Impressions of Impression Management 2

Keywords: Social judgment, self-presentation, trait inference, spontaneous trait inferences, ulterior motivation, relearning paradigm

Many forms of self-presentational behavior are very common, so social perceivers are experienced at observing them. In contrast with existing views, we argue that inferences about ulterior, self-presentational motives may be formed as spontaneously as other trait inferences. Applying a relearning paradigm, we assessed implicit, spontaneous inferences about ulterior motives. Participants read behavior descriptions, some of which could imply ulterior motivation (e.g., "John volunteered to help paint his boss' house", which can imply "ingratiating", or the correspondent trait "helpful") and descriptions that could not ("John volunteered to help paint his friend's house"). We assessed spontaneous inferences about ulterior motives (e.g., ingratiating) and about traits that directly corresponded with the behavior (e.g., helpful). Results showed that participants spontaneously activated the ulterior motive just as much as the correspondent inference. This indicates co-occurring spontaneous inferences of ulterior motives as well as correspondent traits.

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Impressions of Impression Management 3

Impressions of Impression Management:

Evidence of Spontaneous Suspicion of Ulterior Motivation Self-presentational behavior occurs every day and everywhere (e.g., [START_REF] Leary | Self-presentation, Impression management and interpersonal behavior[END_REF], and ingratiation and self-promotion are the most common varieties of it [START_REF] A C C E P T E D M A N U S C R I P T Accepted Manuscript | Toward a general theory of strategic selfpresentation[END_REF]Vonk, 2001). Therefore, social perceivers may be proficient at detecting ingratiation and self-promotion, especially prototypical forms such as ingratiation towards the boss or a beautiful woman (Vonk, 1999a(Vonk, , 1999b)). As a result of everyday practice, perceivers may even recognize this behavior and its motives spontaneously, without much cognitive effort.

The general view in social cognition, however, is that without any effortful thought, behavior is typically taken at face value due to the correspondence bias (see, [START_REF] Gilbert | The correspondence bias[END_REF]. Thus, when a subordinate compliments his boss, our first, spontaneous inference should be that he expresses genuine admiration. Whenever such self-presentational behaviors are observed, theoretically there are three possibilities:

(1) The self-presentational motives go unnoticed and the behavior is taken at face value; this follows from the correspondence bias;

(2) The behavior arouses suspicion of ulterior motivation and is carefully scrutinized. According to [START_REF] Fein | Effects of suspicion on attributional thinking and the correspondence bias[END_REF], suspicion of ulterior motivation evokes sophisticated attributional analysis, that is, conscious and deliberative thought. Thus, self-presentational motives can be detected but this requires cognitive elaboration;

(3) The behavior is spontaneously, without much conscious effort attributed to self-presentational motives. Here, we argue that this occurs more than is predicted by current theories on correspondence bias and suspicion of ulterior motivation.

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Social-cognitive literature indicates that even complex higher mental processes become automatized when frequently exercised [START_REF] Smith | Procedural knowledge and processing strategies in social cognition[END_REF]; see also [START_REF] A C C E P T E D M A N U S C R I P T Accepted Manuscript | Development of automatism of social judgments[END_REF]. Examples are spontaneous trait inferences (STI's; [START_REF] Uleman | People as flexible interpreters: Evidence and issues from spontaneous trait inference[END_REF]; see also, [START_REF] Uleman | Spontaneous inferences, implicit impressions and implicit theories[END_REF], inferences about goals of actors [START_REF] Hassin | Automatic goal inferences[END_REF], about properties of an actor's situation (spontaneous situation inferences, SSI's; [START_REF] Ham | Smart and easy: Co-occurring activation of spontaneous trait inferences and spontaneous situational inferences[END_REF][START_REF] Lupfer | Impact of context on spontaneous trait and situational attributions[END_REF], and about goal-directed behavior [START_REF] Aarts | Habits as knowledge structures: Automaticity in goal-directed behavior[END_REF]. We propose that, even though inferences about ulterior motives interfere with the human tendency toward inferring correspondent traits (e.g., friendly

behavior is guided by a friendly disposition), the process of detecting selfpresentational motives shares important similarities with other frequently exercised higher mental processes. If perceivers regularly observe particular styles of selfpresentation (e.g., flattery), and if they engage in systematic corrective processes each time they do, these corrections may become proceduralized [START_REF] Bassili | Procedural efficiency and the spontaneity of trait inference[END_REF]Smith, 1989) and occur spontaneously (Vonk, 1998, Exp. 5), just as other well-practiced cognitive activities.

Corroborating this assumption, previous studies [START_REF] Vonk | The slime effect: Suspicion and dislike of likeable behavior towards superiors[END_REF](Vonk, , 1999a) ) suggest that some forms of self-presentation are identified with little cognitive effort.

However, in these previous studies, explicit measures were used (see also [START_REF] Fein | Effects of suspicion on attributional thinking and the correspondence bias[END_REF][START_REF] Fein | Suspicion of ulterior motivation and the correspondence bias[END_REF]Vonk, 1999a): Participants judged an actor on explicit (e.g., Likert-type) rating scales. And because explicit questions induce thoughtful, intentional responses [START_REF] Uleman | Spontaneous versus intentional inferences in impression formation[END_REF], these previous studies do not demonstrate that such inferences are made spontaneously.

Spontaneous non-correspondent inferences

The first purpose of the present study is to provide empirical evidence of the

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Impressions of Impression Management 5 spontaneity of inferences about ulterior motivation. Social-cognitive research studied spontaneous inferences in great detail, but methods and findings have never been applied to inferences of ulterior motivation and self-presentational behavior. Various methods have been developed to measure STI's and to guarantee that dependent measures reflect spontaneous inferences (for an overview, see, [START_REF] Uleman | People as flexible interpreters: Evidence and issues from spontaneous trait inference[END_REF]. In the current study, we will adapt such a research paradigm to investigate spontaneous inferences related to suspicion of self-presentational motives.

In the literature on person perception, a fundamental difference is drawn between correspondent and non-correspondent inferences [START_REF] Jones | From acts to dispositions: The attribution process in person perception[END_REF][START_REF] Jones | Correspondent inferences and the attribution cube: A comparative reappraisal[END_REF]. Although both types of inference reflect internal causes of behavior, the term correspondence refers to the extent to which the behavior and the underlying disposition are "similarly described by the inference" (Jones & Davis, 1965, p. 223). So, a correspondent trait inference takes the behavior at face value, whereas a non-correspondent trait inference refers to potential motives other than conveying a true reflection of the self, that is, self-presentational motives in many cases. For instance, in case of helpful behavior, the inference of the motive to help would reflect correspondence; the inference of the motive to ingratiate would reflect non-correspondent inferences (still informative about the target). When forming an impression of others, people can face an attributional dilemma (see [START_REF] Fein | Suspicion of ulterior motivation and the correspondence bias[END_REF]): An actor's behavior corresponds with an internal correspondent trait (e.g., helpful), or the actor aims at gaining some desired end state (e.g., trying to ingratiate).

Multiple spontaneous inferences

If perceivers indeed make spontaneous inferences about ulterior motives, the question arises how these relate to correspondent inferences which, as we already

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Impressions of Impression Management 6 know, are also made spontaneously. Previous research indicates that multiple, sometimes even competing inferences are drawn initially in the impression formation process [START_REF] Reeder | Dispositional Attribution: Multiple Inferences About Motive-Related Traits[END_REF]) and that spontaneous inferences can be activated jointly [START_REF] Ham | Smart and easy: Co-occurring activation of spontaneous trait inferences and spontaneous situational inferences[END_REF]Todd, Molden, Ham, & Vonk, in press). For instance, the behavior "John lifts the stone" can lead to co-occurring activation of the inferences 'strong' (referring to John) and 'light' (referring to the stone), even though they designate internal versus external causes of the behavior [START_REF] Ham | Smart and easy: Co-occurring activation of spontaneous trait inferences and spontaneous situational inferences[END_REF].

Assuming that multiple inferences are drawn in case of self-presentational behavior, this would imply even more inconsistency among the inferences than in the previous studies. For example, when participants read that "John volunteered to help paint his boss' house," they might instantly think of "helpful" and "ingratiating" at the same time. These inferences are evaluatively inconsistent (see also Footnote 1) and they exclude each other more or less as possible causes of the behavior: Unlike inferences about internal and external causes, they do not work in an additive way.

Yet we do assume that both will be spontaneously activated. Investigating this possibility constitutes the second aim of our study.

