Stefano Borgo
email: stefano.borgo@cnr.it

Noemi Spagnoletti
email: noemi.spagnoletti@istc.cnr.it

Laure Vieu
email: laure.vieu@irit.fr

Elisabetta Visalberghi
email: elisabetta.visalberghi@istc.cnr.it

Artifact and Artifact Categorization: Comparing Humans and Capuchin Monkeys

We aim to show that far-related primates like humans and the capuchin monkeys show interesting correspondences in terms of artifact characterization and categorization. We investigate this issue by using a philosophically-inspired definition of physical artifact which, developed for human artifacts, turns out to be applicable for cross-species comparison. In this approach an artifact is created when an entity is intentionally selected and some capacities attributed to it (often characterizing a purpose). Behavioral studies suggest that this notion of artifact is not specific to the human kind. On the basis of the results of a series of field observations and experiments on wild capuchin monkeys that routinely use stone hammers and anvils, we show that the notions of intentional selection and attributed capacity appear to be at play in capuchins as well. The study also suggests that functional criteria and contextualization play a fundamental role in terms of artifact recognition and categorization in nonhuman primates.

Introduction

We aim to show that the human and the capuchin monkey (a far-related primate) are cognitively closer than generally accepted in categorizing natural and artificial entities. Assuming these commonalities are not accidental, the observation of capuchin monkey provides further evidence to identify which elements are most likely primary in both human and non-human primates' artifact categorization. We investigate this issue by using a philosophicallyinspired theory, developed for human artifacts, as a handle for a cross-species comparison. The theory is fairly simple yet complex enough to help in wellknown philosophical puzzles such as the ship of Theseus.

As far as it is known today, capuchin monkeys do not have a specific language for artifacts or artifactual entities, and seem to lack terms even for most natural entities. The comparison is thus based on behavioral evidence gained from capuchin monkeys in captivity and in the wild; this "behavioral" viewpoint is a limitation of our work that should be kept in mind.

A Theory of Physical Artifacts

It can be argued that ontologically physical artifacts, the artifacts most studied in the philosophical literature, see [START_REF] Hilpinen | Artifact[END_REF]Margolis and Laurence 2007), are not simply physical objects that have been physically modified or built by humans, as classically understood. Although the recognition of physical manipulation on an item gives us a strong indication of artifactness, physical manipulation is not essential: for a number of authors, the essence of artifacts lies in intentionality [START_REF] Baker | The ontology of artifacts[END_REF][START_REF] Dipert | Artifacts, art works, and agency[END_REF][START_REF] Thomasson | Artifacts and human concepts[END_REF]. We intentionally select objects in order to use them for a purpose, sometimes physically modifying them to suit our tasks. This gives a broader perspective on artifacts which includes "naturefacts" [START_REF] Oswalt | Habitat and technology: The evolution of hunting[END_REF], such as found art, and at the same time avoids counting as artifacts residues like sawdust and nonintentional animal constructs like wasp nests. [START_REF] Borgo | Artefacts in formal ontology[END_REF] discuss the case of a paperweight made of a pebble, an example of naturefact, and propose that the artifact (paperweight) is a creation obtained by some agent intentionally selecting an object (the pebble) and intentionally attributing to it some capacities [START_REF] Cummins | Functional analysis[END_REF], that is, dispositions or functionalities like holding paper without ruining it, being easily grasped by hand, being firm etc. The theory of physical artifacts proposed in [START_REF] Borgo | Artefacts in formal ontology[END_REF]) presupposes a formal structure in which ontological categories like physical object, agent, quality and event, are characterized and related. The theory itself is formally instantiated relying on the foundational ontology DOLCE [START_REF] Masolo | Wonderweb deliverable 18[END_REF]) which, for simplicity, is also adopted in this paper. An attributed capacity is classified in the theory as an individual quality that characterizes the purpose or function of the artifact as determined by its creator(s). When the artifact does not perform according to the capacities attributed to it, it is said to be flawed or malfunctioning without contending the existence of the artifact itself. Of course, rational agents, on the basis of acquired expertise or knowledge of physics, usually ground their selection and capacity attribution on the observation of actual physical qualities that they assume to be relevant.

The assumption that the artifact (the paperweight in the previous example) is not identical to the physical object of which it is made (the pebble) is also crucial. This approach to physical artifacts contrasts the standard view that an artifact is a physical object with some extra (generally acquired) property, i.e., a function or a role [START_REF] Vieu | Artefacts and roles: Modeling strategies in a multiplicative ontology[END_REF]. In our approach the physical object and the physical artifact are co-located when both present but have different properties and in particular different lifetimes (the paperweight starts existing well after the pebble). In addition, the paperweight depends onmore specifically, is constituted bythe pebble but not vice versa. In fact, physical artifacts and physical objects obey different identity criteria as shown by the fact that artifacts can be repaired and undergo part substitution without losing their identity: they simply are constituted by different physical objects at different times. Such a substitution is ruled out by the view of artifacts as physical objects with extra properties since by altering the physical object one affects the identity of the artifact itself. Hence with our view, in the classical ship of Theseus puzzle [START_REF] Lowe | On the identity of artifacts[END_REF], substituting a plank doesn't destroy the artifact, i.e., the ship, whose attributed capacities continue to persist, although it does destroy the original physical object that constituted it, i.e., the plank assembly.1 When the original planks are assembled again, depending on the identity criteria given to physical objects, one could argue that the original physical object comes back into existence. The artifact that is then created is though a different one with perhaps other attributed capacities.

Similarly, the intentional use of artifacts for purposes not considered at their creation time or the attribution of new capacities to existing artifacts give rise to new artifacts in this theory. This gives a simple answer to a deeply discussed identity issue, see [START_REF] Houkes | Technical functions. On the use and design of artefacts[END_REF], at the cost of multiplying (co-located) entities, a move not to the taste of all ontologists.

Artifacts can be divided in private, i.e., recognized by its creator only, and social (artifact proper à la Dipert 1993) when their intention-based properties take on a social dimension. More specifically, in [START_REF] Borgo | Artefacts in formal ontology[END_REF]) social artifacts are artifacts whose status and type is recognized by members of a specific society. Following Borgo and Vieu, we use artifact type, or simply type, to indicate a description 2 of artifacts sharing some characteristics, i.e., an artifact kind for other authors. An artifact type specifies a number of attributed capacities and constrains the range of values these can take. 3 In our society, most artifacts are social, like knifes and cars. Among the social artifacts, some are recognized because of their observable intrinsic properties: the type "knife" and its subtype "kitchen knife" are recognized in a given knife because of its particular shape and the material it is made of. Others, e.g., paperweights made out of pebbles, can be recognized as such only in their context of use, e.g., when placed on a pile of papers on a desk.

Physical artifacts can of course be distinguished in other ways, e.g. according to whether they are "naturefacts" or manufactured. In the latter case, they can be further divided according to the degree of manufacturing and the complexity of their constituting physical object. Naturefacts are necessarily simple, but manufactured artifacts are often constituted by assemblies. There are different views on the essential characteristics of technical artifacts [START_REF] Borgo | Technical artifact: An integrated perspective[END_REF] and in this paper we take the view that technical artifacts are those manufactured according to a design, specifying a precise artifact type, a production method and a use plan [START_REF] Houkes | Technical functions. On the use and design of artefacts[END_REF].

Finally, tools form a class of physical artifacts whose attributed capacity is related to the purpose of physically acting on some other entity (unlike works of art, for instance). 4 Usually, these are also social artifacts, like the knifes above.

Given this view on artifacts, the next section introduces the non-human primates that serve for our cross-species comparison of the ontological category of artifact.

Capuchin Monkeys and Tool Use

Capuchin monkeys (genus Cebus and Sapajus) are South American primates that separated from the human lineage about 35 million years ago, much earlier than chimpanzees (our closest relatives). Despite the phylogenetic distance the cognitive abilities of capuchins and chimpanzees show many similarities [START_REF] Fragaszy | The complete capuchin: The biology of the genus Cebus[END_REF]). The abilities of capuchins appear cognitively more advanced than those of other New and Old World nonhuman primate species [START_REF] Tomasello | Primate cognition[END_REF]. Very recently, captive capuchins have been shown to be capable of: transferring relations from one situation or set of objects to a different one [START_REF] Kennedy | Analogical reasoning in a capuchin monkey (Cebus apella)[END_REF][START_REF] Spinozzi | Categorization of above and below spatial relations by tufted capuchin monkeys (Cebus apella)[END_REF]), analogical reasoning [START_REF] Truppa | Same/different concept learning by capuchin monkeys in matching-to-sample tasks[END_REF], applying relational structures in tool using tasks and being sensitive to the functional properties of the tools. In these tasks capuchins require more practice than chimpanzees but, eventually, they reach similar levels of performance.

