preceded (modern: symbolic, compositional) language in human evolution, it may be unreasonable to simulate the emergence of phonological systems while ignoring the context in which this emergence process took place.

In that sense, it may not be considered as an asset of the computational models studying the emergence of phonology, that the simulations carried out with these models invariably lead to a shared sound systems which, once established, is absolutely stable over time. A working hypothesis is that imitative interaction games, together with network structuring, although not absolutely necessary for the emergence of stable shared repertoire of sounds, do not prevent them to appear, but have the advantage of carrying the seeds of further, cultural evolution through the propagation of innovative variants, i.e. they allow to model both the stability of shared sound systems and their potential for sound change.

Acknowledgments

The writing of this paper was partially supported by the Brain and Language Research Institute at Aix-Marseille University (Labex BLRI, ANR-11-LABX-0036).


Constantin Udriste

Massimiliano Ferrara

Armando Ciancio

The least-curvature principle of Gauss and Hertz and geometric dynamics

Keywords: Kinematic curvature, Gauss-Hertz curvature, S-shaped curves, geometric dynamics, economic growth, Riemannian manifolds

This paper shows how we can study real life problems in economics, biology and engineering with tools from differential geometry. Section 1 emphasizes the S-shaped time evolutions and underlines that the torse forming vector field reflects economical phenomena. Section 2 analyzes the geometric dynamics on infinite dimensional Riemannian manifolds produced by first order ODEs and the Euclidean metric. Section 3 introduces and studies a least-curvature principle. Section 4 defines and studies the geometric dynamics on infinite dimensional Riemannian manifolds produced by second order ODEs and by the Euclidean metric. Section 5 analyzes the least-curvature principle of Gauss and Hertz in a general setting. Section 6 explores the controllability of the neoclassical growth geometric dynamics and underlines that the theory can be extended to infinite dimensional manifolds. Section 7 contains conclusions.

S-shaped time evolutions

S-shaped (or sigmoid) time evolutions [START_REF] Jarne | Fatas-Villafranca, S-shaped Economic Dynamics, The Logistic and Gompertz Curves Generalized[END_REF] are frequently observed in some dynamic economic phenomena (product life cycles, the gradual diffusion of technological innovations or long-term fluctuations in income, productivity growth etc). It is also useful in biology and demography dynamics. These S-shaped evolutions are usually incorporated into formal models using autonomous ODE systems such as ẋ(t) = X(x(t)), where a solution x(t) has an inflection point x 0 = x(t 0 ). In other words x(t) must satisfy ẍ(t 0 ) = µ ẋ(t 0 ) or D X X(x(t 0 )) = µX(x(t 0 )) and x (t 0 ), x (t 0 ) are linearly independent vectors. This means that X(x(t 0 )) is a proper vector of the Jacobian matrix DX(x(t 0 )) with respect to the proper value µ.

The S-shaped evolutions degenerate in two ways: (i) D X X(x(t)) = µ(t)X(x(t)), ∀t ∈ I ⊂ R, where x(t) is a solution of [START_REF] Jarne | Fatas-Villafranca, S-shaped Economic Dynamics, The Logistic and Gompertz Curves Generalized[END_REF]; in this case the solution x(t) of ( 1) is a straight line;

(ii) D X X(x(t)) = µ(x(t))X(x(t)), for any t ∈ I ⊂ R and for any solution x(t) of [START_REF] Jarne | Fatas-Villafranca, S-shaped Economic Dynamics, The Logistic and Gompertz Curves Generalized[END_REF]; in this case any solution x(t) of ( 1) is a straight line. This remark suggests to introduce a vector field X characterized by the relation D X X(x) = µ(x)X(x), ∀x ∈ R n . Note that a torse forming vector field has this property. Indeed, such a field is defined by [START_REF] Udris ¸te | Geometric Dynamics[END_REF] D

Z X = aZ + (Y, Z)X,
where Z is an arbitrary C ∞ vector field, a is a given C ∞ real function and Y is a given C ∞ vector field.

On the other hand, in the case of torse forming vector field, the relation

D X X = aX + (Y, X)X
is true everywhere, showing the collinearity between D X X and X. In this way, a torse forming vector field used till now only in differential geometry may reflect phenomena from economics, biology, demographics etc (Open Problems).

