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# Perceptual uncertainty and action consequences independently affect hand movements in a virtual environment

## Abstract

When we use virtual and augmented reality (VR/AR) environments to investigate behaviour or train motor skills, we expect that the insights or skills acquired in VR/AR transfer to real-world settings. Motor behaviour is strongly influenced by perceptual uncertainty and the expected consequences of actions. VR/AR differ in both of these aspects from natural environments. Perceptual information in VR/AR is less reliable than in natural environments, and the knowledge of acting in a virtual environment might modulate our expectations of action consequences. Using mirror reflections to create a virtual environment free of perceptual artefacts, we show that hand movements in an obstacle avoidance task systematically differed between real and virtual obstacles and that these behavioural differences occurred independent of the quality of the available perceptual information. This suggests that even when perceptual correspondence between natural and virtual environments is achieved, action correspondence does not necessarily follow due to the disparity in the expected consequences of actions in the two environments.

## Introduction

While there is little doubt about the potential of virtual (VR) and augmented (AR) reality devices for research and training purposes, it is not clear to what degree virtual environments currently provide a good representation of natural environments1,2,3 and how much correspondence between virtual and natural environments can theoretically be achieved. The underlying problem extends beyond the question of the perceptual quality (fidelity) of simulated environments. If we want actions in VR/AR to be an adequate model for actions in natural environments, we must ensure that the same sources of information that influence the planning and execution of movements in natural environments are also available in VR/AR4. While previous research has focussed on the effects of perceptual uncertainty resulting from missing or inadequate implementations of perceptual cues (e.g., binocular cues to depth), the role of higher level factors (e.g., expectations of action related consequences) in virtual environments has so far been investigated to a much lesser extent1. Research on sensorimotor decision making has shown that motor behaviour is strongly influenced by perceptual uncertainty and the expected consequences of actions, i.e., gains and/or costs (for reviews see5,6). Here, we study these two factors, perceptual uncertainty and expected consequences of actions, jointly, and directly measure the effects of acting in virtual environments on visually-guided hand movements.

We manipulated perceptual uncertainty and expected consequences of actions during an obstacle avoidance task. In such a task, both of these manipulations influence the chances of a collision with the obstacles in a given condition, a potentially dangerous consequence of the action. Higher perceptual uncertainty results when there is less (or less reliable) perceptual information and makes it more difficult to accurately perceive action-relevant features, i.e., in our case the height of the obstacles. We manipulated perceptual uncertainty here by presenting the obstacles binocularly (low uncertainty) or monocularly (high uncertainty), because estimates of size and distance have been found to be less accurate in monocular than in binocular viewing7,8,9 due to the absence of binocular cues to depth. Higher perceptual uncertainty is expected to increase the probability of making an error during movement execution and has been shown to result in a larger opening of the hand (maximum-grip-aperture) in grasping tasks10,11,12,13,14 and larger passing distances in obstacle avoidance tasks15.

The pressure to perform actions accurately increases with the expectation of negative consequences (costs) resulting from movement errors (e.g., bumping into an object). Higher error-related costs have been shown to increase the distance participants keep from penalty regions when performing targeted pointing movements16,17. When higher error-related costs were implied by changes in obstacle properties (e.g., empty or filled glasses), participants kept a larger distance from these obstacles in an obstacle avoidance task18. Here, using mirror reflections, rather than computer-generated VR/AR environments (that often contain perceptual artefacts), we created a virtual environment that allowed us to vary the costs of potential movement errors without changing the perceptual information or the appearance of the obstacles19 and, thus, to measure the isolated effects of action consequences on avoidance behaviour.

## Results

In an obstacle avoidance task, we measured the trajectories of participants’ right hand moving over obstacles of four different heights (3.4, 13.0, 22.6, and 32.2 mm). Participants performed the movement without visual feedback (open-loop), i.e., their vision was occluded as soon as the hand started to move (see Methods for details). Figure 1A shows how perceptual uncertainty and the expected costs were systematically varied in the different experimental conditions. Monocular and binocular viewing conditions gave us high (+U) and low (–U) perceptual uncertainty, respectively. The real and mirror conditions allowed us to manipulate the expected costs of mistakes during movement execution, high costs (+C) and low costs (–C), respectively.

Based on previous research, we predict that the most ’dangerous condition’ should be monocular & real (top right in Fig. 1A) where both perceptual uncertainty and costs of mistakes are high (+U & +C). Dangerousness should be lowest when perceptual uncertainty is low and no costs of mistakes are to be expected (–U & –C, bottom left in Fig. 1A), i.e., in the binocular & mirror condition. For the two remaining conditions, binocular & real (–U & +C) and monocular & mirror (+U & –C), we expect ’dangerousness’ to be in between the two extreme conditions.

We measured participants’ hand trajectories when reaching from a start point over the obstacles to the target position (see Fig. 1B). For each trajectory, we determined the peak height of the hand, i.e., the maximal vertical displacement of the hand during the movement. Figure 1C shows the peak height of the hand averaged over participants for the four experimental conditions. We analysed the peak height data using a 2 $$\times$$ 2 $$\times$$ 4 repeated-measures ANOVA with the factors costs (real vs mirror), perceptual uncertainty (binocular vs monocular) and obstacle height (3.4, 13.0, 22.6, and 32.2 mm). We found that all three factors significantly affected the peak movement height: costs ($$F(1,20)=43.643$$, $${p}<.001$$, $$\eta _p^2=0.686$$), perceptual uncertainty ($$F(1,20)=23.032$$, $${p}<.001$$, $$\eta _p^2=0.535$$), and obstacle height ($$F(1.234,24.679)=35.026$$, $${p}<.001$$, $$\eta _p^2=0.637$$). The interaction between perceptual uncertainty $$\times$$ obstacle height was statistically significant ($$F(2.433,48.666)=11.378$$, $$p<.001$$, $$\eta _p^2=0.363$$). Results for all other interactions were not significant (all $${p}\ge .202$$, $$\eta _p^2\le 0.080$$).