We presented participants with descriptions that imply either only a correspondent trait (CT) or can evoke suspicion because they can imply either ulterior motivation (UM) or a correspondent trait (CT). To this end, we manipulated actortarget dependence in descriptions. Dependence is a powerful cue in detecting ulterior motivation [START_REF] Vonk | The slime effect: Suspicion and dislike of likeable behavior towards superiors[END_REF](Vonk, , 1999a)). In our stimulus materials, the actor is either dependent on the target (e.g., "Jake tells the customer that the coat suits him well", where Jake can be considered dependent upon the customer) or not dependent (e.g., "Jake tells his friend that the coat suits him well"). When the actor is dependent on the target, the description can imply either an ulterior motivation (e.g., "sales talk"), or a

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Impressions of Impression Management 7 correspondent trait (e.g., "complimenting"). Without dependence, the ulterior motive is less likely and the description predominantly implies a correspondent trait ("complimenting").

Note that slight variations in the context and target of the behavior allow us to create short sentences, as required to demonstrate spontaneous inferences (e.g., [START_REF] Fazio | Implicit measures in social cognition research: Their meaning and uses[END_REF][START_REF] Uleman | On-line evidence for spontaneous trait inferences at encoding[END_REF], while also including cues pointing to ulterior motivation. As in other STI research, participants were presented with multiple descriptions. To avoid a description activating inferences easily applicable to subsequent descriptions, we selected a variety of selfpresentational behaviors and settings within the ingratiation and self-promotion domains (see Appendix 1).

For descriptions implying both an UM and a CT, we expect to find evidence for both types of spontaneous inferences. For descriptions implying only a CT, we expect to find evidence for activation of a CT only. Also, these descriptions allow us to examine if the strength of a CT is reduced in case of a CT+UM inference.

The generalized relearning paradigm

We measured spontaneous inferences using an implicit measurement paradigm, the "grid relearning paradigm" [START_REF] Ham | Smart and easy: Co-occurring activation of spontaneous trait inferences and spontaneous situational inferences[END_REF]--an adaptation of [START_REF] Carlston | Savings in the relearning of trait information as evidence for spontaneous inference generation[END_REF] relearning paradigm with a broader application scope. In the three tasks of this paradigm, participants are presented with a 4x4 information grid. In the first task, in each cell of the grid, behavior descriptions are presented in the cells, for example, "Bart offered the attractive woman a ride home."

Participants are instructed merely to read the descriptions. In the second task, cue words are presented in each cell and participants are asked to memorize which word is presented in which cell. Finally, in the third task, recall for the words from the second

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Impressions of Impression Management 8 task is tested. In some cases (labeled a relearning trial), the cue word presented in the second task represents an inference (in this experiment, an ulterior motive or a corresponding trait) implied by the description presented in that same cell in the first task. For example, "womanizer" is presented in the same cell where the description "Bart offered the attractive woman a ride home" has been presented. In such cases, assuming that an inference of ulterior motive has already been activated spontaneously during the first task, this implies that participants are now observing a combination they already saw before. In effect, then, they are relearning the combination. In other cases (labeled learning trials), the cue word presented in the second task is not an implication of the description presented in that same cell in the first task. For example, "womanizer" is presented in the same cell where the description "Ben jumped over the fence" has been presented in the first task. So, in the learning trials, relearning does not occur.

In general, the paradigm is based on the idea that relearning is more effective than learning. If the inference represented by the cue word (either an UM or a CT) has been activated spontaneously while reading the description in the first task, recall should be better in relearning trials than in learning trials, because the exposure to the behavior has produced a spontaneous inference with residual effects that facilitate learning in the second task. These facilitation effects (indicated by lower error rates)

were found in several studies on spontaneous social inferences [START_REF] Carlston | Savings in the relearning of trait information as evidence for spontaneous inference generation[END_REF]Carlston et al., 1995;[START_REF] Ham | Smart and easy: Co-occurring activation of spontaneous trait inferences and spontaneous situational inferences[END_REF], and the present study will use them to examine spontaneous activation of ulterior motives along with correspondent traits. We expect to find facilitation effects for both UM and CT cue words when the corresponding description implies both (when the actor is dependent on the target), indicating spontaneous activation of inferences about ulterior motives

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as well as about corresponding traits. When the corresponding description only implies a CT (when the actor is not dependent on the target), we expect to only find facilitation effects for CT cue words, indicating only spontaneous inferences as in previous STI studies.

Method

Participants. One-hundred-and-twenty-four (87 female, 37 male) students at Radboud University Nijmegen (all native Dutch speakers) participated and received 2

Euros for 25 minutes of participation.

Stimulus materials. Sixteen sets of a behavior description with a concurrent cue word (ulterior motive or corresponding trait) were used. Eight of these were experimental material, the other eight were fillers. Each of the eight experimental descriptions was designed in two versions: one implying both an ulterior motive and a corresponding trait (UM+CT) and one implying only a correspondent trait (CT). Each experimental set contained a cue word representing an ulterior motive (UM cue word) and a cue word representing a corresponding trait (CT cue word). 1 UM cue words refer to an inference reflecting a self-presentational motive of the actor (e.g., "bragging" and "sales talk"), indicating that the behavior should not be taken at face value.

Appendix 1 lists all experimental stimulus materials.

The other eight sets of behavior description and concurrent cue words were not related to self-presentation (e.g., "John lifted the stone") and were solely used as fillers. They were selected from earlier research using this paradigm (see Ham & Vonk, 2003, Appendix, Descriptions 1-8).

Overview of the grid relearning paradigm. The grid relearning paradigm consisted of 3 main tasks: an exposure task, a relearning task, and a cued-recall task.

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In each task, a 4x4 grid was displayed on the computer screen. In the exposure task, 16 descriptions were presented: One by one, in random order and randomly distributed across the cells of the grid, each description was displayed for 6 seconds.

Participants were instructed merely to read the descriptions. In the relearning task, a cue word was displayed for 4 seconds in each of the cells. The cues were presented in random order. Participants were instructed to memorize which word was displayed in which cell. In half of the trials--the relearning trials--the cue word fitted an implication of the description that had been presented in the same cell during the exposure task. In the other half of the trials--the learning trials--the displayed cue word was unrelated to the description that had been presented in the same cell. Thus, these learning trials did not allow relearning. Note that relearning trials and learning trials used the same cue words, but in the learning trials they were presented in different cells. Finally, during the cued-recall task, participants were asked to recall which cue word had been shown in which cell. This was done by presenting one cue word at a time at the bottom of the screen below the grid and asking participants to click on the cell of the grid in which this cue word had been presented. Participants

were not given feedback about whether their response was correct or incorrect.

Of all 16 trials, 8 were experimental trials (implying either UM and CT, or only CT inferences), while the other 8 were fillers. Within the 8 experimental trials, 4 trials were relearning trials and 4 trials were learning trials. Facilitation effects (indicating activation of spontaneous inferences) can be observed by within-subjects comparison of recall in the 4 relearning trials to recall in the 4 learning trials.

Design. The design was a 2 (description: UM+CT-implying vs. CT-implying)

x 2 (cue: ulterior motive vs. corresponding trait) x 2 (trial type: relearning vs. learning trial), all manipulated within participants. We used an eight cell within-subjects

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Impressions of Impression Management 11 design.

This design only allowed for one trial per cell. However, because of counterbalancing all stimulus materials (behavior descriptions and cue words), effects found cannot be due to particular stimuli being associated with particular cells of the design (see next paragraph, and also Footnote 3). Furthermore, the descriptions we used described various types of self-presentational behavior in different contexts, indicating the generalizability of the current findings.

All materials were completely counterbalanced between participants. That is, for each participant 4 sets of the 8 sets of a behavior description with a concurrent cue word (ulterior motive or corresponding trait) were randomly selected for the 4 relearning trials, and 4 sets were chosen for the 4 learning trials. Within both the 4 sets serving on relearning trials and the 4 sets serving on learning trials, 2 sets were randomly selected in which the UM+CT-implying description version was used, and 2 sets were chosen in which the CT-implying description version was used. Likewise, within each of these sets of 2, 1 set was randomly selected for which the UM cue word was used whereas for the other set the CT cue word was used. So, across all 8 experimental trials (the 4 relearning trials and the 4 learning trials), each participant saw 4 UM+CT descriptions, 4 CT descriptions, 4 UM cue words, and 4 CT cue words, while no two trials used the same materials.

Procedure. All participants were individually seated behind a computer. After general introductions, they completed a practice task that consisted of an exposure task and a cued-recall task. In the exposure task, 16 famous pop song titles appeared in different cells of the grid and participants were instructed to read the contents of each cell. In the cued-recall practice task, all 16 song titles were presented one by one at the bottom of the screen and participants were asked in which cell each title had

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Impressions of Impression Management 12 been shown during the exposure task.