A few primate species use tools in captivity [START_REF] Shumaker | Animal tool behavior: The use and manufacture of tools by animals[END_REF]). Among monkeys, the capuchins are the best tool users and their abilities are similar to those of chimpanzees. In nature only chimpanzees, among great apes, use tools habitually for diverse purposes and in many varied formats across their geographical distribution [START_REF] Boesch | Complex tool sets for honey extraction among chimpanzees in Loango National Park, Gabon[END_REF]). Tool use is observed much less often in wild orangutans (Pongo abelii), and in the other wild great apes (Western gorillas, Gorilla gorilla, and bonobos, Pan paniscus) though in captivity all the great ape species use tools spontaneously in flexible and diverse ways. Only a few species use tools in natural settings; Macaca fascicularis [START_REF] Gumert | The physical characteristics and usage patterns of stone axe and pounding hammers used by long-tailed macaques in the Andaman Sea region of Thailand[END_REF]) and capuchins excel in this respect. The reports on tool use by wild bearded capuchins (Sapajus libidinosus, formerly Cebus libidinosus) come from seasonally dry Cerrado and Caatinga habitats in the north and east of Brazil. It is worth to mention that given the very wide distribution of capuchin monkeys in Central and South America, tool use has been observed in very few species and only in a few locations. Therefore, it seems that individuals acquire this behaviour only in specific environmental circumstances and that its diffusion among group members is supported by social conditions [START_REF] Fragaszy | Community resources for learning: how capuchin monkeys construct technical traditions[END_REF]. This suggests that the ability is acquired, requires intentionality and can be taken as cultural trait of some groups. At Fazenda Boa Vista, State of Piauí in Brazil capuchin monkeys crack open palm nuts and other encased foods with stone hammers and anvils. In other sites they use stones to access embedded food by percussion and by scraping, and sticks to probe for honey and to flush vertebrate prey [START_REF] Mannu | The enhanced tool-kit of two groups of wild bearded capuchin monkeys in the Caatinga: tool making, associative use, and secondary tools[END_REF].

Using a stone (or log) to crack open an encased food item placed on a solid surface (an "anvil") is considered the most complex form of tool use by nonhuman species routinely seen in nature because it involves producing two spatial relations in sequence between nut and anvil, and between pounding tool and nut [START_REF] Matsuzawa | Primate foundations of human intelligence: A view of tool use in nonhuman primates and fossil hominids[END_REF]. Furthermore, transporting the food item and sometimes the percussor involves costs (time and energy, among others) and may present cognitive challenges, such as anticipating future needs, recalling elements that are out of sight, and planning the course of action. Transporting food resources and repetitive visits to specific places on the landscape to process foods are associated with early Homo and are thought to be important innovations of the Oldowan [START_REF] Potts | Why the Oldowan? Plio-Pleistocene tool making and the transport of resources[END_REF].

Comparative Analysis via the Artifact Theory

Since 2004, we have carried out several observational studies and a series of field experiments on wild capuchins using hammer and anvil tools in Fazenda Boa Vista [START_REF] Visalberghi | Characteristics of hammer stones and anvils used by wild bearded capuchin monkeys (Cebus libidinosus) to crack open palm nuts[END_REF]). Here we report some of the collected evidence that the notion of artifact introduced in Section 2 applies to these monkeys. For a thorough description of the field experiments and their methodology, the interested reader can visit the website EthoCebus.net and check references within it.

In this section we argue that there is evidence that capuchins intentionally select objects on the basis of different qualities that they can perceive, e.g., by tapping and lifting them, and consistently use these objects, depending on the task to solve, according to the capacities they associate to them. This act of capacity association amounts to judge that an object can be used for a purpose, and thus corresponds to what we called capacity attribution in Section 2. As we will see, these judgments are indeed fallible.

As already said, an adult capuchin usually exploits physical objects to crack nuts and other encased food. [START_REF] Spagnoletti | Stone tool use by adult wild bearded capuchin monkeys (Cebus libidinosus)[END_REF] observed wild capuchins routinely using tools to crack open nuts of different resistance. Capuchins select the stone of the appropriate material and weight according to whether they have to crack open nuts (more resistant) or other encased foods (less resistant). Capuchin select stones heavy enough to crack the nut, even if the heavier stone is smaller than the lighter one [START_REF] Visalberghi | Selection of effective stone tools by wild capuchin monkeys[END_REF]. Experimental work has also demonstrated that capuchins choose hard material (quartzite) instead of friable material (sandstone) when about to crack nuts.

Capuchins select stones never encountered before and paired in different assortments. This indicates that choice is intentional and not based on a sort of recognition of what has been successful before. Capuchins use generalized knowledge built upon their lifetime experience with percussive activities. We observed that juvenile capuchins need a lot of experience before acquiring the skills to select the appropriate physical qualities. They do select physical objects that are inadequate, e.g. sandstones that break when cracking hard nuts. This happens also, but rarely, with adult monkeys if there are no suitable stones nearby.

The attribution of capacities to selected objects are kept in capuchins' memory, suggesting artifact persistence. There is evidence that capuchins repeatedly use previously selected stone tools and anvil sites. An anvil has been observed in use 108 times over a period of 12 months. Further indirect evidence of persistence is based on anvils censused regularly and the presence of pits that require many tool strikes to be produced. When wild capuchins have a nut requiring a stone and an anvil for its exploitation, they actively search for anvil sites already used in the past. While doing so, capuchins increase their speed of locomotion also when the site is still out of view, suggesting that they know what to search for and where it is located.

Moreover, the same physical object is repeatedly used without thoroughly checking again its physical qualities. When capuchins are at a site previously used or seen to be used, exploratory behaviours meant to evaluate the affordance of the stone are absent or extremely rare suggesting that they have a notion of social artifact. Exploratory actions such as lifting/moving the hammer stone (to assess its mass) and tapping on it (to infer density/mass by means of the sound produced) were common towards home-made resin stones but not with stones of the material commonly encountered by capuchins in their home-range [START_REF] Visalberghi | Selection of effective stone tools by wild capuchin monkeys[END_REF]).

These observations strongly suggest that capuchins do use tools or, more explicitly, physical artifacts to which they attribute the capacity to crack open nuts. The theory of artifacts presented above is suited to account for capuchin's behavior, since it is essentially based on attributing capacities, and not on manufacturing. A fundamental aspect of the theory, the assumption that there are two distinct entities, namely the artifact and the physical object constituting it, is particularly difficult to justify on the mere basis of capuchin's behavior. Nonetheless, we believe that one of our observations (video recorded by N. Spagnoletti, unpublished) provides some evidence for it. A juvenile capuchin, having selected a friable stone, keeps pounding on a nut while the stone repeatedly breaks up. He is thus holding at each strike a different stone but arguably the same artifact since he doesn't show doubts nor pauses after each breaking of the stone as one would expect. This suggests that a theory in which the identity criteria of artifacts differ from that of the physical objects constituting them is more adequate than one in which artifactualness is just a property of physical objects.

Capuchins have a high ability to select physical objects whose actual capacities very finely match the attributed capacity of the tools they need for a given purpose. They clearly distinguish two types of tools, hammers and anvils. These tools are not manufactured, although other populations do manufacture sticks out of leafed twigs to probe for honey and to flush vertebrate preys. It would even seem that the notion of social artifact may apply to capuchins as well. Preliminary evidence in this direction is that they wait "in queue" to use a hammer and anvil site [START_REF] Spagnoletti | Stone tool use by adult wild bearded capuchin monkeys (Cebus libidinosus)[END_REF]. Once the waiting monkey has access to the anvil it preferably reuses the same pit used by the previous monkey, apparently relying on some notion of social recognition.

Further studies are planned to clarify this issue and to circumscribe the extension of social recognition in capuchins.

So the behavioral evidence in the literature supports the claim that there is a strong similarity of artifact characterization in human and non-human primates. There are several open issues, however, like the capacity of capuchin monkeys to create complex artifacts. One interesting case study would be the assembly of a specific hammer with a specific anvil, thus constituting an instance of complex artifact, and in some sense a manufactured one. Indeed, in one case it has been observed (observation by Spagnoletti, see also [START_REF] Visalberghi | Use of stone hammer tools and anvils by bearded capuchin monkeys over time and space: dynamic construction of an archeological record of tool use[END_REF]) a capuchin bringing a hammer to an anvil from where it had disappeared time before, even though there were other anvils at a closer distance. Further evidence would be necessary to better interpret the importance of this observation in terms of artifact complexity in capuchins.

In the following section we discuss new experimental evidence that reinforces the claim that capuchin monkeys have a category of artifact similar to the one discussed in Section 2. In the rest of the paper, we use the term category in the ethological sense and use ontological category for the philosophical notion, while the term type is as introduced in Section 2. Our aim is to further justify the reading of the ethological categories we encountered in terms of the ontological category of artifact and artifact types.

Tool Selection

Here we address the independence of the capacity attribution by capuchin monkeys from other, arguably irrelevant, properties. More specifically, we ask whether in the context of cracking open nuts there are features, in addition to functional ones such as friability and weight, affecting the tool selection and, if so, whether these can be shown to provide advantages without altering the tool's capacity to achieve the goal. In the experiment described in detail below, capuchins choose between hammer stones (all suitable to crack open nuts) that differ in terms of (a) easiness to be handled (relevant feature) and (b) color (irrelevant feature). In each trial the choice was between a stone with a smooth surface (smooth stone), and a stone with protuberances on its surface (bumpy stone). The two stones differed in colour, with the smooth stone white and the bumpy stone black, or vice versa. We hypothesized that protuberances made the bumpy stone less comfortable for transportation and use than the smooth one. Color was considered an irrelevant feature. In the first condition, each subject received ten trials in which each of the relevant feature was always paired with an irrelevant feature (e.g., the smooth stone was black, whereas the bumpy stone was white). Then, in the second condition, the pairing was switched (e.g., the smooth stone was white, whereas the bumpy stone was black). This procedure allowed us to sort out the feature(s) taken into account by capuchins.