We also analysed two other aspects of the peak height data. First, the rate of change in peak height informs us how sensitive the motor system is to changes in obstacle height and thus provides a measure for perceptual accuracy (given that motor uncertainty remains constant). This can be measured as the slope of the peak height data in Fig. 1C. In this case, a slope of 1 would indicate that an increase in obstacle height by, e.g., 10 mm would result in an increase of the hand’s peak height of 10 mm, too, meaning that the hand is kept at a constant distance from all obstacles. In contrast, a slope of 0 would mean that the hand’s peak height does not change at all with increases in obstacle height. Second, the height of the hand over the ground surface provides an estimate of the inferred dangerousness of a condition resulting from a particular combination of perceptual uncertainty and expected costs. We will refer to the height offset that is common to all obstacles (independent of their height) in a particular condition as the baseline safety margin. It can be measured as the intercept of the peak height data in Fig. 1C. We derived slopes and intercepts by computing linear regression fits to the peak height data for each condition. The lines in Fig. 1C show regression lines based on the averaged regression parameters resulting from fits to individual participants’ data.

Figure 1D (open bars, see also Table S1 in the Supplementary Information) shows that the averaged slopes were close to one for binocular viewing and roughly halved in magnitude for monocular viewing. A 2 $$\times$$ 2 repeated-measures ANOVA with the factors costs and perceptual uncertainty showed that sensitivity to changes in obstacle height was significantly affected by perceptual uncertainty ($${F}(1,20)=17.744$$, $${p}<.001$$, $$\eta _p^2=0.470$$), with slopes being steeper for binocular (less uncertainty) than for monocular viewing (more uncertainty). There was no effect of costs on the slopes ($${F}(1,20)=0.211$$, $${p}=.651$$, $$\eta _p^2=0.010$$). The interaction between both factors was not statistically significant ($${F}(1,20)=0.326$$, $${p}=.574$$, $$\eta _p^2=0.016$$). The finding that slopes were shallower when perceptual uncertainty increased is consistent with the underestimation of sizes and distances under monocular viewing, as reported in size and distance estimation experiments7,8,9. Furthermore, the similarity of the slopes in the real and the mirror conditions indicates that sensitivity to changes in height was similar in these conditions. These results are in line with those from size and distance estimation experiments comparing estimations for real and mirrored objects19, and further confirm that mirrors provide complete and reliable visual information.

Figure 1D (filled bars, see also Table S2 in the Supplementary Information) shows that the averaged baseline safety margins, as measured by the intercepts of the lines shown in Fig. 1C, increased with perceptual uncertainty (larger for monocular than for binocular) as well as with the costs of errors (larger for real than for mirror). The intercept was largest for the monocular & real condition, and it was smallest for the binocular & mirror condition. The intercepts for the two remaining conditions (monocular & mirror and binocular & real) were similar and in between the two other conditions. This order of intercepts, and thus baseline safety margins, exactly reflects the predicted order of ’dangerousness’ for the different conditions based on the different combinations of perceptual uncertainty and error-related costs. A 2 $$\times$$ 2 repeated-measures ANOVA with the factors costs and perceptual uncertainty showed that both main effects were highly statistically significant (costs: $${F}(1,20)=31.243$$, $${p}<.001$$, $$\eta _p^2=0.610$$; perceptual uncertainty: $${F}(1,20)=45.866$$, $${p}<.001$$, $$\eta _p^2=0.696$$). There was no statistically significant interaction ($${F}(1,20)=0.443$$, $$p=.513$$, $$\eta _p^2=0.022$$) between the two factors suggesting that the effects of perceptual uncertainty and costs on safety margins are independent of each other.

## Discussion

In the following, we will first discuss how these findings advance our knowledge about sensorimotor control and sensorimotor decision making, and then elaborate on what they imply for the use of VR/AR as a tool for investigating or training motor behaviour.

### Implications for sensorimotor control

In grasping, increased perceptual uncertainty about the true size and position of a target seems to reliably cause an overall increase in maximum-grip-aperture, i.e., the largest distance between thumb and index finger when approaching an object10,11,20,21. Specifically, the effects that the removal of binocular cues has on hand movements have been relatively well studied in this context10,13,22,23. As grasping movements are very finely tuned to object size24,25, which is underestimated in monocular viewing, one could expect that grip apertures decrease when grasping objects monocularly. While decreased grip apertures were observed by Servos et al.23, subsequent studies have consistently revealed larger apertures in monocular viewing conditions10,12,13,22. A convincing explanation for this seeming paradox was provided by Jackson et al.10 who found that, consistent with the expected underestimation of object size, anticipatory grip forces decreased in monocular grasping while grip apertures increased. Consequently, they argued that the overall increase in the hand opening during monocular viewing constitutes a recalibration of safety margins due to the increased perceptual uncertainty, rather than an overestimation of object size. Our results support this argument by showing that, while increases in obstacle size were indeed underestimated in monocular reaching (as evident in shallower slopes), safety margins increased (as indicated by larger intercepts).

While findings on the hand apertures seem quite consistent across studies, the effects on other components of the hand trajectory have been less robust. That is, studies find conflicting results on whether monocular viewing also disrupts the transport component of the grasp, i.e., the part of the movement where the hand approaches the object to get into grasping range. While some argued that hand transport is unaffected by the removal of binocular information12, others reported prolonged movement times and decreased accuracy in approach parameters13,23,26 suggesting that the reaching movement also critically depends on the availability of binocular information. Here, we found that the effects of sensitivity to obstacle height and safety margins on hand transport in our reaching task reflected exactly those effects previously reported for the maximum grip aperture in grasping tasks. This congruency in action adaptation in response to increased perceptual uncertainty emphasises the role of task relevance. While object size is largely irrelevant for the reaching component of the trajectory during grasping, our variations in obstacle height (which were essential for obstacle avoidance) strongly and consistently influenced behaviour.