After the practice task, the actual experiment began, starting with the exposure task. Participants were asked to read the descriptions that appeared successively in the cells. In each cell of the grid, in random order, a description was presented for 6 seconds. After a description had been presented, the screen turned blank for 2 seconds and the next description was displayed in another cell.

Between the exposure and the relearning task, a filler task was inserted to interfere with participants' recall of specific information presented (cf. [START_REF] Carlston | Savings in the relearning of trait information as evidence for spontaneous inference generation[END_REF]. Participants completed five scrambled-word puzzles. Participants were asked to type in the word these letters formed within 60 seconds.

After the filler task, participants completed the relearning task. As in the practice task, participants were instructed to memorize what cue word was presented in which cell. Each cue word was presented for 4 seconds in a cell of the grid; then the screen turned blank for 2 seconds and the next word was presented in another cell.

Between the relearning and the cued-recall task, a second word puzzle filler task was inserted with other words.

The last task was the cued-recall assessment. Participants were presented with the words from the relearning task and asked in which cell each cue word had been shown during the previous task. This question was asked about all 16 words, in random order. The question was presented underneath the 4x4 grid, and participants answered by clicking a cell with the computer mouse. For all 16 times this question was asked, participants could click one of all cells, and cells showed no indication of having been clicked previously. Thereby, the dependent variable indicates whether the answer was correct or false. Finally, participants were paid and debriefed.

Results

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Error rates were submitted to a 2 (description: UM+CT-implying vs. CTimplying) x 2 (cue: UM vs. CT) x 2 (trial type: relearning vs. learning trial)

MANOVA with all factors varied within-subjects. 2 Overall, spontaneous inferences would be indicated by facilitation effects; more accurate recall on relearning trials than on learning trials, which would produce a significant effect of trial type. This effect was significant, F(1, 123) = 68.43, p < .001. On relearning trials, 38.7% of participants gave the correct answer (SD = 47.2) as compared to 17.7% on learning trials (SD = 38.1).

Importantly, this overall facilitation effect was qualified by a significant threeway interaction of Description x Cue x Trial Type, F(1, 123) = 18.32, p < .001. We analyzed the four separate simple effects of trial type (within the two levels of description type and the two levels of cue type) to examine the interaction (see Figure 1 for an overview). First, confirming our main hypothesis, on the two trials for which the description had been UM+CT-implying, the position of a UM cue word was remembered better on the relearning trial (M = 40.3% correct, SD = 49.3) than on the learning trial (M = 12.9% correct, SD = 33. replicates earlier results using this research paradigm [START_REF] Ham | Smart and easy: Co-occurring activation of spontaneous trait inferences and spontaneous situational inferences[END_REF] and other versions of it (e.g., [START_REF] Carlston | Savings in the relearning of trait information as evidence for spontaneous inference generation[END_REF].

Fourth, we also found evidence of spontaneous CT inferences on the trials on which the cue word was a CT and the description was UM+CT-implying. As expected, on the two trials for which the description had been UM+CT implying, a CT cue word was remembered better on the relearning trial than on the learning trial The current findings are the first to directly demonstrate that inferences about ulterior, self-presentational motives can be activated spontaneously. When participants read short descriptions in which the actor could have a motive to impress or please the target, they showed memory facilitation effects for trait cues referring to the implied self-presentational motive. In contrast, when the same behavior was enacted toward a neutral target, so the possibility of an ulterior motive was less salient, we obtained only evidence for spontaneous inference of the correspondent trait, and not the ulterior motive. These correspondent traits were also activated when the behavior could imply an ulterior motive, but less strongly so.

These findings provide a fundamental starting point for understanding the spontaneous component of cognitive responses when observing behavior possibly driven by ulterior motives. They suggest that inferences about ulterior motives are at least to some extent comparable to other forms of automatic higher mental processes (e.g., [START_REF] Uleman | People as flexible interpreters: Evidence and issues from spontaneous trait inference[END_REF] and can be assessed with research paradigms developed to tap these processes. Just as spontaneous correspondent trait inferences fit [START_REF] Bargh | The four horsemen of automaticity: Awareness, efficiency, intention, and control in social cognition[END_REF] four criteria of automaticity, current results suggest inferences about ulterior motives can be made without the explicit intention to do so, without much time for thought, without any control over the process, and without being aware of it.

Thereby, our study revealed a fundamental finding about trait inference, namely, that spontaneous inferences do not necessarily imply that the behavior is taken at face value, producing correspondence bias: Perceivers can instantly go beyond the information given by identifying hidden motives of the actor. Importantly, they do so when contextual cues (the target towards whom the behavior is enacted) point to such hidden motives, that is, by considering the entire behavioral field, rather than being engulfed only by the behavior (cf. [START_REF] Vonk | The slime effect: Suspicion and dislike of likeable behavior towards superiors[END_REF]Vonk, , 1999a)).

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The second goal of our study was to establish co-occurring inferences about ulterior motives and correspondent traits. Results showed that when a description could imply both, both types of spontaneous inferences were activated. This extends earlier findings [START_REF] Ham | Smart and easy: Co-occurring activation of spontaneous trait inferences and spontaneous situational inferences[END_REF]; see also, Todd et al., in press) of co-occurring activation of inferences (STI's and SSI's), and is consistent with [START_REF] Reeder | Dispositional Attribution: Multiple Inferences About Motive-Related Traits[END_REF] notion of multiple inferences. The present results provide a better understanding of the inference process in case of ulterior motivation. We now know that at an automatic, early stage in the attribution process, spontaneous inferences about both correspondent traits and ulterior motives are activated. This does not necessarily violate [START_REF] Fein | Effects of suspicion on attributional thinking and the correspondence bias[END_REF] view: Because of the evaluative inconsistency of correspondent and ulterior motives, it seems reasonable to assume that their co-occurring activation instigates thoughtful and elaborate attributional analysis. In contrast with earlier theorizing (e.g., [START_REF] Berscheid | Outcome dependency: Attention, attribution and attraction[END_REF][START_REF] Erber | Outcome dependency and attention to inconsistent A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT information about others[END_REF][START_REF] Fein | Effects of suspicion on attributional thinking and the correspondence bias[END_REF][START_REF] Gilbert | On cognitive busyness: When person perceivers meet persons perceived[END_REF], current results indicate that detection of ulterior motivation and consideration of the entire behavioral field does not solely rest on thoughtful attributional analysis, and that STI's do not necessarily disregard the situation producing correspondence bias.

An important implication of our findings is that spontaneous inferences play a role in the cognitive process of detection and perception of ulterior motivation and self-presentational goals. Probably, inferences about suspicious behaviors involve both an automatic stage (i.e., spontaneous, as indicated by the current research) in which the traits and motives related to the self-presentational goal are activated, along with correspondent inferences, and a controlled stage (as indicated by earlier research, e.g. [START_REF] Fein | Effects of suspicion on attributional thinking and the correspondence bias[END_REF] in which the two types of inferences are deliberately weighted against each other.

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The current findings extend earlier research on STI's [START_REF] Crawford | Interfering with inferential, but not associative, processes underlying spontaneous trait inference[END_REF] suggesting that suspicion inhibits STI's in general. That is, results confirm inhibition of correspondent trait activation for descriptions implying both correspondent and ulterior motives (as compared with descriptions that imply a correspondent trait only). However, our results also show that activation of other inferences is not entirely inhibited under suspicion: spontaneous inferences about selfpresentational motives do occur in these cases.

It can be argued that most of the earlier research on spontaneous inferences (e.g., Uleman et al., 1996) relied on behavioral descriptions that contain no ambiguity regarding the underlying trait. This, of course, has been important in identifying the mechanisms of spontaneous inferences, but lacks the natural ambiguity of real behavior. The current research adds to the work on multiple inferences (see [START_REF] Ham | Smart and easy: Co-occurring activation of spontaneous trait inferences and spontaneous situational inferences[END_REF][START_REF] Reeder | Dispositional Attribution: Multiple Inferences About Motive-Related Traits[END_REF]Todd et al., in press) that investigates this type of more ecologically valid behaviors.

To conclude, we have demonstrated that spontaneous inferences of ulterior, self-presentational motives occur. In addition to the SSI's established in earlier research, this indicates that the initial, automatic stage of person perception does not necessarily produce correspondence bias by taking behavior at face value. Contrarily, in this stage, multiple inferences appear to be made that may even be evaluatively inconsistent (e.g., "ingratiating" and "friendly"). By combining the two relatively isolated fields of self-presentation research and person perception research, this opens up new views and research possibilities, improving our understanding of the basics of social inference.