Geometric dynamics on infinite dimensional Riemannian manifolds produced by first order ODEs

The geometric dynamics [START_REF] Udris ¸te | Geometric Dynamics[END_REF]- [START_REF] Udris ¸te | Queue geometric dynamics[END_REF] induced by a flow

ẋ(t) = X(x(t), t), (1)
and by the Euclidean metric is described by the first order Lagrangian

L 1 (x, ẋ, t) = 1 2 || ẋ -X(x, t)|| 2 . Let F = 1 2 ||X|| 2
be the potential energy associated to X. The second order Euler-Lagrange ODE system associated to L 1 , i.e.,

∂L 1 ∂x - d dt ∂L 1 ∂ ẋ = 0 or ẍ(t) = DX(x(t)) -DX T (x(t)) ẋ(t) +∇F (x(t)) + ∂X ∂t , (2)
is an Euler-Lagrange prolongation of the system (1).

Introducing the gyroscopic vector field

Y (x(t), ẋ(t), t) = DX(x(t)) -DX T (x(t)) ẋ(t)+∇F (x(t))+ ∂X ∂t ,
the ODE (2) can be written in the general form of a second order dynamical system. A trajectory in the geometric dynamics (2) is fixed either by initial conditions

x(0) = x 0 , ẋ(0) = v 0
or by boundary conditions

x(0) = x 0 , x(t 0 ) = x t 0 .
If v 0 = X(x(0)), then the trajectory in geometric dynamics coincides with a field line of the vector field X, i.e., a solution of (1). If v 0 = X(x(0)), then the corresponding trajectory is transversal to the field lines. In [3, pag. 5], one asks what is the practical significance of these transversal trajectories; in the Sections 3 and 5, possible answers will be given.

We admit that the theory in [3, pag. 137] can be applied also to the first order differential systems consisting from an infinity of n-vector equations. For that we use the Riemannian (Euclidean) manifold (R n × l 2 , δ ij + δ αβ ) to transform a local flow in a geodesic motion in a gyroscopic field of forces. We start from an arbitrary local flow described by

ẋα (t) = X α (x α (t), t), x α (t) ∈ R n , α ∈ N, (3) on the Riemannian (Euclidean) manifold (R n × l 2 , δ ij + δ αβ ). We introduce the least squares La- grangian L 1 ∞ (x α , ẋα , t) = 1 2 δ ij δ αβ ( ẋi α -X i α (x α , t))( ẋj β -X j β (x β , t)) = 1 2 α || ẋα -X α (x α , t)|| 2
and the associated action on the Sobolev space W 1,2 ([a, b]). By the Euler-Lagrange ODEs, we obtain an Euler-Lagrange prolongation of the system (3),

d 2 x α dt 2 = δ βγ ∂X α ∂x β - ∂X β ∂x α dx γ dt + ∂f ∂x α + ∂X α ∂t , where f = 1 2 δ ij δ αβ X i α X j
β is a density of energy. This prolongation determines a geometric dynamics, i.e., a geodesic motion in a gyroscopic field of forces. Automatically, the least squares Lagrangian determines the Hamiltonian

H 1 ∞ (x α , ẋα , t) = 1 2 δ ij δ αβ ( ẋi α -X i α (x α , t))( ẋj β + X j β (x β , t)).
Of course, L 1 ∞ and H 1 ∞ are sums of series of functions. They ask the uniform convergence, and hence

lim α→∞ ( ẋα -X α (x α , t)) = 0 lim α→∞ ( ẋβ + X β (x β , t)) = 0
along the trajectories in geometric dynamics.

The least-curvature principle

Suppose that, for each index α ∈ N , the point x α (t) represents a particle involved in a first order dynamical system at time t and that we have constraints described by

A k (x) = 0, x = (x α ), α ∈ N, k = 1, ..., m,
where A k are C 1 functions. Admit that the particle is "sensitive" to the velocity vector field X α (x α (t), t), in the sense that without the constraints action, the particle would follow a field line. The velocity vector ẋ(t) = ( ẋα (t)) refers to any kinematically possible path, i.e. solution of the constraints system, for which coordinates at the instant considered are the same as in actual trajectory.

Definition. The least squares Lagrangian (sum of series of functions, uniform convergence)

L 1 ∞ = 1 2 α || ẋα (t) -X α (x α (t), t)|| 2
is called the kinematic curvature of the path x(t) = (x α (t)) on the Riemannian (Euclidean) space (R n × l 2 , δ ij + δ αβ ).

Theorem. From all paths consistent with the constraints, the actual trajectory is that which has the least kinematic curvature.