Materials and Methods

Our field site was located at Fazenda Boa Vista in the southern Parnaíba Basin (9°39 ′ S, 45°25′ W) in Piauí, Brazil. It is a flat open woodland (altitude 420 m asl) punctuated by sandstone ridges, pinnacles and mesas rising steeply to 20-100 m above it.

Subjects were four adult males, one adult female and one juvenile male. Experiments were carried out in an area where capuchins routinely use tools to crack open palm nuts where several anvils are present. For the purpose of the study, we removed all stone hammers.

We used a set of 8 artificial stones all weighing about 600 g; we had two black bumpy stones, two white bumpy stones, two white smooth stones and two black smooth stones (for an example see Fig. 1). Artificial stones were made by filling silicon plasters of different size and shapes with unsaturated liquid polyester resin (Silica synthetic Amorphous Colloidal Silicon Dioxide, aereosil), and Organic Peroxides for curing (Butanox M 50). White or black pigments were added to the resin. To obtain the desired weight, the stone interior was filled with resin containing either calcium carbonate or lead beads.

In each trial, the subject had to choose between two stones to crack open a palm nut. Stones were always positioned in the same location (hereafter, choice location), at 60 cm from one another. The choice location was halfway between a log (wooden anvil, 0.24 m-high) and a sandstone anvil (low anvil, at ground level), at a distance of 4 m from both of them. Other sandstone anvils were also present in the area.

Each subject was tested in two conditions of 10 trials each. In one condition the bumpy stone was black and the smooth one was white, whereas in the other condition the bumpy stone was white and the smooth was black. The order in which the two conditions were presented was balanced across subjects. Each trial started when one of the experimenters provided a nut to the subject; this occurred either by positioning the nut on the wooden anvil (at a distance of 4 m from the choice location), or by throwing the nut near the subject (at a distance of 4 m, or more, from the choice location). This procedure guaranteed that the subject was equidistant from the experimental stones, or approximately so.

Since capuchins like food variety, we presented two species of palm nuts: the tucum (Astrocaryum spp.) and the catulé (Attalea spp.). The peak-force-at-failure of these nuts, routinely cracked open with tools by capuchins in FBV, is not significantly different [START_REF] Visalberghi | Characteristics of hammer stones and anvils used by wild bearded capuchin monkeys (Cebus libidinosus) to crack open palm nuts[END_REF]. Two experimenters scored the data by paper and pencil from a distance of 5.0 m from the choice location, while a third one set the tool pair as required by the trial and provided the subject with the nut. The experiment was videotaped and scored for reliability by a naïve observer who was blind to condition. We scored the first stone touched, transported, and used to strike the nut, anvil used and success. Reliability scored on 25 % of the trials was 100 %.

Results and Discussion

Through the experiment, all subjects almost always contacted and manipulated first the smooth stone regardless of its color. In 59 out of 60 trials of the first condition, they transported and used the smooth stone and, in the second condition, always kept using the smooth stone despite its change in color (see Table 1 and Fig. 2). This demonstrates that capuchins preferred the smooth stone also when the association color-surface was switched. Capuchins were successful in 113 trials out of 120 (94 %). In the only trial in which a bumpy stone was used the subject (Jatobà) succeeded. Furthermore, in a few cases the female Chuchu and Chicao and Tucum (one adult male and one juvenile male) gave up after many strikes, possibly because the nut was too resistant to be opened with a stone of 0.6 kg.

The wooden anvil was the most chosen by all subjects being used in 99 out of 120 trials (see Fig. 2). In 19 trials other anvils present in the area were used; in two trials 2 and 3 different anvils were used by transporting the stone from one to the other.

Table 1 Performance of each subject in the B-smooth condition (in which the smooth stone is black) and in the W-smooth condition (in which the smooth stone is white). F female, M male; Ad Adult, J Juvenile; contact number of trials in which the stone contacted first was the smooth one; manipulation number of trials in which the stone manipulated first was the smooth one: transport number of trials in which the stone transported to the anvil was the smooth one; use number of trials in which the stone used to crack the nut was the smooth one; success number of trials in which the subject succeeded; # anvils number of different anvils used during each condition As noted above the bumpy stone was almost never used. Therefore, to assess its effectiveness we provided some extra trials in which subjects had a choice between two bumpy stones. Results showed that capuchins reached success with a number of strikes similar to those obtained with smooth stones (Fig. 3). This suggests that bumpy stone can be used effectively when needed. In this respect, it is worthwhile mentioning that we repeatedly observed capuchins using siltstones that were bumpy and sharp, when cracking nuts in the forest.

Subject

Overall, the results of this experiment demonstrate that capuchins' choice can take into account also a relevant non-functional feature, such as easiness to handle the Fig. 2 (Left) The adult male (Dengoso) having received the nut at the wooden anvil (in the far back) moves towards the choice location holding the nut in his left hand; (centre) he approaches the smooth black stone and (right) starts to lift it. The bumpy white stone on the left is not even touched (Photo by E. Visalberghi) Fig. 3 The adult male Jatoba is using a bumpy stone to crack a nut on the wooden anvil (Photo by E. Visalberghi) stone while ignoring an irrelevant non-functional feature like color. Wild capuchins' recognition of affordances probably relies on their lifelong experience with a wide variety of stones, some of which are more comfortable on their hands when transported and used than others. This result adds evidence to the claim that selection and capacity attribution in capuchin monkeys are comparable to those of humans.

Artefact Subcategories as Types

As we have seen, capuchins' selection process of stone tools and anvils is quite sophisticated as it examines several properties. From long term observations, it has been definitely established that, when necessary, capuchins select stones of higher weight for nuts of high resistance than for nuts of low resistance; moreover, they choose stones of friable material and very light only for encased food items that are less resistant than low resistant nuts [START_REF] Spagnoletti | Stone tool use by adult wild bearded capuchin monkeys (Cebus libidinosus)[END_REF]. These findings and further experiments by [START_REF] Visalberghi | Selection of effective stone tools by wild capuchin monkeys[END_REF] provide a clear demonstration that capuchins have a rich structure of (ethological) categories into which they categorize artifacts. These include at least the categories of anvil and (stone) hammer; and the latter is further refined into subcategories distinguished by functional and non-functional features (among which material, weight and shape), by purpose (the opening of a hard nut with minimal effort and without ruining the content vs the opening of softer nuts or other food items) and even by context (the distance of the stone from the chosen anvil). We do not have enough evidence to discuss subcategories within the anvil category although it is known that, even in this case, capuchins can make fine distinctions about the anvils' material (stone or wood) and features of the pits in them [START_REF] Liu | Wild bearded capuchin monkeys (Cebus libidinosus) place nuts in anvils selectively[END_REF]. The rest of the paper will focus on the structure of the capuchins' category of hammers which is the most studied and understood capuchins' category as of today.

As we have seen, the refinement of the stone hammer category in capuchins is supported by a series of independent observations. These all confirm that capuchins' stone selection takes into consideration several features of the tool, like material and weight, as well as features of the items to be cracked open with the tool, like shell resistance. The selection process optimizes the distinguishing features of the object to use and the relevant features of the shell to crack open including also evaluations of the effort to bring the objects to the anvil. The concomitant examination of a large number of parameters during the selection process is further corroborated by a recent field experiment in which Massaro and Visalberghi (unpublished results) tested five capuchins with a low resistant nut, or with a high resistant nut. In each trial the subject could choose between a 0.6 kg hammer stone already positioned on the anvil and a 2 kg hammer stone positioned at distance of 4 m from the anvil (2 m for one subject). In short, whereas the use of the light stone was readily available, the use of the heavy stone had a cost in terms of transport to the anvil. Results indicate that all subjects chose the light stone in the trials with low resistance nuts, whereas they chose and transported the heavy stone in trials with the high resistant nuts.

In conclusion, capuchins categorize stone tools on the basis of several features. Some of them are hierarchically more important (e.g., weight and material) since they strongly affect tool functionality, whereas others (such as smoothness) are less important because only mildly affecting functionality. They also take into account the resistance of the nut during the process of tool selection and other contextual elements like the distance of the preferred hammer from the anvil, which affects the total costs of tool use. If we separate the contextual elements from the characterizing features, we can conclude that the capuchins' (stone) hammer category is structured into at least three subcategories: hammers for high resistant nuts, hammers for low resistant nuts and hammers for food items encased in even less resistant shells. The analysis of these subcategories and their consistent recognition across capuchins in the studied groups suggest that these subcategories correspond to types as introduced in Section 2. This correspondence is quite natural: the capuchins' subcategories are determined just by looking at some properties of the tools (like hardness and weight) which must have values within some optimal range. Remaining features of the physical object under consideration (like color and smoothness) are irrelevant to its categorization although may determine preferences for usage. This shows that the subcategory is consistently associated to a set of features and that attribution of capacities depend on their values only.