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  7), indicated by a simple effect of trial type, F(1, 123) = 26.87, p < .001. Second, on the two trials for which the description had been CT-only implying, a UM cue word was not remembered better on the relearning trial than on the learning trial (M = 21.8% correct, SD = 41.4, vs. M = 17.7% correct, SD = 38.4), F<1 for the effect of trial type. Within the level of UM cue words, a significant simple interaction of Trial Type x Description indicates that the memory advantage for UM+CT-implying descriptions was superior to the effect for CT-only descriptions, F(1, 123) = 10.95, p < .01. Confirming expectations, this finding indicates that spontaneous UM inferences occur when the description implies a UM by referring to a target that the actor depends on. 3 examined simple effects of trial type for CT cue words, which would indicate spontaneous CT inferences. As expected, on the two trials for which the description had been CT-only implying, the position of a CT cue word was remembered better on the relearning trial than on the learning trial (M = 56.5% correct, SD = 49.8 vs. M = 17.7% correct, SD = 38.4), indicated by a simple effect of trial type, F(1, 123) = 46.25, p < .001. This demonstration of activation of STI's

(

  M = 36.3% correct, SD = 48.3 vs. M = 22.6% correct, SD = .42), indicated by a simple effect of trial type, F(1, 123) = 5.66, p < .05. Within the level of CT cue words, a significant simple interaction of Trial Type x Description indicated that the memory advantage for UM+CT-implying descriptions was smaller than for CT-only descriptions, F(1, 123) = 8.89, p < .01. This suggests that, even though activation of CT inferences does occur in UM+CT-implying descriptions, it is stronger for descriptions that imply CT only. This finding fits earlier findings of inhibitory effects in trait activation[START_REF] Dijksterhuis | The knife that cuts both ways: Facilitated and inhibited access to traits as a result of stereotype activation[END_REF]: Activation of a stereotype can decrease the retrieval probability of traits that are inconsistent with that stereotype. Similarly, the activation of a CT may have been partially suppressed because of the co-occurring activation of the evaluatively inconsistent UM--an effect which does not occur in case of a CT-only description.

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M A N U S C R I P T ACCEPTED MANUSCRIPT Vonk, R. (1999b). Impression formation and impression management: Motives, traits, and likeability inferred from self-promoting and self-deprecating behavior.

Social Cognition, 17, 390-412. Vonk, R. (2001). Aversive self-presentations. In R. M. Kowalski (Ed. The Netherlands. Electronic mail may be sent to j.r.c.ham@tue.nl.

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Footnotes 1 Overall, valence of ulterior motives is more negative than of correspondent motives. This is inherent to the process of trait attribution: People often present themselves in positive and socially desirable ways (leading to positive correspondent inferences), and inferring an ulterior motive implies misleading actor behavior and fake positive qualities (leading to a more negative, ulterior motive inference).

However, positivity can be ruled out as a confound: The current experimental design does not examine differences between correspondent inferences versus ulterior motives, but interaction effects between cue type (UM vs. CT) and description type (UM+CT-implying versus CT-implying). The analyses will compare activation of UM cue words with activation of the same UM cue words (because all materials were completely counterbalanced between participants), after presentation of the two description types. Likewise, we will compare activation of CT cue words with the same CT cue words, after presentation of the two description types. Note that these descriptions are identical except for the words used to describe the relationship between the actor and the target (e.g., "John volunteered to help paint his boss'

house", versus "…a fellow student's house").

2 When analyzing error rates, we will analyze binomial data--that is, answers were either correct or wrong. In line with Kirk (1982, pp. 75-76; see also, [START_REF] Lunney | Using analysis of variance with a dichotomous dependent variable: An empirical study[END_REF], we used analysis of variance to analyze them.

3 There might be differences in the rememberability (being able to remember where this word was presented in the grid) of UM cue words versus CT cue words (e.g., because of conceptual overlap such as between "bragging", "show-off" and "flaunt"). However, facilitation effects that indicate activation of UM (or CT) inferences are a comparison between relearning trials and learning trials on which the

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cue word always was a UM (or a CT) cue word. Thereby, a comparison is made between correctly remembering the same set of (UM or CT) words on relearning trials and on learning trials. So, the facilitation effect that indicates spontaneous activation of UM inferences is independent of the facilitation effect that indicates spontaneous activation of CT inferences, and any differences in rememberability of UM versus CT cue words are irrelevant to the differential effects of the type of description on these two facilitation effects.

4 Although in English "complimenting" might be seen as a description of the behavior, the Dutch word for it ("complimenteus") is a trait. The Dutch word for "womanizer" ("versierder") is more positive (has connotations with charmer) than the English word. "Rich" and "in love" do not refer to stable personality traits, but describe inferences about underlying causes of behavior comparable to other characteristics used in the current and earlier research of STI's. We were unable to generate only trait terms to denote underlying causes of self-presentational behaviors, but note that all words describing inferences refer to more or less stable characteristics of an actor that can be reflected in behavior. In everyday life, people not only use trait terms to describe the stable characteristics of others, but also many other inference terms (e.g., interests, values, chronic states, and abilities; [START_REF] Beach | A free response approach to the study of person cognition[END_REF]. Pete used much jargon when talking with his fellow students.

A C C E P T E D M A N U S C R I P T

Flaunt Intelligent

Note. Participants saw all four combinations of UM+CT or CT-implying descriptions and UM or CT cue word. In relearning trials, the cue word fitted an implication of the description that had been presented in the same cell, whereas in learning trials a cue word was selected that was unrelated to the description that had been presented in the same cell. Descriptions and cue words are best possible translations from Dutch. 4

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E Aubry
email: e.aubry@essaim.univ-mulhouse.fr-dfendel@club-internet.fr-f.m.schmitt@essaim.univ-mulhouse.fr-m.renner@univ-mulhouse.fr

-D M Fendeleur -F

-M Schmitt

Renner

Measurement of the natural frequencies and critical speed of roll-tensioned discs

Thin steel discs can be used to make circular saw plates. The process of roll-tensioning is traditionally used to increase the stability of rotating discs at high rotation speed, and so dynamic properties and critical speed, which determines the maximum stable rotation speed of the disc. Rolltensioning consists in introducing a residual stress state in the plate. The structure is locally deformed between two opposite pinch rollers in rotation which rotate the plate. This creates a plastic deformation along a circular zone at constant radius. Manufacturers are generally able to roll tension acceptably through know-how gained from experience. However, this is done, very often, without any scientific knowledge. In parallel, many efficient theories have been developed by several scientists to explain the dynamic phenomena which occur. Our principal target in this paper is to expose how we developed a reliable method to determine the critical speed in connection with the roll-tensioning process applied to the structure. This method is to allow the quantification of roll tensioning during the production process, in industrial situation.

To perform our tests, we adhered as closely as possible to the normal working conditions of the type of tool : a circular rotating tool, with a bore, fixed on an arbor with flanges of a given diameter. We used modal analysis methods to evaluate the natural frequencies of each mode shape when the disc is not rotating . Then their evolution with the increase of the rotation speed were followed. Experimental results and theoretical results described in the bibliography were compared. This article is composed of three parts. The first part describes the experimental bench used for the tests. It presents also the software developed for the determination of the modal properties of the plate. In the second part, we show that the results obtained for a disc which has not been roll-tensioned and the theoretical values are comparable. This proves the effectiveness of our method. In the third part, by evaluating the natural frequencies and the critical speed of several discs roll-tensioned differently, we clearly point out the influence of the roll-tensioning process on the dynamic properties of the structure.

Then, we conclude with a description of the advantages and the limits of our method.

Description of the test stand :

The test stand (Figure 1) is composed of the different parts described below :

• a D.C. motor with a speed variator allows any rotation speed between 0 and 3000 rpm,

• a pair of flanges (outside diameter = 70 mm) to fix the disc,

• an eddy current proximity system to measure the vibrations,

• an impact hammer fitted with a force sensor to excite and measure the excitation force),

• a signal analyzer to estimate the frequency response function of the structure,

• a computer with Matlab® software to extract the data, to propose an interpretation of the measure and to plot the Campbel's diagram.

All the equipment described assists in finding the natural modes when the structure is running or not by the use of the classical modal analysis methods. In each case, the impact hammer, fitted with a force sensor, provides the excitation and the response is measured by the proximity sensor. The sensor measures the lateral displacements at the rim of the plate, where the vibration have the largest amplitudes.

The spectrum analyzer stores the time signal then proceeds to the signal analysis (windowing, averaging, FFT, etc...) and calculates the Frequency Response Function (real and imaginary parts, phase and magnitude). Then, a Matlab® script has been written to perform a post-analysis and to display the resulting curves.