Previous research has shown that potential consequences of actions have an effect on the magnitude of the avoidance response18,27,28, but in all these studies the manipulation of consequences resulted from a change in the appearance of the obstacles. Here, we were able to separate the effects of consequences and appearance and found that both independently influenced the safety margins. This independence of the two effects would not necessarily have been expected, because when there are no consequences to the inaccurate execution of movements, perceptual uncertainty might matter less, or not at all. Our results fit with those from studies on sensorimotor decision-making where, in the absence of penalties, increased motor uncertainty or decreased spatial distances between reward and penalty region did not influence the endpoints of pointing movements, whereas in conditions with penalties they did29. Our results show that perceptual uncertainty does not become irrelevant to actions in the absence of error-related costs, suggesting that there are two separate processes influencing the safety margin: One that adjusts movements due to purely perceptual criteria and one that adjusts for other—higher level—task relevant factors, that we have described as the consequences of actions. The perceptual process is oblivious to the differences between mirrored and real presentations because perceptually they are identical. This is supported in our results by the similar slopes for the mirror and real conditions. The second process only evaluates the higher-level aspects.

With respect to sensorimotor decision making6, our results show that even in a task without a speed-accuracy trade-off, feedback, explicit payoff values or a specific instruction to maximize gains, participants still seem to strive for behaviour that balances safety concerns and efforts. Of course, we cannot make any specific predictions about the optimality of these movements, but we can make a couple of related observations. Since participants performed the task without tight time constraints (see Methods for details), it would have been possible for participants to choose fixed, broad safety margins, across the whole experiment, that were large enough for the effects of perceptual uncertainty on the estimation of (real) obstacle height to be safely ignored. They did not. Similarly, in the mirror condition, participants could have ignored differences in height between the obstacles because there would have been no consequences if they had been moving too low. However, this is also not what their movements showed. They seemed to avoid making unnecessarily high movements, which would require more effort, by only making the adjustments necessary to compensate for differences in sensitivity or inferred costs. It is difficult to say to what degree the adjustment due to differences in costs is under deliberate control, but the persistent difference between trajectories in the monocular and binocular viewing conditions independent of the costs suggests an automatic behaviour. Even if participants in the mirror condition somehow felt compelled to mimic passing real obstacles, it would be very unlikely that they could deliberately have simulated the differences between monocular or binocular viewing.

### Implications for VR/AR

Our results showed that, first, avoidance behaviour differed in natural and virtual environments and second, that this difference was not alone the result of impoverished visual stimulation. From this it follows that the behavioural differences between natural and virtual environments will not simply disappear with improvements in display hard- and software.

The concept of (virtual) presence, i.e., the “experience of being there”, has been suggested as a metric for the quality of virtual environments although there seems little agreement about how virtual presence should be operationalised and measured30,31. Meehan et al.32 explored physiological parameters as operationalisations of virtual presence. They investigated whether stressful virtual environments can induce changes in physiological parameters and found that changes in heart rate satisfied their requirements for a measure of presence. Here, we used the correspondence of actions (obstacle-avoidance behaviour) in natural and virtual environments as a measure of how well a virtual environment could potentially simulate a natural environment. While the appropriateness of a metric for presence will likely be task-dependent, for many practical applications action-correspondence might be a relatively easily attainable metric with high ecological validity.

While the majority of behavioural research into VR/AR has focussed on perceptual aspects of virtual environments, particularly the implementation of naturalistic 3D information, much less is known about the effects of ’virtuality’ itself, i.e., the immaterial nature of environments and objects, on behaviour1. The systematic investigation of these effects on behaviour with currently available VR/AR devices is difficult because the effects of impoverished visual stimulation and virtuality are confounded. For example, if we find an increase in safety margins in an obstacle avoidance task when using a VR/AR device, we cannot easily determine if and to what degree this increase was caused by the impoverished visual stimulation and/or by different expectations of consequences due to the virtual nature of the obstacle. Our mirror reflection manipulation was specifically designed to isolate these effects and investigate them under otherwise identical experimental conditions. This comparison of behaviours in real and mirrored environment can be thought of as a ’benchmark’ of how much similarity between actions in natural and simulated environments can possibly be achieved if identical visual information was available in natural and simulated environments. The lower baseline safety margins that we found for the mirrored obstacles compared to the real obstacles indicate that the virtual nature of the obstacles influences movement trajectories independently from the effects of perceptual uncertainty. We suggest that the critical aspect that causes this difference in behaviour is the different expectations of error-related costs. In particular, the absence of any direct consequences resulting from a collision with virtual obstacles (when viewing mirror reflections), resulted in riskier behaviour, i.e., smaller baseline safety margins. The issue of expected consequences therefore has to be considered when VR/AR devices are used to study behaviour and—more importantly—when they are used to train or control actions that have to be executed precisely and where the consequences of potential mistakes can be severe, e.g., medical application like robot-assisted surgery33,34.

A challenge for the development of VR/AR is whether it is possible to compensate for the absence of direct consequences to movement errors. For example, in the case of our experiment, would it be possible to achieve similar movement trajectories for real and mirrored obstacles, e.g., by simulating the missing haptic feedback in case of a collision with the mirrored obstacle using a force-feedback device or by introducing monetary penalties? A further aspect, which our experiment did not address directly, are the influences of motor uncertainty and biomechanical effort. Both have been shown to shape movements as well15,16,17. This will be especially relevant if users need to move to control virtual representations, e.g., pointers or simulated limbs. In this case, the relation between effort and movement magnitude or direction will differ from their natural relation. Obviously, for many applications of VR/AR exactly this is intended and highly desirable. However, for situations where we want behaviour acquired and trained in VR/AR to transfer in the expected way to their corresponding real-world counterparts, these differences need to be considered. For example, if in an obstacle avoidance task making a large avoidance movement required a smaller increase in effort in VR than in natural environments, then safety margins might be trained to be larger despite similar perceived danger. To sum up, isolating relevant sensory information is only one part of the challenge that VR/AR faces, identifying other task relevant factors like error-related costs or biomechanical effort is also crucial when, for specific purposes, we require similar actions in natural and virtual environments.

In conclusion, our results strikingly demonstrate that even when exact perceptual correspondence between natural and simulated environments is achieved, action correspondence does not necessarily follow due to a disparity in the expected consequence of actions in the two environments.