Proof. Differentiating the constraints we respect to t, we obtain

∂A k ∂x i α (x(t)) ẋi α (t) = 0.
Let ẋi α be a typical component of the velocity in the path considered and ẋi α0 be the corresponding component of the velocity in the actual trajectory. Then, by the previous equality,

∂A k ∂x i α (x(t))( ẋi α (t) -ẋi α0 (t)) = 0,
i.e., a small displacement δx i α (t) of the system is proportional to ( ẋi α (t) -ẋi α0 (t)). The components of the velocities exercised by the curvature are typified by ( ẋi α0 (t) -X i α (x α0 (t), t)). Since these components must be normal to the submanifold defined by the constraints, we have (orthogonality condition)

δ ij δ αβ ( ẋi α0 (t) -X i α (x α0 (t), t))( ẋj β (t) -ẋj β0 (t)) = 0.
This relation can be written as follows

α || ẋα (t) -X α (x α0 (t), t)|| 2 = α || ẋα0 (t) -X α (x α0 (t), t)|| 2 + α || ẋα (t) -ẋα0 (t)|| 2 . Consequently α || ẋα (t) -X α (x α0 (t), t)|| 2 > α || ẋα0 (t) -X α (x α0 (t), t)|| 2 .
Now, the result from the statement of the theorem follows. This result shows that the trajectories of the Euler-Lagrange prolongation (2), which are not field lines of (1), can be trajectories of the dynamical system ẋ(t) = X(x(t), t) subject to some constraints.

Geometric dynamics on infinite dimensional Riemannian manifolds produced by second order ODEs

The general form of a second order dynamical system (Newton Law) is

ẍ(t) = Y (x(t), ẋ(t), t). (4)
A trajectory of this second order dynamical system is fixed either by initial conditions

x(0) = x 0 , ẋ(0) = v 0
or by boundary conditions

x(0) = x 0 , x(t 0 ) = x t 0 .
The second order ODEs (4) suggests that giving a vector field Y (x(t), ẋ(t), t), we can introduce a second order least squares Lagrangian

L 2 = 1 2 ||ẍ(t) -Y (x(t), ẋ(t), t)|| 2 ,
using the Euclidean metric. The fourth order Euler-Lagrange equations produced by L 2 , i.e.,

∂L 2 ∂x - d dt ∂L 2 ∂ ẋ + d 2 dt 2 ∂L 2 ∂ ẍ = 0
are Euler-Lagrange prolongations of a system of type (4). To simplify, we accept

Y (x(t), ẋ(t), t) = Y (x(t))
and we denote F = 1 2 ||Y || 2 . Then the Euler-Lagrange prolongation of the system (4) is

x (4) (t) = DX(x(t)) -DX T (x(t)) ẍ(t) + ẋT (t) (Hess(Y )(x(t))) ẋ(t) -∇F (x(t)).
This prolongation determines a geometric dynamics.

We admit that the theory in [3, page. 137] can be applied also to the second order differential systems consisting from an infinity of n-vector equations. For that we use the Riemannian (Euclidean) manifold (R n × l 2 , δ ij + δ αβ ) to transform a Newton law in an Euler-Lagrange prolongation of fourth order. We start from an arbitrary local second order dynamical system described by

ẍα (t) = Y α (x α (t), ẋα (t), t), (5)
x α (t) ∈ R n , α ∈ N, on the Riemannian (Euclidean) manifold (R n × l 2 , δ ij + δ αβ ).
We introduce the second order least squares Lagrangian

L 2 ∞ (x α , ẋα , ẍα , t) = 1 2 δ ij δ αβ (ẍ i α -Y i α (x α , ẋα , t))(ẍ j β -Y j β (x β , ẋβ , t)) = 1 2 α ||ẍ α -Y α (x α , ẋα , t)|| 2
and the associated action on the Sobolev space W 1,2 ([a, b]). By the Euler-Lagrange ODEs, we obtain an Euler-Lagrange prolongation of the system (3), which is a fourth order system. This prolongation determines a geometric dynamics. Automatically, the geometric dynamics determines the Hamiltonian

H 2 ∞ (x α , ẋα , ẍα , t) = 1 2 δ ij δ αβ (ẍ i α -Y i α (x α , ẋα , t))(ẍ j β + Y j β (x β , ẋβ , t)).
Of course, L 2 ∞ and H 2 ∞ are sums of series of functions. They ask the uniform convergence, and hence

lim α→∞ (ẍ α -Y α (x α , ẋα , t)) = 0 lim α→∞ (ẍ β + Y β (x β , ẋα , t)) = 0
along the trajectories in geometric dynamics.