Natural frequencies and critical speed of a rotating disc without tensioning :

When a steel disc is not rotating, the natural frequencies of this disc clamped at the bore and free at the rim are given by Blevins 1 :

( )

2 / 1 2 2 2 ij 2 ij 1 12 Eh a 2 f         - = ν ρ π λ (1)
where a is the outside diameter, h the thickness, E the elasticity module of the material, ρ the specific mass and ν the Poisson's coefficient. The boundary conditions allow to calculate the coefficient λ ij by imposing given relations on the bending moments equation and on shearing stresses equations 1, 2

. λ ij is related to the ratio of the diameters (outside diameter / clamping diameter). The mode shapes for each natural frequency are characterized by some nodal diameter and/or by some nodal circles. The number of nodal diameter is indicated by the letter i, and the number of nodal circle is j.

When the disc is in motion, some rotation stresses are induced by the centrifugal forces 3, 4, 5 .

These modify the natural frequencies of the disc according to the equation :

2 2 2 s 60 K f f       Ω + = ( 2
)
where f is the natural frequency calculated as indicated in Eq.1, Ω the rotation speed in rpm and K dimensionless value.

The phenomenon of stationary waves is represented for a given natural frequency by a mode shape. This is due to the existence of two different waves around the disc, with the same frequency but travelling with the rotation speed in two opposite directions. When the disc rotates, the rotation speed is added to the propagation speed of the travelling wave in the direction of rotation (forward waves) and, is substracted from the propagation speed of the travelling wave in the opposite direction (backwards waves). This leads to a splitting of each mode shape into two different frequencies. Only mode shapes with certain nodal diameters are affected. Therefore, the frequencies of the forward modes increase and the frequencies of the backward modes decrease in relation to the rotation speed [START_REF] Shajer | Simple formulas for natural frequencies and critical speeds of circular saws[END_REF] :

60 i f f s f Ω + = and 60 i f s b f Ω - = ( 3 )
where f f and f b respectively are the frequencies associated with the forward and the backward waves; i is the number of nodal diameter.

We measured the mode splitting phenomenon. The disc used was not roll-tensioned, had an outside diameter of 350 mm, a thickness of 2,6 mm and the flanges had a diameter of 70 mm. The natural frequency shown on the figure 2 was 470 Hz when the disc was stopped. Through a modal analysis, it was determined that the mode shape with four nodal diameters (i=4).

The split modes exist for all the natural frequencies. A theoretical special rotation speed where the backward frequency tends toward zero can be found. This leads to the determination of the critical speed which is «the first rotation speed for a backward frequency reaching zero» 6, 7, 8, 9

. A special representation is used to visualize the evolution of each frequency depending on the rotation speed : it is called the Campbell's diagram, which can be theoretically calculated too [START_REF] Shajer | Simple formulas for natural frequencies and critical speeds of circular saws[END_REF] .

For the measurement, we intentionally divided the frequency range into two parts to have a better accuracy. The figure 3a shows both the results theoretical and experimental on a range of 50 to 250 Hz and the figure 3b both the results theoretical and experimental on a range of 200 to 1000 Hz . On the fist range, the three first modes can be isolated. In addition, the unbalance due to the run-out defect of the disc disturb the measurement and affects the FRF. It is more difficult to extract the interesting values into the FRFs' signal. But, by comparison with the theoretical simulation, the splitting of the second mode is clearly visible. On the higher range of frequency (200 to 1000 Hz), the signal are less disturbed and the measure can more easily used for a post-treatment. In this case, the relation theory-experimentation is obvious. The critical speed can be found by extrapolating the experimental curves. The first frequency

Natural frequencies and critical speed for a roll-tensioned idling disc :

The roll-tensioning process significantly modifies the natural properties of the disc, whether it is stopped or rotating. For several discs rolled with different pressure applied to each, the natural frequencies were measured. The results given in the chart (Tab.1) are classified by increasing pressure. The natural frequencies for the clamped disc change with the increased application of pressure. A specific evolution can be found (Fig. 4). Other tests performed on several discs with various dimensions confirm these trends.

Some modelings cited in the bibliography 6, 8, 9, 10 confirm this too.

When a tensioned disc rotates, the global state of stresses inside the disc is a combination of the residual stresses coming from the tensioning, of the centrifugal stresses and of the thermal stresses. This modifies the dynamic behavior of the plate and changes the critical speed. Actually, roll-tensioning is done on purpose, in order to improve the stability of the rotating disc.

The equation of motion taking into account these phenomena has been solved in numeric form years ago by Shajer 11, 12 . Because of the number of tensioning methods and because of the difficulty to check the quality of the tension, it is very hard to use the theoretical simulation to qualify the tensioning and to estimate the critical speed.

After several measurements, the Campbell's diagrams were established for the five previous discs.

In this article, the evolution of the backward and the forward frequencies are shown Fig. 5 for two given modes (mode shape involving two and four nodal diameters). The backward and forward frequencies are systematically in a special order : B5 -B99 -B4 -B6 -B8. B5 is less tensioned than B99, B99 is less tensioned than B4, etc... An example can be given : the B8 disc has his natural frequencies always over the B5's frequencies for every rotation speed. This means that B8 has a higher critical speed than B5.

we found the critical speed at 3900 rpm. For B99, it is 4000 rpm; for B4, 4350 rpm; for B6, 4400 rpm and for B8, 5150 rpm.

Conclusion :

At first and as expected, we have shown that the roll-tensioning process has an obvious effect to increase the critical speed and consequently the stability of the rotating disc. But, our target was to point out, through the measurements of the dynamic properties, the relation between the amount of the « tension » applied and the positive variation of the critical speed. Also, with the Campbell's diagram, it is possible to determine the critical speed and to guess the tension. This makes a quality control of the rolltensioning process possible.

Although the techniques of dynamic analysis and modal analysis are quite easy to use in the laboratory, it is difficult to apply the method developed in this article, to industry. The complexity of the measurements and the use of the experimental data require very skilled people. Nevertheless, based on our experience, we can imagine a simplest method to make the measurements. Actually, if it is absolutely necessary to determinate the critical speed to appreciate the influence of the tension on the stability of a rotating disc, the effect of the tensioning can be shown by measuring the first natural frequencies of simple clamped disc.

On our test bench, we evaluated the evolution of these first frequencies when the disc is clamped with flanges (the rim is free). The figure 4 shows this particular evolution. To measure the tension, the literature usually proposes to focus on the frequencies related to the modes f 00 and f 04 . Some devices using this method can be found on the market [START_REF] Szymani | Used and maintenance of thin circular saws at california Cedar products Company[END_REF] . But in working conditions, we showed that this technique while interesting, has some disadvantages : the reliability of the clamping conditions are poor, the measurement

List of figures :

Fig. 1 :Fig. 2 :Fig. 4 :Fig. 5 :Tab. 1 :Fig. 1 :Fig. 2 :

 1245112 Fig.1 : Testing bench

Fig. 4 :Fig. 5 :

 45 Fig.4 : Natural frequencies depending on the roll-tensioning amount


Bernard Bonnard
email: bernard.bonnard@u-bourgogne.fr

Monique Chyba
email: chyba@hawaii.edu

Two applications of geometric optimal control to the dynamics of spin particles

The purpose of this article is to present the application of methods from geometric optimal control to two problems in the dynamics of spin particles. First, we consider the saturation problem for a single spin system and second, the control of a linear chain of spin particles with Ising couplings. For both problems the minimizers are parameterized using Pontryagin Maximum Principle and the optimal solution is found by a careful analysis of the corresponding equations.

Summary. The purpose of this article is to present the application of methods from geometric optimal control to two problems in the dynamics of spin particles. First, we consider the saturation problem for a single spin system and second, the control of a linear chain of spin particles with Ising couplings. For both problems the minimizers are parameterized using Pontryagin Maximum Principle and the optimal solution is found by a careful analysis of the corresponding equations.

Introduction

The past few years have witnessed an intense recent research activity on the optimal control of the dynamics of spin systems controlled by a radio frequency magnetic field (RF-magnetic field) with application to Nuclear Magnetic Resonance (NMR) spectroscopy and Medical Resonance Imaging (MRI). A pioneer application of geometric optimal control was the contribution to the saturation problem of a single spin system [?], where the authors replaced the standard inversion recovery sequence by the time minimal solution formed by a sequence of bang RF-pulses with maximal amplitude and pulses with intermediate amplitude corresponding to singular trajectories in optimal control [?]. This result is a consequence of the Pontryagin Maximum Principle [?] after performing a reduction to a two dimensional system which allows a complete analysis of the problem.