Within the three subcategories, capuchins make consistent choices with regard to other features like smoothness vs bumpiness. These features play however a secondary role since they affect selection only within one of the above subcategory. Clearly, the description of the three subcategories is based on a particular classification of food items. This result could be a byproduct of the investigation based on behavioral studies only. While acknowledging this limitation, one can suggest that the development of capuchins' categorical structure might be strongly influenced by categorization of the natural objects of interest, i.e., the food items. An interesting question is whether these artifact subcategories do in fact reflect the environmental features and the individuals living experiences, or are somehow independent from these. Provided we accept that artifact categorizations in primates share a common nature, an answer to this issue would have important consequences on the debate about the nature of artifact categorization in humans. In the case of capuchin monkeys, findings from other field sites clearly suggest that these specific categories are far from being fixed across capuchin populations. In an environment in which the resistance of the food items processed with tools are less extreme than in our site, [START_REF] Ferreira | On the occurrence of Cebus flavius (Schreber 1774) in the Caatinga, and the use of semiarid environments by Cebus species in the Brazilian state of Rio Grande do Norte[END_REF] found that capuchins still use stones of different weight to process food items of different resistance, see also [START_REF] Falótico | Uso de ferramentas por macacos-prego (Sapajus libidinosus) do Parque Nacional Serra da Capivara -PI. Tese de Doutorado em Psicologia Experimental[END_REF]. Therefore, if artifact categorization is acknowleged in all studied capuchin groups that use tools, the category structures they form are sensitive to what the individual has experienced in its environment given the resources available.

Conclusion

The results reported here indicate that a philosophically-motivated definition of artifact developed for humans applies equally to capuchin artifacts, and so it fails to characterize human artifacts, apart from adding the simple restriction that the creator must be human. While we recognize that a deep understanding of capuchin categorization is still not at our reach, what we know today suggests that human characterization needs to be strengthened with new conditions or that, as we suspect, we should accept that the notion of artifact is not specific to the human kind.

When specific sub-classes of artifacts are concerned, the matter becomes less clear. We mentioned above that the status of complex artifacts and social artifacts is unclear in capuchins; studies specifically focused on these notions are still missing. For social artifacts, this study requires to assess capuchins' intentionality. We generally attribute intentionality to others in our species relying on our rich communication and behavioral system as well as on similarity and co-evolution factors. The situation is different when considering capuchins, and capuchin behavioral studies regarding the attribution of intentionality to their kin concerning artifacts still need to be done. Nevertheless, ongoing experiments are expected to shed some light on relevant social aspects: in particular we are currently studying how the social context influences juveniles' learning processes in tool selection and tool handling.

One could think that no capuchin artifact instantiates the notion of technical artifact in the sense used here, i.e., artifacts manufactured according to a design specifying a precise artifact type, a production method and a use plan. Wild capuchins have not been observed to manufacture hammers and anvils or to produce flakes for a specific goal but it seems they also have no actual reason to produce them. Indeed, captive capuchins can be trained to do so [START_REF] Westergaard | A simple stone-tool technology in monkeys[END_REF]. Also, wild capuchins do manufacture other tools, stripping leaves off twigs and, as described above, they certainly handle artifact types and have clear use plans. So the hypothesis that capuchin's do not have a notion of technical artifact is not really confirmed, unless we take designs to be informational entities that can be explicitly and intentionally communicated to others. This suggestion constitutes a restriction of the notion of design and brings in another aspect we have ignored in this paper: information transmission. Capuchins have a communication system but we ignore if they have terms to refer to specific individual objects, as the stone tools. Currently studies are in progress. Furthermore, capuchins do not carry actions aimed at teaching others. Similarly, capuchins do not even hand food at offsprings. So enriched, the notion of design and the derived notion of technical artifact would likely not be applicable to capuchins and would, perhaps, be proper to human primates. And it would support the fact that technical artifacts seem to evolve, new designs superseding earlier ones, much rapidly in humans than in other primates.

To conclude, our studies show that human and capuchins are cognitively closer than usually accepted and give hints on how to enrich the adopted formal theory of artifacts to account for more specifically human notions.

Fig. 1

 1 Fig. 1 Exemplars of the smooth stones (lower left and upper right) and the bumpy stones (upper left and lower right) presented during the experiment, in white and black (Photo by E. Visalberghi)

The importance of constitution in this theory makes it closer to Baker's approach(Baker

2004).Differently from Baker's, artifacts here are not aggregates and the distinction between the artifact (ship) and the physical object (aggregate of planks) is preserved.2 In the DOLCE framework, types are non-physical entities of the ontological category of concepts, as introduced in[START_REF] Masolo | Social roles and their descriptions[END_REF]).3 See[START_REF] Masolo | Wonderweb deliverable 18[END_REF] for further clarifications of the terminology regarding qualities among which we include attributed capacities.

This characterization corresponds to the usual sense of tool in the lexicon as well as the use in ethology, but is quite different from Dipert's definition[START_REF] Dipert | Artifacts, art works, and agency[END_REF] as in particular they are not limited to manufactured artifacts.

Acknowledgments This interdisciplinary work has arisen within the Institute of Cognitive Sciences and Technologies of the Italian CNR, across the Laboratory for Applied Ontology and the Unit of Cognitive Primatology & Primate Center.

Permission to work in Brazil was granted by IBAMA and CNPq to N.S. and E.V. Thanks to the Oliveira family for permission to work at Boa Vista and logistical support. Thanks to Elsa Addessi to have collected the data of the Experiment here reported with E.V. and N.S.


Robustness analysis and Evaluation of a PMSG-based Marine Current Turbine System under Faulty Conditions

This paper deals with a Permanent Magnet Synchronous Generator (PMSG) driven by a Marine Current Turbine (MCT) through a PWM power rectifier and used for isolated sites. The generator model is established in the synchronous rotating dq reference frame. The control of the speed, the d-axis current, and the q-axis current is achieved using PI correctors. Robustness and performance of the considered system are evaluated in healthy and faulty conditions. The faulty mode deals with the study of single and multiple openswitch damages appearing in the PWM power rectifier. Simulations are carried-out to highlight the proposed PMSG-based MCT performance in both cases.

INTRODUCTION

Because of oceans cover more than 70% of the earth, many new renewable energy sectors are emerging and interesting development are taking place such as thermal energy, wave energy and marine tidal energy [START_REF] Ben Elghali | Marine tidal current electric power generation technology: State of the art and current status[END_REF]. Indeed, the potential of electric power generation from marine tidal currents is very interesting; it has been shown that 48% of the European tidal resource in the UK, 42% in France and 8% in Ireland [START_REF] Ben Elghali | Modeling and control of a marine current turbine driven doubly-fed induction genrator[END_REF] [START_REF] Ben Elghali | High-order sliding mode control of a marine current turbine driven doubly-fed induction generator[END_REF].

Unfortunately, in the literature there is no study about defaults in the hydraulic conversion chain. But, certainly, hydrodynamic energy conversion systems are exposed to environmental and functional constraints. Firstly, environmental constraints are due to the severe weather conditions because of the geographic location (installation under the sea and existence of harmonic current speeds caused by the swell and wave), second, functional constraints are due to the increase in power implies to a higher voltages and / or currents transited. These constraints provide the degradation of performance of the various functional blocks of the system and its accelerated aging process, which leading to many faults essentially related to the blades, to the PMSG (short-circuit between turns, between phases and between phase and neutral) and even to the rectifier (short-circuit faults, open-circuit faults and intermittent gate misfiring faults).

This paper describes the hydrodynamic conversion system by giving the modeling of each component: the MCT, the PMSG and the PWM power rectifier. The control of this chain is achieved by using: MPPT control for extracting of the maximum power and proportional integral correctors to control the speed and the dq-axis currents. Besides, we are focus on the impact of an open-circuit fault in the PWM power rectifier.

This paper is structured as follows; in section II, the hydrodynamic conversion chain structure is given. In section III, the models of the marine current turbine and the PMSG are developed. In section IV, control of the system will be presented. In section V, performance evaluation results are presented. The conclusion is given in section VI.

II. HYDRODYNAMIC CONVERSION CHAIN STRUCTURE

As shown in fig. 1, the conversion chain is composed by marine turbine, a permanent magnet synchronous machine coupled to a DC Bus through a PWM power rectifier. S. Toumi, S.E. Ben Elghali, M. Trabelsi, E. Elbouchikhi, M.E.H. Benbouzid, M.F. Mimouni

III. ENERGY CONVERSION SYSTEM MODELING

A. Marine current turbine modeling

The mechanic power for a marine current turbine has a similar dependence to that of a wind turbine and is given by the following equation [START_REF] Bahaj | Fundamentals applicable to the utilisation of marine current turbines for energy production[END_REF] [5]: For typical MCTs, the optimal C p value for normal operation is estimated to be in the range of 0.35-0.5 [START_REF] Ben Elghali | A simulation model for the evaluation of the electrical power potential harnessed by a marine current turbine[END_REF]. Indeed, basing on the experimental results and for a given turbine, the C p can be approximated as a function of the tip speed ratio λ and the blade pitch angle β [START_REF] Slootweg | General model for representing variable speed wind turbines in power system dynamics simulations[END_REF].

Fig. 3 illustrates the C p curve for simulations.