The least-curvature principle of Gauss and Hertz

Suppose that, for each index α ∈ N , the point x α (t) represents a particle involved in a second order dynamical system at time t. Admit that the constraints are described by Pfaff equations

δ αβ A αki (x)dx i β = 0, x = (x α ), α, β ∈ N, k = 1, ..., m, i = 1, ..., n,
where the coefficients A αki are C 1 functions. We add the external force Y α (x α (t), ẋα (t), t) which acts on the particle. The acceleration vector ẍ(t) = (ẍ α (t)) refers to any kinematically possible path, i.e. solution of the previous Pfaff system, for which coordinates and velocities at the instant considered are the same as in actual trajectory. Definition. The least squares Lagrangian (sum of series of functions, uniform convergence)

L 2 ∞ = 1 2 α ||ẍ α (t) -Y α (x α (t), ẋ(t), t)|| 2
is called the Gauss-Hertz curvature [START_REF] Whittaker | A Treatise on the Analytical Dynamics of Particles and Rigid Bodies[END_REF] of the path

x(t) = (x α (t)) on the Riemannian (Euclidean) space (R n × l 2 , δ ij + δ αβ ).
Theorem. From all paths consistent with the constraints, the actual trajectory is that which has the least Gauss-Hertz curvature.

Proof. Differentiating the constraints

δ αβ A αki (x(t)) ẋi β = 0
we respect to t, we obtain

δ αβ A αki (x(t))ẍ i β (t) + ∂A αki ∂x j γ (x(t)) ẋi β (t) ẋj γ (t) = 0.
Let ẍi α be a typical component of the acceleration in the path considered and ẍi α0 be the corresponding component of the acceleration in the actual trajectory. Then, by the previous equality,

δ αβ A αki (x(t))(ẍ i β (t) -ẍi β0 (t)) = 0,
i.e., a small displacement δx i α (t) of the system is proportional to (ẍ i α (t) -ẍi α0 (t)). The components of the forces exercised by the Gauss-Hertz curvature are typified by (ẍ i α0 (t) -Y i α (x α0 (t), ẋα0 (t), t)). Since these forces do no work, we must have (orthogonality condition)

δ ij δ αβ (ẍ i α0 (t) -Y i α (x α0 (t), ẋα0 (t), t)) (ẍ j β (t) -ẍj β0 (t)) = 0.
This relation can be written as follows

α ||ẍ α (t) -Y α (x α0 (t), ẋα0 (t), t)|| 2 = α ||ẍ α0 (t) -Y α (x α0 (t), ẋα0 (t), t)|| 2 + α ||ẍ α (t) -ẍα0 (t)|| 2 . Consequently α ||ẍ α (t) -Y α (x α0 (t), ẋα0 (t), t)|| 2 > α ||ẍ α0 (t) -Y α (x α0 (t), ẋα0 (t), t)|| 2 .
Now, the result from the statement of the theorem follows.

Neoclassical growth geometric dynamics

We start with the neoclassical growth flow

k(t) = X(k(t)) -δk(t) -c(t), k(0) = k 0 > 0,
where k = (k 1 , ..., k n ) is the capital intensity vector, c = (c 1 , ..., c n ) is the control vector and δ is a real number. The Jacobian matrix of the growth vector field is DX -δI. Therefore, the geometric dynamics induced by the growth vector field and the Euclidean metric is described by the second order ODE system k(t) = DX(k(t)) -DX T (k(t)) k(t) +∇F (k(t)) -ċ(t),

where F = 1 2 ||X(k) -δk -c|| 2 .
Obviously, the geometric dynamics is controlled by the command c(t) and its derivative ċ(t). Now, we underline that the growth flow can be analyzed as in Sections 2-5 (see also [START_REF] Udris ¸te | Economic Geometric Dynamics[END_REF]), but the sense of index α is still an open problem.

Conclusions

The S-shaped evolutions, the geometric dynamics on infinite dimensional Riemannian manifolds, the leastcurvature principles and the neoclassical growth flow are suitable mathematical objects and models for describing the real life problems in economics, biology and engineering with tools from differential geometry.

As example, the least-curvature principles, in fact the distance between a point and a subset, can describe the evolution of a dynamical system subject to some constraints.

Acknowledgements: Partially supported by Grant CNCSIS 86/ 2008 and by 15-th Italian-Romanian Executive Programme of S&T Cooperation for 2006-2008, University Politehnica of Bucharest. The actual form of the paper was obtained by the first author between two o'clock and six o'clock, in the morning of March 24, 2008, at "Friendly Inn at Harvard" Hotel, Cambridge, Massachusetts, USA, with occasion of American Conference on Applied Mathematics (Math'08) organized by World Scientific and Engineering Academy and Society (WSEAS).