The first objective of this article is to present in details the geometric tools to analyze this saturation problem as well as some extensions of it in relation with the contrast problem in MRI [?, ?]. The second objective of this article is to discuss the time optimal control of a linear chain of spin particles with Ising couplings [?] in relation with quantum computing. Restricting to the case of three spins we introduce the geometric framework to analyze the problem and we improve the preliminary results contained in [?, ?]. As for the single spin system, the minimizers are parameterized using the Maximum Principle and the optimal solution is computed using the framework of recent developments of invariant sub-Riemannian geometry (SR-geometry) on SO [START_REF]Arnold Mathematical methods of classical mechanics, Translated from the Russian[END_REF] [?].

Preliminary: The Pontryagin Maximum Principle

In this section, we recall the necessary optimality conditions [?] which allows us to parameterize the optimal solutions. For our purpose it is sufficient to formulate them in the time minimum case.

Necessary optimality conditions

Consider the minimum time problem for a smooth control system:

dx dt (t) = f (x(t), u(t)), (1)
where x(t) ∈ M, with M a n-dimensional manifold, the control domain U is a subset of R m and the state variable satisfies the boundary conditions x(0) ∈ M 0 and x(t f ) ∈ M 1 , with M 0 , M 1 being smooth submanifolds of M.

We introduce the pseudo-Hamiltonian:

H(x, p, u) = p, f (x, u) , (2)
where , is the scalar product and p ∈ T * M denotes the adjoint vector. The Maximum Principle states that if the trajectory t → x(t), t ∈ [0,t f ] associated to the admissible control u : [0,t f ] → U is optimal, then there exists p : [0,t f ] → T * M non zero and absolutely continuous such that the following equations are satisfied almost everywhere on [0,t f ]:

dx dt (t) = ∂ H ∂ p (x(t), p(t), u(t)) (3) d p dt (t) = - ∂ H ∂ x (x(t), p(t), u(t)), (4)
as well as the maximum condition:

M(x(t), p(t)) = max v∈U H(x(t), p(t), v) (5)
where

H(x(t), p(t), u(t)) = M(x(t), p(t)). Moreover M(x(t), p(t)
) is constant along the trajectory and the following boundary conditions are satisfied:

x(0) ∈ M 0 , x(t f ) ∈ M 1 , (6)
p(0) ⊥ T x(0) M 0 , p(t f ) ⊥ T x(t f ) M 1 (Transversality conditions). ( 7
)
3 The Saturation Problem

In this section, we recall prior results on the single spin system and introduce the model for multiple spins.

3.1 The case of a single spin and its extensions to systems of spins

The single spin system is modeled by the Bloch equation written in a moving frame and adapted coordinates:

dx dt (t) = -Γ x(t) -u 2 (t)z(t) (8
)

dy dt (t) = -Γ y(t) + u 1 (t)z(t) (9
)

dz dt (t) = γ(1 -z(t)) -u 1 (t)y(t) + u 2 (t)x(t), (10)
where q = (x, y, z) represents the magnetization vector restricted to the Bloch ball: | q |≤ 1, (Γ , γ) are the physical parameters which are the signature of the chemical species and satisfies 2Γ ≥ γ > 0, and

u = (u 1 , u 2 ) is the bounded RF-applied magnetic field | u |≤ m.
The objective of the saturation problem is to bring the magnetization vector q from the north pole : N = (0, 0, 1) (which is the equilibrium point of the free system) to the center O = (0, 0, 0) of the Bloch ball. The physical interpretation is related to the fact that in MRI, the amplitude | q | corresponds to a grey level, with | q |= 1 corresponding to white and | q |= 0 to black. A direct generalization of the statement above is to bring the system from a forced equilibrium position (associated to a nonzero fixed constant control) to the center O.

Equations (??)-(??) can be written in a compact form as an affine control system

dq dt (t) = F(q(t)) + u 1 (t)G 1 (q(t)) + u 2 (t)G 2 (q(t)) (11)
where F represents the dissipation of the system and the G i 's the controlled vector fields:

F(q(t)) =   -Γ x(t) -Γ y(t) γ(1 -z(t))   , G 1 (q(t)) =   0 z(t) -y(t)   , G 2 (q(t)) =   -z(t) 0 x(t)   . ( 12
)
An extension with application to MRI is to consider an ensemble of N spin systems, associated to the same chemical species (i.e. with the same physical parameters Γ and γ). We denote by q s (t), s = 1, ..., N, the solutions of the system:

dq s dt (t) = F(q s (t)) + (1 -ε s ){u 1 (t)G 1 (q s (t)) + u 2 (t)(G 2 (q s (t))} (13)
where the control u

= (u 1 , u 2 ) satisfies | u |≤ m.
The saturation problem for the ensemble of spins is to steer the system from the north pole N = ((0, 0, 1), ..., (0, 0, 1)) to the center O = ((0, 0, 0), ...(0, 0, 0)) of the products of the Bloch balls. This is equivalent to the saturation problem in MRI, where the amplitude m(1 -ε s ) corresponds to the variation of the applied RF-field induced by the spatial position of the spin s in the image.

The Saturation problem in minimum time for a single spin

First of all, we observe that due to the symmetry of revolution of the problem with respect to the polarizing z-axis we can restrict our analysis to a 2D-single-input system:

dy dt (t) = -Γ y(t) + u(t)z(t) (14) dz dt (t) = γ(1 -z(t)) -u(t)y(t), ( 15
)
where | u(t) |≤ m. The system can be written as dq dt = F(q) + uG(q), with q = (y, z). Applying the Maximum Principle, an optimal solution is found by concatenation of regular and singular arcs defined as follows.

Definition 1. For a system of the form dq dt = F(q) + uG(q), the control is said to be:

• singular if p(t), G(q(t)) ≡ 0, and is determined by differentiating this implicit equation.

• regular if p(t), G(q(t)) = 0, and is given for a.e. t by u(t) = m sign p(t), G(q(t)) .

One denotes by σ s a singular arc, and by σ ±m bang arcs such that the control is given by u = ±m and σ 1 σ 2 an arc σ 1 followed by an arc σ 2 . The first step is to compute the singular arcs. They satisfy p(t), G(q(t) = 0, and differentiating this equation with respect to time one gets the relations:

p(t), G(q(t)) = p(t), [G, F](q(t)) = 0, (16)
p(t), [[G, F], F](q(t)) + u(t)[[G, F], G](q(t)) = 0, (17)
where [, ] denotes the Lie bracket.The singular trajectories are therefore located on the set

S = {q; det(G, [G, F])(q) = 0}
, which is given in our case by y(-2δ z + γ) = 0 with the notation δ = γ -Γ . It is formed by the z-axis of revolution y = 0 and the horizontal direction:

z 0 = γ/2δ .
The singular control is computed as a feedback using (??). We have u

(t) = p(t), [[G, F], F](q(t))/[[G, F], G](q(t)) which implies:

• For y = 0, the singular control is zero and the corresponding system is dz dt (t) = γ(1z(t)). It relaxes to the equilibrium state.

• For z 0 = γ/2δ , the feedback singular control is u s = γ(2Γ -γ)/2δ y and the dynamics is given by:

dy dt (t) = -Γ y(t) - γ 2 (2Γ -γ) 4(γ -Γ ) 2 y(t) , (18)
and u s → ∞ when y → 0. The time to steer (-1, * ) to (0, * ) is given by the integral T =

1 0 dy Γ y+ γ 2 (2Γ -γ) 4(γ-Γ ) 2 y
which can be easily computed.

The interesting case in the saturation problem is when the horizontal line z 0 = γ/2δ is such that -1 < z 0 < 0 which imposes the following condition on the physical parameters: 2Γ > 3γ, and we shall restrict our analysis to this case.

To complete the analysis, we must determine the optimality status of the singular line. It can be small time maximizing (slow) or minimizing (fast). To distinguish the two cases one uses the generalized Legendre-Clebsch condition [?] as follows. Let D ′′ = det(G, F) = γz(z -1) + Γ y 2 and C = {D ′′ = 0} be the collinear set. If γ > 0, this set is not reduced to a point, but forms an oval curve joining the north pole (0, 1) to the center (0, 0) of the Bloch ball and the intersection with the horizontal singular line is empty. Denoting D = det(G, [[G, F], G]), the singular lines are fast displacement directions if DD ′′ > 0 and slow if DD ′′ < 0. From this condition one deduces that the z-axis of revolution is fast if 1 > z > z 0 and slow if z 0 > z > -1, while the horizontal singular line is fast.

From this analysis, we deduce that the standard inversion sequence: u(t) = m to steer (0, 1) to (0, - * ) followed by u(t) = 0 to relax the system along the z-axis is not time optimal. It has to be replaced by a policy using the horizontal singular line. The complete analysis is not straightforward since | u s |→ ∞ when y → 0 along the singular horizontal line and the singular control saturates the constraint | u |≤ m.