B. Marine current generator modeling

The generator chosen for the marine current system was the PMSG [START_REF] Conroy | Frequency response capability of full converter wind turbine generators in comparison to conventional generation[END_REF], it has be chosen because of its advantages such as high efficiency, compact structure and the possibility to eliminate the gearbox that reduce the maintenance and this very favorable in terms of underwater application [START_REF] Zhou | Grid-connected marine current generation system power smoothing control using supercapacitors[END_REF]. Dynamic modeling of PMSG can be described in d-q reference system as follows [START_REF] Errami | Modelling and Control Strategy of PMSG Based Variable Speed Wind Energy Conversion System[END_REF]

di R v i p i dt l l di v R p i p i dt l l l dw p p p T T f dt J J J T p i = - + Ω + Φ = - -Ω - Ω + = - - Ω = Φ

IV. ENERGY CONVERSION SYSTEM CONTROL

Fig. 4 shows the control scheme for the energy conversion chain. The d-axis current reference i minimize resistive losses and there current for a given torque, so, we c generator torque directly by the qua component [START_REF] Ansel | Modelling and simulation of an autonomous variable speed micro hydropower station[END_REF]. The q-axis curren calculated by the speed loop controller.

The required d-q components voltage vector are derived from tw controllers. Then, we added the compen improve the dynamic response [START_REF] Liyong | A Novel Control Strategy of Power Converter Used To Direct Driven Permanent Magnet Wind Power Generation System[END_REF] [START_REF] Mansour | Study and Control of a Variable-Speed Wind-Energy System Connected to the Grid[END_REF] Finally, PWM is used to gener signal to implement the vector c generator. PERFORMAN RE

PMSG

A. Simulation paramet

Tidal currents are prov moon. Indeed, the moon gr ocean to bulge outwards on o which causes a rise in the wa aligned with the moon and halfway between those two level is accompanied by a water called the tidal current.

B. Considered faults

Although IGBTs are rugged, they suffer failure due to excess thermal and electrical stress, these failures can be broadly categorized as short-circuit faults, open-circuit faults and intermittent gate misfiring faults. Indeed, IGBT's open-circuit faults are usually linked to the loss of bonding wires of the control signal or to a short-circuit fault causing rupture of the transistor [START_REF] Lu | A literature review of IGBT fault diagnostic and protection methods for power inverters[END_REF] [START_REF] Trabelsi | PWM-Switching pattern-based diagnosis scheme for single and multiple open-switch damages in VSI-fed induction motor drives[END_REF]. Also, this fault can arise when the switches are destructed by an accidental over current or a fuse connected with series for short protection is blown out [START_REF] Bolognani | Experimental fault-tolerant control of a PMSM drive[END_REF]. Because of the opencircuit faults, the converter cannot synthesize desired output voltages and it provides a large torque ripple, an increased harmonic component in current [START_REF] Kim | Fault model and performance evaluation of an inverter-fed permanent magnet synchronous motor under winding shorted turn and inverter switch open[END_REF].

In this section, we will consider an open circuit of the upper switch, the lower switch and the two switches on the same time of the first PWM power rectifier leg and we will consist in studying the influence of this fault on the PMSG phase currents. Fig. 6 shows the three case of an open circuit.

C. Simulation parameters C. Simulation results

In this section, a simulation software has been set up by using the Matlab/Simulink/SimPowerSystem environment.

The parameters used in simulation tests are displayed in Table I. The PMSG power (fig. 7) and the PMSG speed (fig. 8) are deeply correlated with the current marine speed (fig. 5).

The direct current is maintained to zero (fig. 9), the quadratic current (fig. 10) and The PMSG rotor speed (fig. 8) are maintained to theirs references. Thus proves the effectiveness of the established regulators.

Fig. 11 shows that the DC bus voltage is constant.

The PMSG voltages and currents are given by fig. 12 and fig. 13; they are three sinusoidal voltages and currents with constant frequency equal to 50Hz. Under normal operating conditions and for faulty states, the PWM power rectifier has been modeled with a real model of the power switches and the by-pass diodes. A dead time of 4 μs has introduced to prevent short-circuits within the rectifier legs. A fault condition is caused at t = 1.02s. In fig. 14, the fault is applied to the upper switch (T1), it is observed that the phase current i a lost its negative direction and it is flowing only in a positive direction but in fig. 15, where the fault is applied to the lower switch (T4), we show the opposite result because in this case the phase current i a lost its positive direction and conserve only its negative direction. In fig. 16, where we applied a multiple opencircuit fault for the upper switch (T1) and the lower switch (T4), the phase current i a drops to zero.

VI.

CONCLUSION This paper presents the modeling and the control of an energy conversion chain in isolated site based on a permanent magnet synchronous generator. We have used the d-q rotor reference to model overall the system. The main focus has been the study of the performance of the considered system in healthy and faulty conditions such as an open-circuit fault of a single and a multiple switch in the PWM power rectifier and its influence on the three phase currents. The proposed system has been performed by using Matlab/Simulink.

Fig. 2

 2 Fig. 2 presented the PWM power rectifier, this converter uses the three-phase voltage sources provided by the PMSG. It composed by a three legs, each leg features two semiconductor switches (T k , T k+3 k=1, 2, 3) with antiparallel connected freewheeling diodes (D k , D k+3 ). The switches of the same leg are controlled by a PWM bloc in the form of a logic control signals S k (k = 1, 2, 3) also known gate signals, it defined by:

  the percentage of mechanical power that can be extracted from the fluid stream by the turbine.

Fig. 1 .Fig. 3 . 1 )Fig. 2 .

 1312 Fig.1. General scheme for a hydraulic conversion chain in isolated site

Fig. 5 Fig. 4 .

 54 Fig.5shows the sim current speed.

Fig. 6 .Fig. 8 .

 68 Fig. 7. PMSG power

Fig. 10 .Fig. 15 .Fig. 16 .

 101516 Fig. 10. Quadratic current and its reference


L Rizzi

O F Piattella

S L Cacciatori

V Gorini

The step-harmonic potential

We analyze the behavior of a quantum system described by a one-dimensional asymmetric potential consisting of a step plus a harmonic barrier. We solve the eigenvalue equation by the integral representation method, which allows us to classify the independent solutions as equivalence classes of homotopic paths in the complex plane. We then consider the propagation of a wave packet reflected by the harmonic barrier and obtain an expression for the interaction time as a function of the peak energy. For high energies we recover the classical half-period limit.

1

I. INTRODUCTION

The harmonic oscillator plays a central role in physics because it is exactly solvable and provides a simple model for a host of physical phenomena. Besides the simplest case of the one-dimensional free oscillator, modifications of the harmonic oscillator are of interest. A typical variant is to confine the oscillator in a box. This system has been investigated in one dimension 1-5 and for arbitrary dimensions. [START_REF] Montgomery | Degeneracy of confined D-dimensional harmonic oscillator[END_REF] The purpose of these investigations is to calculate the corrections to the energy levels caused by the presence of infinite barriers at a finite distance. This analysis has also been done for spherically symmetric potentials, such as the hydrogen atom. [START_REF] Barton | The influence of distant boundaries on quantum mechanical energy levels[END_REF] In addition, the behavior of a wave packet propagating in a generic power-law one-dimensional potential well has been considered in terms of its "collapse and revival," namely of its scattering over the well and its subsequent reforming. 7,[START_REF] Robinett | Wave packet revivals and quasirevivals in one-dimensional power law potentials[END_REF] The truncated harmonic oscillator has been used to study how the presence of discrete levels in the energy spectrum affects tunneling through such a well. [START_REF] Chalk | Tunneling through a truncated harmonic oscillator potential barrier[END_REF] In the present paper we study the bound states and the propagation of a wave packet in a one-dimensional potential consisting of a half-space harmonic oscillator plus a step. In Sec. II we solve the Hamiltonian eigenvalue equation using the integral representation method, which allows us to classify the independent solutions as equivalence classes of homotopic paths in the complex plane. In Sec. III we calculate the energy of the bound states and compare them with the standard harmonic oscillator and the half-space harmonic oscillator with an infinite barrier. In Sec. IV we study the properties of the propagation of a wave packet which, coming from infinite distance, is reflected by the harmonic barrier. Our analysis is based mainly on the investigation of the delay time in the reflection, which can be interpreted as the duration of the interaction with the harmonic barrier. We express the interaction time as a function of the peak energy and study its asymptotic behavior. We show that in the high energy limit the delay approaches the classical value, namely the half period of the harmonic oscillator. Finally, we comment on how this behavior changes in the presence of a stronger or weaker confinement.

The problem that we address has been studied by Mei and Lee 2 to test the adequacy of a perturbation scheme on an exactly solvable model. Our analysis, which has a different purpose, has the advantage of being based on the integral representation method which can be applied to a wider class of problems. In addition, we do not confine ourselves to the study of the bound states, but also investigate the motion of continuous spectrum wave packets.

II. THE STEP-HARMONIC POTENTIAL

Consider a particle subject to the one-dimensional potential:

U(x) =      U 0 (x ≥ 0) 1 2 κx 2 (x < 0), (1)
where κ and U 0 are real positive constants (see Fig. 1). The proper and improper eigenfunctions of the Hamiltonian operator are ordinary solutions of the eigenvalue equation outside of the discontinuity of the potential. Such solutions must be continuous together with their first derivatives across the singularity. [START_REF] Caldirola | Introduction to Theoretical Physics[END_REF][START_REF] Griffiths | Introduction to Quantum Mechanics[END_REF] We will solve the eigenvalue equation for x > 0 and x < 0. For x > 0 the eigenvalue equation is

- 2 2m d 2 u(x) dx 2 + U 0 u(x) = Eu(x), (2)
where m is the particle mass and E is the energy eigenvalue. The general solution of Eq. ( 2)

has the form:

u(x) =      Ae ikx + Be -ikx (E > U 0 ) A ′ e kx + B ′ e -kx (0 < E < U 0 ) , (3)
where k ≡ 2m|E -U 0 |.