## Methods

### Setup

We used a custom-made mirror setup (Fig. 2) to present the stimuli. The mirror setup consists of a box (W 56 cm $$\times$$ D 56 cm $$\times$$ H 68 cm). The box is separated into a top and bottom part by a semi-transparent mirror (56 cm $$\times$$ 40 cm) placed in the vertical centre of the box, i.e., the workspace below the mirror in which in the participants’ hand moved had a height of approximately 34 cm. Obstacles were placed below the mirror (real condition, Fig. 2 left) or upside-down above the mirror (mirror condition, Fig. 2 middle). Participants sat in a height-adjustable chair in front of the setup with their head on a chin-rest looking down into the mirror. Regardless of whether the stimuli were placed below or above the mirror, the participants always saw them in the same location and orientation below the mirror. In the real condition, two battery-operated LED lamps were placed above the obstacles but below the mirror illuminating the obstacles from the front and the right, while in the mirror condition the lamps were placed above the mirror illuminating the mirror images of the obstacles from the front and the right. The start position for the participants' index finger was 3 cm in front of the obstacles and the target location, indicated by a yellow circular marker, was located 6 cm beyond the obstacles. The distance from the start position to the target position was 28 cm. Through an opening in the left side of the box, hand movements were measured using an infra-red based Optotrak 3020 system (Northern Digital Incorporation, Waterloo, Ontario, Canada) with a sampling rate of 200 Hz tracking one infra-red light emitting diode that was attached to the nail of the index finger of the right hand. Participants’ vision was occluded using liquid crystal shutter goggles (PLATO Translucent Technologies, Toronto, Ontario35). The experiment was programmed in MATLAB (Mathworks, Natick, MA, USA) using the Optotrak Toolbox36.

### Stimuli

We used obstacles of four different heights (Table 1 and Fig. 2, right). Based on pilot experiment, we chose the height of the highest obstacle so that the height of the avoidance movements would not be limited by the height of the workspace (34 cm). The obstacles were constructed from LEGO material. The lowest obstacle consisted of an empty grey LEGO base board (W 383 mm $$\times$$ D 190 mm $$\times$$ H 3.4 mm). The studs on the base board had a height of 1.8 mm. The plane part of the base board had a height of 1.6 mm. The other obstacles were created by erecting six rows of 10 LEGO towers consisting of one, two, or three white square LEGO bricks (W 16 mm, D 16 mm, H 11.4 mm), respectively, on top of the base board. The studs on the bricks had a height of 1.8 mm. The bricks without the studs had a height of 9.6 mm. The distance between towers was 16 mm. Overall the area covered by the bricks was W 303 mm $$\times$$ D 175 mm.

### Procedure

Ten of the participants started with the mirror condition and 11 started with the real condition. Mirror and real conditions were tested in separate sessions (separated by at least one day). Within each condition, participants always started with the monocular viewing condition followed by the binocular condition. In the monocular condition, the participants’ non-dominant eye was occluded with an eye-patch. The shutter goggles were closed between trials when the experimenter positioned the obstacle boards. They opened together with an auditory go-signal at the beginning of each trial. If in a trial participants started before the auditory signal or within 100 ms after the signal, the trial was excluded and repeated at a random position within the experiment. As soon as the participants’ right hand started to move, i.e., the 3D Euclidean distance between the marker and the start position exceeded 25 mm, the glasses closed again. Thus, all movements were performed visually open-loop. The four different obstacle heights were presented in pseudo-randomized order in each condition. There were ten repetitions per obstacle height. In total, each participant performed 160 trials. Before the start of each trial, the experimenter changed the obstacle board placing it either above or below the mirror. The participants were instructed to move their hand over the obstacles presented in front of them and to touch the target patch placed beyond the obstacle board and return to the start position. Participants were told to wait for the auditory signal and start moving their hand as soon as possible. There was no instruction regarding the speed of the movements but the movement should have been finished within 3 s. Trials where the movement duration exceeded 3 s were excluded and repeated. Participants were encouraged to be as accurate as possible in hitting the target and to make natural hand movements, but there was never any feedback regarding their accuracy. Participants were not informed about the different presentation conditions, i.e., mirror and real, because telling them explicitly about the difference between the mirror and real conditions could potentially have biased their avoidance behaviour in the hypothesised direction. By not informing participants, we reduced the chances of detecting a difference between these conditions. However, we deliberately did not try to conceal this difference from the participants. There were several ways in which they could have detected the differences between the mirror and real conditions. Obviously, they could just have found out by trying to touch the obstacles in the mirror condition. Although vision was occluded between trials, auditory cues could have indicated whether the experimenter exchanged the stimuli at the top or the bottom of the mirror setup (see Fig. 2). Another difference between the mirror and the real conditions was the placements of the two lamps which illuminated either the real obstacles or the mirror and hence were placed either below or above the mirror. Since the mirror and real conditions were tested in different sessions on different days, participants might not have noticed this difference. After the experiment, we asked participants whether they had noticed a difference between the mirror and real conditions. Of 21 participants, 16 participants reported having noticed the difference, and five reported having not noticed the difference. We decided to present the data of the complete dataset since we think it would be inappropriate to separate the dataset based on the informal interview of the participants after the experiment. Moreover, we cannot be absolutely certain whether the five participants who did not report a difference between the mirror and real conditions were not aware of the difference between the conditions, or whether they were aware of it but did not perceive a (visual) difference. In the Supplementary Information, we present the mean slopes and intercepts as well as statistical analysis separately for the groups of participants who reported a difference between the mirror and real conditions and the participants who did not report a difference (Figure S1 and Tables S1-S3 in the Supplementary Information).

### Participants

We collected data from 21 right-handed participants (age range: 19–39 years, mean (SD) age: 26.5 (5.62)). At the beginning of the first experimental session, the participants’ dominant eye was determined. Participants were asked to extend their arms out in front of them and create a triangular opening between their thumbs. With both eyes open, participants were asked to centre this triangular opening on a red light on the wall (distance from participant to the wall was three meters). Then participants were asked to close their left eye. If the object (the red light) stayed centred, their right eye (the one that was open) was determined as the dominant eye, otherwise the left eye was determined as dominant. For 16 participants the right eye was dominant, for five the left eye. Participants were naive as to the purpose of the experiment and were compensated £10 for their time. The experimental procedures used were in accordance with the Ethics Code of the British Psychological Society, the declaration of Helsinki and approved by the Psychology Ethics Committee of the University of Aberdeen (Ethics code: PEC/4250/2019/7). All participants gave written informed consent.