The final result is given in Theorem ??, see [?] for more details.

Theorem 1. In the time minimal saturation problem, the optimal policy is of the form σ m σ s σ m σ s .

Interpretation

The first bang arc is used to move the system from the equilibrium point (0, 1) to the horizontal singular line while the second bang arc σ m connects the horizontal singular arc to the vertical and this occurs before saturating the singular control.

4 The geometry of a linear three spin system with Ising couplings

Mathematical model

We restrict ourselves to the optimal control of three coupled spins, but the problem can be generalized to a chain with any number of spins. We follow here the presentation of [?, ?].

We introduce the spin 1/2 matrices σ α , where α represents the number of the particle carrying spin, related to the Pauli matrices by a 1/2 factor. Such matrices satisfy:

[σ x , σ y ] = iσ z , σ 2 x = σ 2 y = σ 2 z = 1/4. ( 19
)
The Hilbert space of the system consists of a dimensional space formed by the tensorial product of the three two dimensional spin 1/2 Hilbert spaces. The Hamiltonian of the system can be written as follows:

H = H C + H F , (20)
where H C the Hamiltonian of the free system and H F the Hamiltonian corresponding to the RF-magnetic field are given by

H C = 2J 12 σ 1z σ 2z + 2J 23 σ 2z σ 3z (21)
H F = u(t)σ 2y , (22)
with the coefficients J i j representing the coupling constants between the spins i and j.

We consider the time evolution of the vector X = (x 1 , x 2 , x 3 , x 4 ) where x 1 = σ 1x , x 2 = 2σ 1y σ 2z , x 3 = 2σ 1y σ 2x , x 4 = 4σ 1y σ 2y σ 3z where denotes here the expectation value. To compute the dynamics, we introduce the density matrix ρ, a 8 × 8-matrix which satisfies :

dρ dt = -i[H, ρ]. (23)
Using the definition of the expectation value of a given operator:

O = Tr(Oρ), (24)
one gets:

d dt σ 1x = Tr(σ 1x dρ dt ) = -iTr(σ 1x [H, ρ]) = -iTr([σ 1x , H]ρ). ( 25
)
Hence we deduce that:

dx 1 dt = -J 12 Tr(2σ 1y σ 2z ρ). (26)
By rescaling the time by the factor J 12 , we obtain:

dx 1 dt = -x 2 . ( 27
)
Similar computations lead to the evolution of X given by:

dX dt =     0 -1 0 0 1 0 -u 0 0 u 0 -k 0 0 k 0     X, k = J 23 /J 12 . (28)
The optimal control problem is to transfer in minimum time the initial position (1, 0, 0, 0) to the position (0, 0, 0, 1) as an intermediate step to realize the transfer in minimum time from σ 1x to σ 3x . Indeed it connects the first spin to the third one by controlling the second spin.

Introducing the coordinates

r 1 = x 1 , r 2 = x 2 2 + x 2 3 , r 3 = x 4 , (29)
and

tan α = x 3 /x 2 , (30)
we obtain the system:

d dt   r 1 r 2 r 3   =   0 -cos α 0 cos α 0 -k sin α 0 k sin α 0     r 1 r 2 r 3   , (31)
where r = (r 1 , r 2 , r 3 ) ∈ S 2 (the two dimensional sphere), and u 1 = -k sin α, u 3 =cos α.

In those coordinates the minimum time problem is equivalent to find the fastest transfer on the sphere from (1, 0, 0) to (0, 0, 1). It can be written:

dr 1 dt = u 3 r 2 , dr 2 dt = -u 3 r 1 + u 1 r 3 , dr 3 dt = -u 1 r 2 , (32)
min u(.) T 0 (I 1 u 2 1 + I 3 u 2 3 )dt, k 2 = I 1 /I 3 . ( 33
)
This problem is equivalent to a Riemannian problem on the sphere S 2 , with a singularity at the equator r 2 = 0, the metric being

g = dr 2 1 + k 2 dr 2 3 r 2 2 . ( 34
)
Introducing the spherical coordinates

r 2 = cos ϕ, r 1 = sin ϕ cos θ , r 3 = sin ϕ sin θ , (35)
where ϕ = π/2 is the equator, the metric g takes the form

g = (cos 2 θ + k 2 sin 2 θ )dϕ 2 + 2(k 2 -1) tan ϕ sin θ cos θ dϕdθ + tan 2 ϕ(sin 2 θ + k 2 cos 2 θ )dθ 2 ,
with the associated Hamiltonian

H = 1 4k 2 {p 2 ϕ (sin 2 θ + k 2 cos 2 θ ) + p 2 θ cotan 2 ϕ(cos 2 θ + k 2 sin 2 θ ) -2(k 2 -1)p ϕ p θ cotanϕ sin θ cosθ }.
If k = 1, the Hamiltonian takes the form H = 1 4 (p 2 ϕ + p 2 θ cotan 2 ϕ) and describes the standard Grushin metric on S 2 .

Connection with invariant metrics on SO(3) and integration

A first approach consists in lifting the problem on SO(3). We introduce the matrix R(t) = (r i j (t)) of SO(3) where r 1 = r 11 , r 2 = r 12 , r 3 = r 13 are the components of the first row and we consider the right-invariant control system:

d dt R t =   0 u 3 0 -u 3 0 u 1 0 -u 1 0   R t . ( 36
)
The first column equation corresponds to our problem. Hence our optimal control problem becomes:

min u(.) T 0 (I 1 u 2 1 + I 3 u 2 3 )dt (37)
for the right invariant control system with the following boundary conditions:

R t (0) =   1 * * 0 * * 0 * *   , R t (T ) =   0 * * 0 * * 1 * *   , (38)
and we want to steer the first axis of the frame R t from e 1 to e 3 , where e i denotes the canonical basis. Equivalently, it is transformed into a left-invariant control problem to use the computations in [?]:

dR dt = R   0 -u 3 0 u 3 0 -u 1 0 u 1 0   , min u(.) T 0 (I 1 u 2 1 + I 3 u 2 3 )dt (39)
with the corresponding boundary conditions. This defines a left-invariant SR-problem on SO(3) depending on the parameter k 2 = I 1 /I 3 . Upon an appropriated limit process I 2 → +∞, it is related to the Euler-Poinsot rigid body motion [?]:

dR dt = R   0 -u 3 u 2 u 3 0 -u 1 -u 2 u 1 0   , min u(.) T 0 (I 1 u 2 1 + I 2 u 2 2 + I 3 u 2 3 )dt (40)
which is well-known model for invariant metrics on SO(3) depending on 2 parameters, the ratio I 2 /I 1 and I 3 /I 1 . There are two special cases:

1. The bi-invariant case I 1 = I 2 = I 3 where the geodesics solutions are the rotations of SO(3).

The case of revolution where

I 1 = I 2 .
The optimal solutions can be parameterized by the Pontryagin maximum principle and thanks to the explicit formula given in [?], the solutions can be computed in both the Riemannian and the sub-Riemannian cases using the elliptic functions. See [?] for the details of the computations. We here only sketch the main points. We introduce the following matrices:

A 1 =   0 0 0 0 0 -1 0 1 0   , A 2 =   0 0 1 0 0 0 -1 0 0   , A 3 =   0 -1 0 1 0 0 0 0 0   (41)
with the Lie brackets relations:

[A 1 , A 2 ] = -A 3 , [A 1 , A 3 ] = A 2 , [A 2 , A 3 ] = -A 1 . ( 42
)
The optimal control problem is then stated as:

dR dt = 3 ∑ i=1 u i RA i , min u(.) T 0 3 ∑ i=1 I i u 2 i dt. ( 43
)
Applying the Maximum Principle and denoting by H i the symplectic lifts of the vector fields RA i leads to introduce the pseudo-Hamiltonian:

H = 3 ∑ i=1 u i H i - 1 2 3 ∑ i=1 I i u 2 i . (44)
From the maximization condition ∂ H ∂ u = 0 we deduce that u i = H i /I i , i = 1, 2, 3. Plugging back this expression for the controls u i in the Hamiltonian H defines the true Hamiltonian:

H n = 1 2 ( H 2 1 I 1 + H 2 2 I 2 + H 2 3 I 3 ). (45)
Then we derive the Euler equation:

dH i dt = {H i , H n }, (46)
where {, } is the Poisson bracket. Using Lie brackets computations, one obtains:

dH 1 dt = H 2 H 3 ( 1 I 3 - 1 I 2 ), (47)
dH 2 dt = H 1 H 3 ( 1 I 1 - 1 I 3 ), (48)
dH 3 dt = H 1 H 2 ( 1 I 2 - 1 I 1 ). (49)
The SR-case can be derived formally by setting u 2 = εv 2 , ε → 0 to obtain a bi-input system. Since u i = H i /I i , this is equivalent to I 2 → +∞. The Hamiltonian reduces to:

H n = 1 2 ( H 2 1 I 1 + H 2 3 I 3 ), (50)
with the corresponding Euler equation where the parameter k 2 = I 1 /I 3 is the invariant classifying the SR-metrics.