We must choose A ′ = 0. Otherwise, for 0 < E < U 0 , u(x) would diverge exponentially for x → +∞ and therefore it would neither belong to L 2 (R) (i.e. the space of the square summable functions over R), nor satisfy the eigenpacket condition for improper eigenfunctions. [START_REF] Caldirola | Introduction to Theoretical Physics[END_REF][START_REF] Griffiths | Introduction to Quantum Mechanics[END_REF] For x < 0 the eigenvalue equation is

d 2 u(y) dy 2 + (ǫ -y 2 )u(y) = 0, (4)
where

y = αx, α ≡ 4 mκ 2 , ǫ ≡ 2E m κ , ω ≡ κ m . (5)
We set u(y) = F (y) exp (-y2 /2) and obtain from Eq. ( 4) the following equation for F (y)

F ′′ (y) -2yF ′ (y) + (ǫ -1)F (y) = 0, (6)
which is the Hermite equation. [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF][START_REF] Hochstadt | The Functions of Mathematical Physics[END_REF] The solutions of Eq. ( 6) are entire functions and can be found by the method of integration by series. We prefer to employ the integral representation method. [START_REF] Hochstadt | The Functions of Mathematical Physics[END_REF] We start by looking for solutions of Eq. ( 6) that have the form

F (y) = γ dt f (t)e -t 2 +2ty , ( 7
)
where γ is a path in the complex plane C and f is a suitable function which is holomorphic in a region which contains the graph of γ. We substitute Eq. (7) into Eq. ( 6) and obtain

γ dt [4t 2 + (ǫ -1)]f (t)e -t
From Eq. ( 9) it follows that Eq. ( 7) is a solution of Eq. ( 6) if

tf (t)e -t 2 +2ty ∂γ = 0 and f (t) = t -ǫ+1 2 . (10)
Therefore, we can write a solution of Eq. ( 6) in the form

F (γ) (y) = γ dt t -ǫ+1 Γ 1 Γ 2 Γ 3 Re(t) Re(t) Γ 4
FIG. 2: Possible paths for ǫ = 2n + 1 (n = 0, 1, . . . ).

A. The case ǫ = 2n + 1

In this case we can rewrite Eq. (7) as

F (γ) n (y) = γ dt e -t 2 +2ty t n+1 , (12)
where the integrand is holomorphic on C but the origin. Possible choices of γ for which the contour condition in Eq. (10) holds are shown in Fig. 2; Γ 1 and Γ 3 have a real part that goes to infinity, Γ 2 is a closed path circling the origin, and Γ 4 is a closed path that does not contain the origin. By virtue of Cauchy's theorem, F

= 0, and because the paths can be deformed so that Γ 1 + Γ 3 = Γ 2 , the other three solutions satisfy the relation

F (1) n + F (3) n = F (2)
n , where F

n is the solution corresponding to the path Γ j (j = 1, 2, 3, 4). Then we have, as expected, two linearly independent solutions for Eq. (6).

As an exercise, we show that the solution corresponding to Γ 2 , namely

F (2) n (y) = dt e -t 2 +2ty t n+1 , (13)
corresponds to the Hermite polynomial of order n. By completing the square in the integrand of Eq. ( 13), we find

F (2) n (y) = e y 2 dt e -(t-y) 2 t n+1 . ( 14
)
We take advantage of Cauchy's formula and rewrite Eq. ( 14) as

F (2) n (y) = 2πi n! (-1) n e y 2 d n dy n (e -y 2 ) = Im(t) Re(t) Γ 1 Γ 2 Γ 3
FIG. 3: Possible paths for ǫ = 2n + 1. Note that in this case Γ 2 cannot be closed at infinity.

B. The case ǫ = 2n + 1

In the generic case ǫ ∈ R, ǫ = 2n + 1, we rewrite Eq. ( 11) as

F (γ) ǫ (y) = γ dt e -t 2 +2ty t β , (16)
where β ≡ (ǫ+ 1)/2. If β is not a positive integer, t = 0 is a branch point for the multivalued function t β . In this case we must cut the complex plane, for example along the positive real axis. In the latter case, the classes of possible paths are depicted in Fig. 3.

In the following we show that the solutions corresponding to Γ 1 and Γ 3 (F

(1) ǫ and F

ǫ ) diverge as e y 2 for y = ±∞ and therefore the corresponding eigenfunction u(y) cannot be either proper or improper. We are thus left with the solution corresponding to Γ 2 [F (2) ǫ ] which again diverges as e y 2 for y → +∞. It also diverges for y → -∞, but the corresponding u(y) and its derivatives vanish more rapidly than any polynomial thanks to the presence of the exp(-y 2 /2) factor.

According to standard results in the theory of integrals depending on a parameter, it is easy to show that F (γ) ǫ (y) is an entire function. Furthermore, the derivatives of F (γ) ǫ are obtained by differentiating with respect to y under the integral sign of Eq. ( 16). Thus we obtain the relation

d m F (j) ǫ dy m = 2 m F (j) ǫ-2m . (17)
We next address the asymptotic behavior of two independent solutions, for example, F

(1) ǫ and F

(2)

ǫ .

The Γ 1 solution. We rewrite Eq. ( 16) for the path Γ 1 by introducing the variable z = t-y:

F (1) ǫ (y) = e y 2 Γ 1 dz e -z 2 (y + z) β . ( 18
)
The branch point is now z = -y and the cut is shifted as well. We extract |y| from the integral in Eq. ( 18) [START_REF]if we were to extract y β for negative values of[END_REF] :

F (1) ǫ (y) = e y 2 |y β | G 1 (y), (19)
where

G 1 (y) ≡ Γ 1 dz e -z 2 [sgn(y) + z |y| ] β . ( 20
)
An elementary calculation shows that

lim y→±∞ G 1 (y) = lim y→±∞ Γ 1 dz e -z 2 [sgn(y) + z |y| ] β = - √ π sgn(y) β , (21)
where we have taken the limit under the integral sign by virtue of the dominated convergence theorem [START_REF] Cunningham | Taking limits under the integral sign[END_REF] (see Appendix B). The asymptotic behavior of F

(1) ǫ (y) for y → ±∞ is therefore

F (1) ǫ (y) ∼ - √ π e y 2 y β . (22)
The corresponding eigenfunction u (1) (y) = F

(1) ǫ (y) exp (-y 2 /2) cannot be either proper or improper.

The Γ 2 solution. The solution corresponding to Γ 2 [F (2) ǫ (y)] has the form

F (2) ǫ (y) = Γ 2 dt e -t 2 +2ty t β , (23)
where, as shown in Fig. 3, Γ 2 circles around the branch point in an anti-clockwise sense.

This solution has different behavior for y → +∞ and y → -∞.

Asymptotic behavior for y → +∞. By virtue of Cauchy's theorem we can deform Γ 2 to split the integral in Eq. ( 23) into a sum of two integrals over the paths Γ 1 and Γ 3 :

F (2) ǫ (y) = Γ 1 ∪Γ 3 dt e -t 2 +2ty t β . ( 24
)
We again introduce z = t -y and extract y β from the integral. We obtain where

F (2) ǫ (y) = e y 2 y β G 2 (y), (25)
δ 3 δ 1 Re(t)
G 2 (y) ≡ Γ 1 dz e -z 2 (1 + z y ) β + Γ 3 dz e -z 2 (1 + z y ) β . ( 26
)
Thanks to the dominated convergence theorem we find

lim y→+∞ G 2 (y) = -2ie -iπβ √ π sin(πβ). ( 27
)
The asymptotic behavior of F

ǫ (y) for y → +∞ is thus

F (2) ǫ (y) ∼ -2ie -iπβ √ π sin(πβ) e y 2 y β . (28)
Note that, if ǫ = 2n + 1, Eq. ( 28) is incorrect because G 2 (y) → 0 for y → +∞. We already know that, in this case, F [START_REF] Mei | Harmonic oscillator with potential barriers[END_REF] (y) is as a polynomial of degree n.

Asymptotic behavior for y → -∞. By taking advantage of Cauchy's theorem, we deform and split Γ 2 in the 3 sub-paths shown in Fig. 4. We choose for simplicity r = 1; from Eq. ( 23) we obtain

F (2) ǫ (y) = I β (y) -2ie -iπβ sin(πβ) ∞ 1 dt e -t 2 +2ty t β , (29)
where

I β (y) ≡ i 2π 0 dθ e i(1-β)θ e -cos(2θ)-i sin(2θ)+2y cos θ+2iy sin θ . ( 30
)
For y < 0 and t > 0 the following inequality holds

t -β e -t 2 +2ty < t -β e -t 2 , (31)
and, moreover, t -β exp(-t 2 ) is integrable in [1, +∞). Therefore, the integral on the righthand side of Eq. ( 29) vanishes for y → -∞ by virtue of the dominated convergence theorem.