### Data analysis

The 3D position data of the index finger marker was filtered offline using a second-order Butterworth-filter with a low-pass cut-off frequency of 15 Hz. Trials were rejected when the hand movement started before the auditory start signal, the movement was not finished within 3 s, or the marker was not visible for longer durations during the movement resulting in missing data. Of a total of 3360 trials, 32 trials were excluded from the data analysis. For data analysis, we used the trajectories in the vertical direction (z-dimension) and determined for each trajectory the maximum z-value (peak height) reached between the start and end of the reaching movement. Start and end of the movement were determined using a resultant velocity threshold of 0.05 m/s. For each participant and condition, we computed the simple linear regression using a least-squares criterion to predict peak movement height based on obstacle height. The resulting slopes and intercepts were then averaged over participants separately for each condition.

For statistical analysis, movement data were analysed using a 2 $$\times$$ 2 $$\times$$ 4 repeated-measures ANOVA with the factors perceptual uncertainty (binocular vs monocular), costs (mirror vs real), and obstacle height (3.4, 13.0, 22.6, and 32.2 mm). Intercepts and slopes were analysed using a 2 $$\times$$ 2 repeated-measures ANOVA with the factors perceptual uncertainty and costs. If Mauchly’s Test of Sphericity indicated that the assumption of sphericity had been violated ($${p}<.05$$), Greenhouse–Geisser corrected values were used (indicated by fractional degrees-of-freedom).

To test whether there was a difference between those participants who started with the mirror condition and those who started with the real condition, for both intercepts and slopes 2 $$\times$$ 2 repeated-measures ANOVAs with the within-subject factors perceptual uncertainty and costs and the between-subject factor order-of-presentation were run. There was neither a significant main effect of order-of-presentation on intercepts ($${F}(1,19)=1.646$$, $${p}=.215$$, $$\eta _p^2=0.080$$; all interactions: $${p}\ge .234$$, $$\eta _p^2\le 0.074$$) nor slopes ($${F}(1,19)=0.076$$, $${p}=.786$$, $$\eta _p^2=0.004$$). For slopes, there was a significant interaction between costs and order-of-presentation ($${F}(1,19)=15.083$$, $${p}<.001$$, $$\eta _p^2=0.443$$; all other interactions: $${p}\ge .123$$, $$\eta _p^2\le 0.121$$).

## Data availability

The data are available online from the Open Science Framework: https://osf.io/vskxy/.

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Hesse, C. & Franz, V. H. Memory mechanisms in grasping. Neuropsychologia 47, 1532–1545. https://doi.org/10.1016/j.neuropsychologia.2008.08.012 (2009).

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Keefe, B. D., Hibbard, P. B. & Watt, S. J. Depth-cue integration in grasp programming: no evidence for a binocular specialism. Neuropsychologia 49, 1246–1257 (2011).

23. 23.

Servos, P., Goodale, M. A. & Jakobson, L. S. The role of binocular vision in prehension: a kinematic analysis. Vis. Res. 32, 1513–1521 (1992).

24. 24.

Smeets, J. B. J. & Brenner, E. A new view on grasping. Mot. Control 3, 237–271 (1999).

25. 25.

Jeannerod, M., Long, J. & Baddeley, A. Intersegmental Coordination During Reaching at Natural Visual Objects Vol. 9, 153–168 (Erlbaum, Hillsdale, 1981).

26. 26.

Marotta, J. J., Perrot, T. S., Nicolle, D., Servos, P. & Goodale, M. A. Adapting to monocular vision: grasping with one eye. Exp. Brain Res. 104, 107–114 (1995).

27. 27.

Kangur, K., Billino, J. & Hesse, C. Keeping safe: intra-individual consistency in obstacle avoidance behaviour across grasping and locomotion tasks. i-Perception8, 2041669517690412 (2017).

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Giesel, M., Kangur, K., Harris, J. M. & Hesse, C. Investigating the influence of surface properties on reaching movements. J. Vis. 19, 252a–252a (2019).

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Trommershäuser, J., Gepshtein, S., Maloney, L. T., Landy, M. S. & Banks, M. S. Optimal compensation for changes in task-relevant movement variability. J. Neurosci. 25, 7169–7178 (2005).

30. 30.

Sheridan, T. B. Musings on telepresence and virtual presence. Presence Teleoper. Virtual Environ.1, 120–126 (1992).

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Ellis, S. R. Presence of mind: a reaction to thomas sheridan’s “further musings on the psychophysics of presence”. Presence Teleoper. Virtual Environ. 5, 247–259 (1996).

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Meehan, M., Insko, B., Whitton, M. & Brooks, F. P. Jr. Physiological measures of presence in stressful virtual environments. ACM Trans. Graph. 21, 645–652 (2002).

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Peters, B. S., Armijo, P. R., Krause, C., Choudhury, S. A. & Oleynikov, D. Review of emerging surgical robotic technology. Surg. Endosc. 32, 1636–1655 (2018).

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## Acknowledgements

Supported by Leverhulme Trust Grant RPG-2017-232 awarded to CH and JH.

## Author information

Authors

### Contributions

M.G. and C.H. conceived the experiment, M.G. and C.H. coded the experiment, M.G. and C.H. designed the analysis, A.N. built the stimuli, A.N. conducted the experiment, M.G., A.N., and C.H. analysed the results. M.G., J.H., and C.H. wrote the manuscript. All authors reviewed the manuscript.

### Corresponding author

Correspondence to Martin Giesel.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

### Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Giesel, M., Nowakowska, A., Harris, J.M. et al. Perceptual uncertainty and action consequences independently affect hand movements in a virtual environment. Sci Rep 10, 22307 (2020). https://doi.org/10.1038/s41598-020-78378-z

• Accepted:

• Published:


text:
# Intersection graphs for “conflicting” directed paths in trees

Given an undirected tree and a set of directed paths in this tree (or equivalently, ordered $s$–$t$ pairs), we construct a graph with the paths as vertices and an edge between two paths if they traverse an edge of the tree in opposite directions. Is anything known about the resulting graphs? I found some similar settings, but none that exactly matches. Is this class equivalent to a known one, or does it have interesting characterizations or properties? The only thing I could find is that these graphs cannot have a $K_4$ as subgraph.