To get an uniform integration procedure we use the following result from [?].

Proposition 1. For each invariant Hamiltonian on SO(3) (the Riemannian and the SR-case)

the system is integrable by quadrature using the four first integrals: the Hamiltonian H n and the Hamiltonian lifts of the right invariant vector fields A i R.

Integration

The general algorithm consists in integrating the Euler equation, while the remaining quadrature is deduced using the Euler angles Φ i defined by taking the following decomposition of a matrix R in SO(3):

R = (expΦ 1 A 3 ) • (expΦ 2 A 2 ) • (expΦ 3 A 3 ) (51)
while the angles Φ 2 , Φ 3 can be found from the relations:

H 1 = -| H | sin Φ 2 cos Φ 3 , H 2 =| H | sin Φ 2 sin Φ 3 , H 3 =| H | cos Φ 2 (52)
and the angle Φ 1 is computed by integrating the differential equation

dΦ 1 dt = | H | (H 2 sin Φ 3 -H 1 cos Φ 3 ) (sin Φ 3 ∂ H n ∂ H 2 -cos Φ 3 ∂ H n ∂ H 1 ). (53)
In the SR-case, the Euler equation is integrated as follows. We fix the level set of the true Hamiltonian to 1/2: H n = 1/2, and we introduce α such that cos α = H 1 / √ I 1 and sin α = H 3 / √ I 3 . We obtain the pendulum equation: In conclusion, in the spin time optimal problem the optimal solutions can be found among extremals solutions of the Maximum Principle.

d 2 α dt 2 = k 2 -1 2I 1 sin 2α, k 2 = I 1 /I 3 . (54)

Direct integration on the sphere

We identify S 2 as the homogenous space SO(3)/SO [START_REF] Agrachev | Sigalotti A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds[END_REF] . In this interpretation, the Hamiltonian

| H | 2 = H 2 1 + H 2 2 + H 2
3 corresponds to the bi-invariant case and represents the Casimir function which commutes with the Hamiltonian associated to every invariant Riemannian and sub-Riemannian metrics.

On the homogeneous space this defines the round sphere with constant curvature +1, whose metric in spherical coordinates is given by :

g = dϕ 2 + sin 2 ϕdθ 2 (55)
and with Hamiltonian

F = p 2 ϕ + p 2 θ sin 2 ϕ . ( 56
)
We have the following result. Expressing F in the (u, v) coordinates we have:

F(u, v) = p 2 u b ′ 1 (u, v) + 2p u p v b ′ 2 (u, v) + p 2 v b ′ 3 (u, v). ( 69
)
An easy computation shows that:

S = (b ′ 1 -b ′ 3 + 2ib ′ 2 ) = D 2 (ϕ u -iψ u ) 2 (b 1 -b 3 + 2ib 2 ) (70) = (ϕ u + iψ u ) -2 (b 1 -b 3 + 2ib 2 ). (71)
We choose the change of coordinates such that we have the following normalization: S = 1. Hence, we must solve the equation

(ϕ u + iψ u ) = R(z). ( 72
)
In the new coordinates the metric takes the Liouville normal form:

g(u, v) = ( f (u) + g(v))(du 2 + dv 2 ), (73)
and the integration is standard see for instance [?].

Grushin Singularity

The family of metrics g is defined as a metric on the distibution ∆ = Span{F 1 , F 3 } where F 1 , F 3 are respectively the vector fields corresponding to rotations with axis e 1 and e 3 . By construction such a metric has a Grushin singularity [?] at the equator E where rank∆ is one and the distribution is transverse to the equator. The local normal form near a point of the equator is described in the aforementioned article. Hence our family of metrics defines an almost-Riemannian metric on the sphere with Grushin singularity at the equator.

The optimality problem

We discuss briefly in this section the optimality problem which can be handled using the technical framework developed in [?] combining geometric analysis and numerical techniques. We use the following concepts. On the almost-Riemannian manifold (S 2 , g), the cut point along a geodesic curve γ, projection of an extremal curve solution of the Maximum Principle, emanating from q 0 ∈ S 2 is the first point where it ceases to be minimizing and we denote C cut (q 0 ) the set of such points forming the cut locus. The first conjugate point is the point where it ceases to be minimizing among the geodesics C 1close from γ and we denote C(q 0 ) the set of such points, forming the conjugate locus.

Our optimality problem amounts to transfer with minimum length the point q 0 = (1, 0, 0) given by ϕ 0 = π/2, θ 0 = 0 in spherical coordinates to the point q 1 = (0, 0, 1) defined by ϕ 1 = π/2, θ 1 = π/2. In particular the problem is solved by computing the cut locus C(q 0 ) of the equatorial point.

First, we have the following lemma.

Lemma 2. The family of metrics g depending upon the parameter k have a discrete symmetry group defined by the two reflexions : H(ϕ, p ϕ ) = H(π -ϕ, -p ϕ ) (reflexion with respect to the equator) and H(θ , p θ ) = H(-θ , -p θ ) (reflexion with respect to the meridian θ = 0).

The next step is to use the Grushin singularity resolution described in [?] and the previous symmetries.

Proposition 3. Near q 0 identified to 0, the conjugate and cut loci for the metric restricted to a neighborhood of 0 can be computed using the local model dx 2 + dy 2

x 2 . The cut locus is a segment [-ε, +ε] minus 0 while the conjugate locus is formed by four symmetric curves of the form x = cy 2 minus 0 and tangential to the meridian θ 0 = 0 (although the Gaussian curvature is strictly negative and tends to -∞ at the equator).

In [?] the cut and conjugate loci in the Grushin case k = 1 on S 2 of an equatorial point are completely described making an homotopy g λ = dϕ 2 + G λ (ϕ)dθ 2 , G λ (ϕ) = sin 2 ϕ

(1-λ sin 2 ϕ) , λ ∈ [0, 1] from the round metric to the Grushin case which explains in particular the curvature concentration at the equator in the Grushin case. We get the following result. Proposition 4. In the Grushin case k = 1, the cut locus of the equatorial point ϕ 0 = π/2, θ 0 = 0 is the whole equator minus this point while the conjugate locus has a double heart shape, with four meridional singularities, two at the origin described previously and two cusps on the opposite meridian.

The general case is studied in [?] using a continuation method on the conjugate locus starting from a Grushin case (observe also that the cut locus is decribed in [?] using the equatorial symmetry). From the geometric point of view the neat framework is given by recent works to describe the conjugate and cut loci on Liouville surfaces generalizing the ellipsoid case [?, ?].

Note the following result that can be easily proved .

Proposition 5. For every k * , the cut locus of the equatorial point is the equator minus the point.

Proof. A simple computation shows that the Gaussian curvature in each hemisphere is strictly negative. Hence there is no conjugate point for a geodesic starting form the equatorial point before returning to the equator. Due to the reflectional symmetry with respect to the equator, two geodesics starting from the equatorial point intersects with same length when returning to the equator. This proves the result.

  Details of the parameterizations of the solutions are given in [?]. The computations in the Riemannian case are standard [?, ?].

Lemma 1 .+ p 2 θ

 12 Consider the standard Grushin metric on S 2 with Hamiltonian H 0 = p 2 ϕ + p 2 θ cotan 2 ϕ. Then {H 0 , F} = 0 where F is the Hamiltonian of the round metric on S 2 , F = p 2 ϕ sin 2 ϕ .

Proof. Using the symmetry of revolution of the Grushin metric, we have that p θ is a constant (Clairaut relation). Hence p 2 θ is a constant. Therefore p 2 ϕ + cos 2 ϕ sin 2 ϕ p 2 θ + p 2 θ is a constant, which means that F is a first integral .

Up to a constant renormalization the Hamiltonian becomes:

where

We have the following.

Proposition 2. The following relations are satisfied:

Therefore, {H, F} = 0 for each k * .

Integration

To integrate the geodesic flow we use the standard Birkhoff method, see [?, ?] for the details.

The Hamiltonian H admits a first integral F which is quadratic in p and corresponds to a Liouville metric on S 2 . The metric is written in the isothermal form:

outside the equator using a rescaling of r 3 .

which is preserving the isothermal form and the orientation satisfies the Cauchy-Riemann relation:

In the isothermal coordinates, the first integral becomes: (68)