In contrast, for I β (y) we have

|I β (y)| ≤ 2π 0
dθ e i(1-β)θ e -cos(2θ)-i sin(2θ)+2y cos θ+2iy sin θ = 2π 0 dθ e -cos(2θ) e 2y cos θ , (32) so that |I β (y)| ≤ 2πe 2|y|-1 . Therefore, for y → -∞, the absolute value of F

ǫ (y) is dominated by 2πe 2|y|-1 , and u II (y) = F [START_REF] Mei | Harmonic oscillator with potential barriers[END_REF] ǫ (y)e -y 2 /2 (33) is rapidly decreasing. Hence it is square summable on the positive real axis. Thus, it follows that for ǫ = 2n + 1 the full-space harmonic oscillator does not admit proper or improper eigenfunctions. More general theorems [START_REF] Bowen | Methods of establishing the asymptotic behavior of the harmonic oscillator wave functions[END_REF] allow one to obtain our results indirectly, for example, by studying the asymptotic behavior of the power series expansion of the solutions of Eq. ( 4). Here we have adopted a more direct approach.

III. EIGENFUNCTIONS AND ENERGY LEVELS

From the results of Sec. II we can write the energy eigenfunctions as

E < U 0 u(x) =      AF ǫ (αx)e -α 2 x 2 2 (x < 0) Be -kx (x > 0) (34)
E > U 0 u(x) =      CF ǫ (αx)e -α 2 x 2 2 (x < 0) De ikx + Ee -ikx (x > 0), (35)
where we have dropped the superscript from F

ǫ . The integration constants A, . . . , E must be chosen such that u(x) and its first derivative are continuous at x = 0 (the junction conditions).

A. The case E < U 0

If the energy is smaller than the step height U 0 , the junction conditions imply that

B -F ǫ (0)A = 0 (36a) kB + αF ′ ǫ (0)A = 0. ( 36b
)
The condition for the existence of a nontrivial solution is the vanishing of the system determinant. We define J(β) ≡ F ǫ (0) [see Eq. (A1)]. In Appendix A we obtain the following expression for J(β) [see Eq. (A3)]:

J(β) = sin(πβ) ie iπβ Γ 1 -β 2 . ( 37
)
The recurrence relation for the derivatives of F ǫ , Eq. ( 17), can be rewritten in terms of J(β)

as

F ′ ǫ (0) = 2J(β -1). (38)
Equation ( 38) implies that the junction conditions can be rewritten as

-2αJ(β -1) = kJ(β), (39)
or, equivalently, as

Γ 1 - β 2 Γ 1 -β 2 = - β 0 -β 2 , ( 40
)
where β 0 = U 0 /( ω) + 1/2. By using the relation 13

Γ(z)Γ(1 -z) = π sin(πz) , (41)
we can rewrite Eq. (40) as

Γ β + 1 2 Γ β 2 cot π 2 β = - β 0 -β 2 , (42)
For a given value of β 0 Eq. ( 42) is an implicit relation determining the energy levels. The advantage of Eq. ( 42) is that the singular behavior is contained in the cotangent function.

For 0 ≤ U 0 < ω/2 (1/2 ≤ β 0 < 1) the step is too small to allow for the existence of discrete energy levels. The first level (the ground state) appears for β 0 = 1 at the value E 0 = ω/2, which is the ground state of the full-space harmonic oscillator. For 1 ≤ β 0 < 3 there is only one level, whose energy grows with increasing β 0 starting from its minimum value ω/2. When β 0 crosses the value 3 a second level appears at the energy E 1 = 5 ω/2.

By further increasing β 0 (and hence U 0 ) the subsequent levels appear as U 0 crosses the values

E k = ω(2k + 1/2) (k ∈ N)
, corresponding to the (k + 1)th even level of the oscillator. Thus, for a fixed value of β 0 such that 2k +1 < β 0 < 2k +3 there are exactly k +1 bound states with energies E n (n = 0, 1, . . . , k) satisfying the inequalities ω(2n + 1/2) < E n < ω(2n + 3/2).

An example with k = 1 is shown in Fig. 5. Each E n is a monotonically increasing function of U 0 which asymptotically approaches the value ω(2n + 3/2) (the (n + 1)th odd state of the oscillator) as U 0 → ∞. The (unnormalized) eigenfunctions corresponding to the eigenvalue E n are

u n (x) =      F ǫn (αx)e -α 2 x 2 2 (x < 0) J(β n )e -knx (x ≥ 0), (43)
where the β n are the solutions of Eq. (42);

ǫ n = 2β n -1, k n = 2m(U 0 -E n )
, and

E n = ωǫ n /2.

B. The case E > U 0

The junction conditions on the eigenfunctions of Eq. ( 35) are

D + E -CF ǫ (0) = 0 (44a) ik(D -E) -CαF ′ ǫ (0) = 0, ( 44b
)
implying the normalized (with respect to k) improper eigenfunctions are given by

u ǫ (x) = 1 √ 2π      Π(β)F ǫ (αx)e -α 2 x 2 2 (x < 0) e -ikx + ζ(β)e ikx (x ≥ 0), (45)
where, as usual, 2β ≡ ǫ + 1, k ≡ 2m(E -U 0 ), 2E ≡ ωǫ, and

Π(β) ≡ 2 J(β) + i 2 β-β 0 J(β -1) -1 , (46)
ζ(β) ≡ J(β) -i 2 β-β 0 J(β -1) J(β) + i 2 β-β 0 J(β -1) = Γ 1 -β 2 -i 2 β-β 0 Γ 2 -β 2 Γ 1 -β 2 + i 2 β-β 0 Γ 2 -β 2
, ( 47
)
where we have used Eq. (A1). Note that |ζ(β)| = 1. As expected, the continuous part of the spectrum (E > U 0 ) is simple.

IV. REFLECTION AND DELAY

To study reflection phenomenon, we consider the following superposition of continuous states:

ψ(x, t) = ∞ 0 dk c(k)u ǫ(k) (x)e -i E(k)t . (48)
From Eq. ( 45) we have

ψ(x, t) = 1 √ 2π      ∞ 0 dk c(k)Π(β(k))F ǫ (αx)e -α 2 x 2 2 - i E(k)t (x < 0) ∞ 0 dk c(k) ζ(β(k))e ikx + e -ikx e -i E(k) t = ψ ref + ψ in (x > 0) . ( 49
)
We write ψ in and ψ ref in the form:

ψ in (x, t) = 1 √ 2π +∞ 0 dk |c(k)|e -i[kx+Ω(k)t-γ(k)] , (50)
ψ ref (x, t) = 1 √ 2π +∞ 0 dk |c(k)|e i[kx-Ω(k)t+δ(k)+γ(k)] , (51)
where we have defined

e iδ(k) ≡ ζ(β(k)) and Ω(k) ≡ E(k) = U 0 + k 2 2m . ( 52
)
If c(k) is sufficiently regular and non-vanishing only in a small neighborhood of k, then ψ in and ψ ref represent wave packets that move according to the equations of motion [START_REF] Caldirola | Introduction to Theoretical Physics[END_REF][START_REF] Griffiths | Introduction to Quantum Mechanics[END_REF] x in = -

dΩ dk k= k t + dγ dk k= k = - k m (t -t 0 ) = - p m (t -t 0 ), ( 53
)
for the "incoming" wave packet, and for the reflected "outgoing" one. The solution represents a particle of well defined momentum p = k which approaches the origin from the right, interacts with the harmonic potential (at t = t 0 ), and is totally reflected. The phase shift results in a delay in the time the wave packet bounces back, which is caused by the interaction with the confining harmonic barrier. Because the phase shift δ depends only on k through β, we can write the delay as

x ref = dΩ dk k= k t - dγ dk k= k - dδ dk k= k = p m (t -t 0 ) - m p dδ dk k= k , (54)
τ ( β) = 1 ω dδ dβ β= β , (55)
where β = β( k).

We prove that

lim β→∞ δ ′ (β) = π, (56)
lim

β→β 0 δ ′ (β) =                +∞ β 0 ∈ k∈N (2k, 2k + 1) -∞ β 0 ∈ k∈N (2k + 1, 2k + 2) 0 β 0 ∈ N = 1, 2, . . . , (57)
where the prime denotes the derivative with respect to β. To this purpose, note that by employing Eq. (41) δ ′ (β) can be cast in the form:

δ ′ (β) = 1 2 √ β -β 0 sin (βπ) 1 β -β 0 + Ψ β 2 -Ψ β + 1 2 + 2π sin (βπ) (β -β 0 ) Γ (β/2) Γ (β/2 + 1/2) √ 2 sin 2 βπ 2 + Γ (β/2 + 1/2) √ 2 Γ (β/2) cos 2 βπ 2 , ( 58
)
where Ψ is the Digamma function (that is, the logarithmic derivative of the Gamma function). [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] In Fig. 6 we plot τ versus β for different values of β 0 . Note the resonances located at β ≃ 3, 5, 7, 9, . . . , corresponding to the formation of metastable states at the respective energies E ≃ 5 ω/2, 9 ω/2, 13 ω/2, 17 ω/2, . . . . These states have lifetimes which decrease as the corresponding energies increase and move farther away from the threshold energy U 0 . Conversely, as U 0 increases, the lifetime of the resonance closest to the height of the step becomes progressively longer and then infinite when the resonance turns into the next bound state. This behavior is evident in Fig. 6, in which the first three plots correspond to values of β 0 for which there is only one bound state. In the successive three plots the resonance at β = 3 has disappeared, having turned into the second bound state.