-

## 1 Answer

This may count only as a "similar setting," but Martin Golumbic has studied what he calls the edge intersection graph of a tree $T$, which has a node for each path in $T$, and an arc between two nodes if their paths share at least one edge. So his paths are not directed. He and coauthors have established a number of structural, coloring, and complexity properties of these "EPT" graphs, under various assumptions (e.g., on vertex degrees). Here are three references:

• M.C. Golumbic, R.E. Jamison, Edge and vertex intersection of paths in a tree, Discrete Mathematics 55 (1985), 151-159. (Elsevier link)

• Golumbic, Lipshteyn, and Stern, "Representing Edge Intersection Graphs of Paths on Degree 4 Trees," Discrete Mathematics, 2008. (DeepDyve link).

• Golumbic, Lipshteyn, and Stern, "The k-edge intersection graphs of paths in a tree," Discrete Applied Mathematics, Volume 156, Issue 4, 2008. (Elsevier link)

-

format: plain


text:
Usted está aquí: The BG simulation: an algorithmic tool for impossibility results

# The BG simulation: an algorithmic tool for impossibility results

Damien Imbs (IM-UNAM) - martes 4 de septiembre, 12 horas
Ponente: Damien Imbs (IM-UNAM)
Cuándo 04/09/2012 de 12:00 a 13:00 Salón "Graciela Salicrup" vCal iCal

Resumen:

In distributed computing, some models do not allow solving all problems. There are various ways to prove such impossibilities. In this talk, we will see a purely algorithmic tool for proving such impossibilities: the BG simulation. This simulation shows that, in an asynchronous system, the important parameter with regard to calculability is not the total number of cooperating processes; it is how many of them can fail.


text:
# Synopsis: The nuclear option

Optical clocks based on nuclear energy levels may offer unprecedented precision and a chance to find out whether fundamental constants change over time.

Atomic clock makers look for two important characteristics when they choose atoms to build their time pieces. First, the energy levels that generate the clock signal must be narrow and well defined. And second, the energy levels should be immune from external electromagnetic fields so that the clock rate doesn’t vary with, for example, position near Earth.

Several atomic species with these qualities exist, but researchers would like to take optical clocks to an even higher level of precision and stability. One approach would be to find nuclear energy levels with the right features for clocks. Nuclei have extremely sharp energy levels, but typically the excitation energies range from kilo-electron-volts to mega-electron-volts—far above the several-$\text{eV}$ photon energies that are available to make optical clocks. Now, Corey Campbell, Alexander Radnaev, and Alexander Kuzmich of the Georgia Institute of Technology report in Physical Review Letters their studies of thorium nuclei that may be just right for clock building.

One likely candidate, thorium-$229$, has a pair of low-lying nuclear states separated only by $7.6\phantom{\rule{0.333em}{0ex}}\text{eV}$. As a first step in harnessing these levels, the authors trap and cool the triply charged form of $\text{Th}$-$229$ into a regular array called a Wigner crystal, which allows spatial localization of each nucleus. Future work will include addressing single ions in the crystal and spectroscopic studies of the nuclear clock transition. This could not only lead to clocks of unusual precision, but it might also help answer questions about whether fundamental physical constants are changing over time. – David Voss

### Announcements

More Announcements »

## Subject Areas

Atomic and Molecular Physics

## Previous Synopsis

Biological Physics

Soft Matter

## Related Articles

Atomic and Molecular Physics

### Synopsis: Losing Light in a BEC

The index of refraction in a gas of bosons is enhanced relative to its value in a classical gas, a predicted quantum effect that has now been observed in ultracold sodium atoms. Read More »

Atomic and Molecular Physics

### Focus: Atoms As Thermometers

A small number of atoms in repeated trials can accurately measure the temperature of an ultracold gas cloud—a step toward measuring temperature on the micrometer scale. Read More »

Optics

### Synopsis: Bad Cavities for Precise Lasers

The frequency of a laser based on trapped ultracold atoms can be made insensitive to fluctuations in the laser cavity’s length. Read More »


text:
Opened 3 years ago

Closed 2 years ago

# Make a top-level table/Table function

Reported by: Owned by: kcrisman was major sage-5.10 user interface jason, jhpalmieri, abmasse, novoselt, andrew.mathas sage-5.10.beta0 John Palmieri Jason Grout, Karl-Dieter Crisman, Sébastien Labbé N/A

As the summary says.

See this sage-devel thread for more suggestions, including that we should have a Table class that table calls, which uses something like html.table in the notebook, but also has a _latex_ method, makes ascii art in the command line, etc.

Just a few examples from the doctests in the patch:

        sage: table([['a', 'b', 'c'], [100,2,3], [4,5,60]])
a     b   c
100   2   3
4     5   60
sage: table([['a', 'b', 'c'], [100,2,3], [4,5,60]], frame=True)
+-----+---+----+
| a   | b | c  |
+-----+---+----+
| 100 | 2 | 3  |
+-----+---+----+
| 4   | 5 | 60 |
+-----+---+----+

sage: latex(table([['a', 'b', 'c'], [100,2,3], [4,5,60]], frame=True))
\begin{tabular}{|l|l|l|} \hline
a & b & c \\ \hline
$100$ & $2$ & $3$ \\ \hline
$4$ & $5$ & $60$ \\ \hline
\end{tabular}

You can also specify alignments of 'left', 'center', or 'right'.

### comment:2 Changed 2 years ago by jhpalmieri

• When defining a table, the user should be able to specify position/alignment (left, right, center). Should they be able to do this for each column, and do we like LaTeX notation for this? Should they be able to specify positioning for each individual entry?
• Is it too much work to try to align numbers at their decimal places? LaTeX and LaTeX ways to do this. html and html ways to do this, but perhaps complicated or messy.
• Should we refactor the matrix printing code so that it uses this class?
• html.table expects a nested list as input, and so the entries in [1,2,3] get plotted as separate rows. is that the right thing, or should [1,2,3] get plotted as a single row?
• We should probably steal code from html.table, matrix._repr_, matrix.str, matrix._latex_
• Should we allow subtables (like subdivisions in matrices)?

### comment:3 Changed 2 years ago by jason

Sure. But minimally, how about: table(list of rows, header=header row). I think we can definitely start simple here.

### comment:4 Changed 2 years ago by jhpalmieri

• Authors set to John Palmieri
• Description modified (diff)
• Status changed from new to needs_review

Okay, here's a first attempt. For some reason I find the syntax table(array, header=another_row) strange, so this just allows for the header argument to be True or False, and the actual row needs to be the first row of array.

### comment:5 Changed 2 years ago by jhpalmieri

To get header_column to have any effect in HTML, we also will need to patch the Sage notebook, to modify the file devel/sagenb/sagenb/data/sage/css/main.css:

• ## main.css

 old table.table_form * td { padding-top: 5px; padding-bottom: 5px; padding-left: 15px; padding-right: 15px; } /* line 899, ../../../../sass/src/main.scss */ table.table_form * th { background: #a6ba4e; height: 29px; padding-left: 15px; padding-right: 15px; padding-top: 5px; padding-bottom: 5px; color: white; text-align: left; } table.table_form * th.ch { background: #7c8b3a; } table.table_form * td.ch { background: #a6ba4e; color: white; } /* line 909, ../../../../sass/src/main.scss */ table.table_form * tr.row-a { background: #f8f8f8; text-align: left; } /* line 912, ../../../../sass/src/main.scss */

I guess this means a change to devel/sagenb/sass/src/main.scss, but I'm not sure of the syntax in that file. Maybe this:

• ## main.scss

 old padding-top: 5px; padding-bottom: 5px; padding-left: 15px; padding-right: 15px; } padding-right: 15px; &.ch { background: #a6ba4e; color: white; } } th { background: #a6ba4e; height: 29px; padding-top: 5px; padding-bottom: 5px; color: white; text-align: left; } text-align: left; &.ch { background: #7c8b3a; } } tr { &.row-a { background: #f8f8f8;

This is not a big deal and the ticket can be merged independently of such a change. I may get around to putting up a pull request for the notebook, but no guarantees.

### comment:6 Changed 2 years ago by jhpalmieri

Oh, one more thing, at least for now: the previously existing technique of using html.table(...) treated a single list as a column, whereas this patch treats a single list as a row instead. This feels more natural to me. Anyone who really wants either a single row or a single column should use nested lists properly anyway.

### comment:7 Changed 2 years ago by kcrisman

My refrain of the last few months to your patches has been "I don't have time to give it a proper review, but this looks great on the face of it", so why not repeat it? I look forward to using this, and I know a lot of other people who rely on the equivalent way too much in Mma will love it.

Last edited 2 years ago by kcrisman (previous) (diff)

### comment:8 Changed 2 years ago by jason

Right now html.table(data, header=True) treats the first row as the header, but html.table(data, header=['label1', 'label2']) will let you specify the header independently. This is *extremely* useful. Usually I have a table of numbers or something, and being able to just specify the header row separately makes a huge difference in clarity.

### comment:9 Changed 2 years ago by jhpalmieri

I think that

table([head] + data, header_row=True)

is just about as clear as

table(data, header_row=head)

especially since head comes first in the first version, consistent with its position in the table. But here's a new patch which implements both, and the same for header_column.

### comment:10 Changed 2 years ago by jason

Thanks! I was going to submit a follow-up patch to add that, but thanks for taking care of it!

### comment:11 Changed 2 years ago by jhpalmieri

By the way, doing something like

table(data, header_row=['a', 'b', 'c'], header_column=['x', 'y'])

is just strange. I can explain what it would do – first add the header row at the top of the data, then add the header column on the left – but in the docstring I just said that doing this is "not supported". Someone can specify True for one argument and a list for the other, or True for both, but I don't want to document using a list for both.

### comment:12 Changed 2 years ago by jhpalmieri

New patch, expanding on the documentation of the options method, and also running a Testsuite.

### comment:14 follow-ups: ↓ 17 ↓ 21 Changed 2 years ago by kcrisman

• Reviewers set to Jason Grout, Karl-Dieter Crisman
• Status changed from needs_review to needs_work

Some dumb questions.

• elif header_row is not False: why not just elif header_row? Presumably a Python subtlety.
• Do we need something in html.table() (what remains of it) to change it so that the previous behavior of a single list giving a column and not a row is preserved? I know you removed the example, but technically speaking one should deprecate this...
• Is there an extra
self._options['header_column'] = header_column

in there? It should be set already above that.
• I don't see any error catching. This is particularly important in making sure someone doesn't (contra the doc, but still) try to add in a header row later on via header_row=[1,2,3] and in making sure that the table is, in fact, a rectangle. I think that for sanity, especially with very large tables, this would be helpful to have this error instead of who-knows-what-crazy-error Sage would raise otherwise.
• Feature request; take a list of lists and just fill in the rest! I had to make my own code for this in some sense, filling in empty slots, if I recall correctly. But presumably this is like the list in comment:2 and for a future ticket.

### comment:15 follow-up: ↓ 16 Changed 2 years ago by kcrisman

I was just speaking with Andrew Mathas of the Sage-combinat group at Sage Days at ICERM and it turns out that he has a pretty impressive (minor bugs remaining) table object as well, which takes any iterable as a header column with indexing and has a slew of formatting options. Although the current functionality is almost ready for showtime, it would be useful to have some dialogue about the relation between these things, especially because some of the formatting stuff is nice. He also has a nice hook to ask for displaying a table in a browser automatically (html gets sent to the browser).

Last edited 2 years ago by kcrisman (previous) (diff)

### comment:16 in reply to: ↑ 15 Changed 2 years ago by kcrisman

I was just speaking with Andrew Mathas of the Sage-combinat group at Sage Days at ICERM and it turns out that he has a pretty impressive (minor bugs remaining) table object as well, which takes any iterable as a header column with indexing and has a slew of formatting options. Although the current functionality is almost ready for showtime, it would be useful to have some dialogue about the relation between these things, especially because some of the formatting stuff is nice. He also has a nice hook to ask for displaying a table in a browser automatically (html gets sent to the browser).

See #14103.

### comment:17 in reply to: ↑ 14 Changed 2 years ago by kcrisman

• Feature request; take a list of lists and just fill in the rest! I had to make my own code for this in some sense, filling in empty slots, if I recall correctly. But presumably this is like the list in comment:2 and for a future ticket.