It is simple (using steepest descent or Stirling's formula, for example) to show that

Γ(z + 1/2) Γ(z) = √ z 1 + O 1 z , (59)
for z ≫ 1. By using one of the integral formulas for the Digamma function, [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] we can also show that

lim z→∞ Ψ (z) -Ψ z + 1 2 = 0. (60)
Thanks to Eqs. ( 59) and (60) it is straightforward to derive Eqs. ( 56) and (57). In particular Eq. ( 56) implies that

lim β→∞ τ (β) = π ω = T 2 . ( 61
)
The wave packet undergoes half an oscillation during the interaction with the harmonic potential before being reflected, which results in a delay of half a period compared with the reflection from a perfect mirror (that is, when the confining barrier is an infinite wall).

Thus, as expected, the high energy limit reproduces the classical behavior.

V. CONCLUSIONS

The main features of the discrete part of the spectrum can be summarized as follows.

For sufficiently small U 0 (the height of the step) there is no discrete spectrum. When U 0 increases and approaches the value ω/2 from below, there appears a resonance at energy E 0 ≃ ω/2. This resonance converts into a bound state when U 0 reaches the value ω/2

and the corresponding eigenfunction is proportional, at x < 0, to the eigenfunction of the ground state of the free harmonic oscillator and is flat otherwise. By further increasing the height of the step the ground state energy increases monotonically with U 0 and, as U 0 → ∞, approaches asymptotically from below the first odd level 3 ω/2 of the full-space harmonic oscillator. A new discrete energy level appears at each energy E k = ω(2k + 1/2) (k ∈ N) whenever U 0 crosses the value E k . In the limit of infinite U 0 (leading to the halfspace oscillator), the energy levels become the odd levels of the oscillator itself, as expected from the symmetry of the problem. Loosely speaking, the levels are born as "even" and, upon increasing the height of the step, end up as "odd" (see Fig. 5). This behavior is not peculiar to the step problem associated with the harmonic oscillator, but is typical of the corresponding step variant of every symmetric confining potential.

The continuous spectrum is simple and extends from U 0 to ∞. A wave packet coming from infinity collides with the confining harmonic branch and is thereby entirely reflected.

The interaction with the potential results in a delay of the reflected packet which, as is well known for problems of this kind, is proportional to the derivative of the phase shift of the plane wave component evaluated at β = β( k). This delay can be interpreted as the interaction time with the harmonic barrier. When the confining part of the potential is infinite (U(x) = +∞ at x < 0) the delay vanishes, and the reflection on a perfect mirror is instantaneous. The desirable feature of our example is that we can derive an exact analytic expression for the delay [Eq. ( 58)] as a function of the step height and of the peak energy of the incoming packet.

Although these characteristics are typical of the step variants of all symmetric confining potentials, the harmonic oscillator potential is "more equal" than the others. It is the only analytic (except possibly at x = 0), convex or concave locally bounded symmetric and confining potential that gives rise to classical isochronous oscillations and thus to evenly spaced energy levels. [START_REF] Cariñena | Isochronous classical systems and quantum systems with equally spaced spectra[END_REF][START_REF] Asorey | Isoperiodic classical systems and their quantum counterparts[END_REF] In our step variant of the problem we recover both these features in the limit U 0 → ∞, when the potential reduces to the half-space harmonic oscillator. Thus, it is possible that the harmonic potential is the only confining barrier that displays a constant nonvanishing interaction time in the limit of high energies. For steeper barriers we expect the interaction time τ to vanish at high energies, while for milder potentials we expect the delay to become infinite in this limit, in accordance with the corresponding classical behavior.

Similarly, we expect that, as U 0 → ∞, the spacing between two neighboring discrete levels approaches infinity in the former case and zero in the latter.

In a forthcoming paper we corroborate this conjecture by analyzing two examples using the integral representation method. This method can be employed to analyze a wide class of "step-something" potentials in which the harmonic part is replaced by another type of barrier. We encourage readers to investigate, for example, the step-linear (sl) and the stepexponential (se) potentials

U sl (x) =      U 0 (x ≥ 0) -Mx (x < 0) (62)
U se (x) =      U 0 (x ≥ 0) Me -x/σ (x < 0) , (63)
where M, σ, and U 0 are real positive constants.

to a function f ∞ : R → R. Suppose that there exists a positive I-summable function g : R → R + that dominates every f k (that is, |f k (x)| ≤ g(x) ∀k ∈ N). It follows that f ∞ is I-summable and that we can take the limit under the integral, that is,

lim k→∞ I dx f k (x) = I dx lim k→∞ f k (x) = I dx f ∞ (x). (B1)
It is noteworthy that, in our case, we do not need to invoke Lebesgue integration and the dominated convergence theorem, and our calculations are based on Riemannian integration.

Even though Riemann integration theory lacks theorems regulating the interchange between limit and integration operations, the following theorem is sufficient for our purposes: 15

FIG. 1 :

 1 FIG.1: The step-harmonic potential.

FIG. 4 :

 4 FIG.4: A standard trick in contour integration.

FIG. 5 :

 5 FIG.5:The solid and the dashed lines represent respectively the left-and the right-hand side of Eq. (42). The intersections determine the energy levels. Here β 0 = 4.5.

FIG. 6 :

 6 FIG. 6: Plots of the delay time τ (in units of T /2, where T = κ/m is the period associated to the harmonic oscillator) versus the "energy" β of the incoming wave packet for six values of the step height β 0 . (a) β 0 = 1.5, (b) β 0 = 2, (c) β 0 = 2.5, (d) β 0 = 3.5, (e) β 0 = 4, and (f) β 0 = 4.5.

Theorem 2 .

 2 Let f n be a sequence of functions defined on [a, ∞) and Riemann integrable on [a, b] for all b > a. Assume that (i)f n (x) → f ∞ (x) almost everywhere in [a, ∞), f ∞ beingRiemann integrable on every finite interval. (ii) There exists a positive function g definedon [a, ∞) such that ∞ a g is convergent and |f n (x)| ≤ g(x) for all n. Then ∞ a f n → ∞ a f ∞ .Note that, in this theorem, the integrability of the limiting function is part of the hypothesis, whereas in the dominated convergence theorem it is a consequence of the theorem itself.

e -t 2 +2ty ,[START_REF] Griffiths | Introduction to Quantum Mechanics[END_REF] where γ must be chosen according to the first condition in Eq.[START_REF] Caldirola | Introduction to Theoretical Physics[END_REF] and such that the integral in Eq. (11) is well defined. The classification of the appropriate γ's allows us to classify all solutions of Eq. (6).The integrand in Eq. (11) is singular at t = 0. For ǫ = 2n + 1 (n = 0, 1, . . . ), the point t = 0 is a pole of order n + 1, otherwise it is a branch point.In the following we distinguish two classes of paths, which correspond to two linearly independent solutions of Eq. (6).

2πi n! H n (y),(15)where H n (y) is the Hermite polynomial of order n.[START_REF] Hochstadt | The Functions of Mathematical Physics[END_REF]

A. Consortini and B. R. Frieden, "Quantum-mechanical solutions for the simple harmonic os-

R. W. Robinett, "Visualizing the collapse and revival of wave packets in the infinite square well using expectation values," Am. J. Phys. 68 (5),410-420 (2000).

Acknowledgments

We are grateful to Carlo Garoni for having inspired the topic that we have analyzed in this paper.

Appendix A: Calculation of J(β)

In this appendix we prove Eq. (37). According to the definition given in Sec. III we have J(β) ≡ F [START_REF] Mei | Harmonic oscillator with potential barriers[END_REF] ǫ (0) =

where 2β ≡ ǫ + 1. Assume β ∈ C. For |β| ≤ R, the integrand function in Eq. (A1) is bi-continuous and holomorphic with respect to β in any compact disc. Furthermore, its absolute value is bounded by a summable positive function: |e -t 2 t -β | ≤ e -2ℜ(t 2 ) |t| 2R . These properties imply that the integral in Eq. ( A1) is uniformly convergent. Therefore, J(β) is an entire function. If we change the variable in Eq. (A1) to u = t 2 , we must cut the complex plane and define t = √ u on its complete Riemann surface, which is composed of two sheets.

The new path Γ ′ 2 is shown in Fig. 7. On both sheets of the u-plane, there is an integral along a straight line and an integral on a semi-loop. It is easy to show that the integral on the semi-loop vanishes when shrunk to a point, provided that β ∈ (-∞, 1). The integrals along the straight lines can be deformed, using Cauchy's theorem, to become integrals along the positive real axis. We thus have

which we can write as

Because the poles of the Gamma function in Eq. (A3) are cancelled by the zeroes of the sine, the right-hand side is an entire function so that Eq. (A3) holds on the entire complex plane by analytic continuation.

Appendix B: Taking limits under the integral sign

For convenience we give here a very useful elementary theorem which we have used in the paper:

Theorem 1: The dominated convergence theorem. Let f k : R → R be summable on an interval I, that is, I f k < ∞, ∀k ∈ N. Moreover, let f k converge almost everywhere