This already works. I think I was wanting to have this work better, so that I didn't have to use map(None,...) and then replace with ''.

@interact
def _(n=7):
L = map(None,[p for p in prime_range(n+1) if p%4==1],[p for p in prime_range(n+1) if p%4==3])
L = [['',l[1]] if l[0] is None else l for l in L]
T = [['$p\equiv 1\\text{ mod}(4)$','$p\equiv 3\\text{ mod}(4)$']]

But probably that's irrelevant.

### comment:18 Changed 2 years ago by jason

Whew. I would probably do that either of these two ways (the second way is more efficient, but really you could even do better by caching the table from run to run)

from itertools import izip_longest
from collections import defaultdict

@interact
def _(n=7):
L = list(izip_longest([p for p in prime_range(n+1) if p%4==1],[p for p in prime_range(n+1) if p%4==3],fillvalue=''))
html.table(L,header=['$p\equiv 1\\text{ mod}(4)$','$p\equiv 3\\text{ mod}(4)$'])

@interact
def _(n=7):
m = defaultdict(list)
for p in prime_range(n+1):
m[p%4].append(p)
L = list(izip_longest(m[1], m[3],fillvalue=''))
html.table(L,header=['$p\equiv 1\\text{ mod}(4)$','$p\equiv 3\\text{ mod}(4)$'])

### comment:19 Changed 2 years ago by kcrisman

Hmm, so you're saying it's not really a failing in html.table?

### comment:20 Changed 2 years ago by jason

If html.table took a list of columns, then I think you could argue your case. Since it takes a list of rows, you have to pad the columns with the empty cells.

If you formatted your table as two rows, then I think html.table should (but probably doesn't) format different-sized rows.

On the other hand, it seems useful to have a columns option of some sort to the table. Or maybe we can just use pandas (eventually) for more sophisticated formatting.

### comment:21 in reply to: ↑ 14 Changed 2 years ago by jhpalmieri

Some dumb questions.

• elif header_row is not False: why not just elif header_row? Presumably a Python subtlety.

I think that can be changed.

• Do we need something in html.table() (what remains of it) to change it so that the previous behavior of a single list giving a column and not a row is preserved? I know you removed the example, but technically speaking one should deprecate this...

Technically speaking, I think you're right. I don't like the old behavior at all; it doesn't make sense to me. It might actually make sense to just raise an error if you pass a single list, not a nested list. I'm not sure what the right thing to do is.

• Is there an extra
self._options['header_column'] = header_column

in there? It should be set already above that.

Yes, you're right.

• I don't see any error catching. This is particularly important in making sure someone doesn't (contra the doc, but still) try to add in a header row later on via header_row=[1,2,3]

So you're suggesting that

sage: T = table([[1,2,3], [4,5,6]])

should raise an error instead of just silently making [1,2,3] the header? Okay, I guess that makes sense.

and in making sure that the table is, in fact, a rectangle. I think that for sanity, especially with very large tables, this would be helpful to have this error instead of who-knows-what-crazy-error Sage would raise otherwise.

• Feature request; take a list of lists and just fill in the rest! I had to make my own code for this in some sense, filling in empty slots, if I recall correctly. But presumably this is like the list in comment:2 and for a future ticket.

I'm not sure how these last two items relate to each other: the last one sounds like you want to be able to pass lists of different lengths and have the gaps filled in, and the previous one sounds like you want to raise an error if you pass lists of different lengths. But maybe I don't understand.

Would it be useful to have a transpose method to switch rows and columns? Here's a new patch which adds that, and also addresses all of the issues except the old behavior of html.table() and the last issue about a non-rectangular situation.

### comment:22 Changed 2 years ago by jhpalmieri

I also added documentation (in the eval method in html.py) for calling the _html_ method for an object, if it has one. This is analogous to the _latex_ method. I think this could be useful in other situations.

### comment:23 Changed 2 years ago by jhpalmieri

• Status changed from needs_work to needs_review

### comment:24 follow-ups: ↓ 26 ↓ 27 ↓ 28 Changed 2 years ago by slabbe

Thanks for clarifying the documentation of html(_) and for making html calling _html_ if it exists.

I have two questions:

1. Doc of html now says:
In any case, *print* the resulting html string. This method
always *returns* an empty string.

What if I want to print the output in a file and not in the shell. What should I do? This is something I came across in the past.

1. I really like the table function. I am defining such a function in a patch I posted just yesterday at #13069. In my case, the function takes columns as input and not rows. So, if I want to use your code, would you propose an interface that takes columns as input:
sage: columns = [('a', 100, 4), ('b', 2, 5), ('c', 3, 60)]
sage: table(columns=columns)

or the user should transpose columns into rows himself?

sage: columns = [('a', 100, 4), ('b', 2, 5), ('c', 3, 60)]
sage: rows = zip(*columns)
sage: rows
[('a', 'b', 'c'), (100, 2, 3), (4, 5, 60)]
sage: table(rows)

### comment:25 Changed 2 years ago by novoselt

I would prefer a more straightforward method then going through zipping (which would not immediately occur to me if I was thinking of it).

### comment:26 in reply to: ↑ 24 Changed 2 years ago by kcrisman

1. I really like the table function. I am defining such a function in a patch I posted just yesterday at #13069. In my case, the function takes columns as input and not rows. So, if I want to use your code, would you propose an interface that takes columns as input:

See yet another implementation at #14103.

+1 to being straightforward.

### comment:27 in reply to: ↑ 24 Changed 2 years ago by slabbe

1. I really like the table function. I am defining such a function in a patch I posted just yesterday at #13069.

To make the reference easier, here is the function I coded at #13069 (the patch is 90 Kb).

def table_from_columns(col1, col2, space=3):
r"""
Return a string of a table of two columns.

This is used by the _repr_ method of DoubleSquare class.

INPUT:

- col1 - list
- col2 - list
- space - integer (default 3), number of space characters
between

EXAMPLES::

sage: from sage.combinat.double_square_tile import table_from_columns
sage: col2 = ['11', '1313', '131313', '1313']
sage: print table_from_columns(col1, col2)
ab         11
asdfasdf   1313
1313
"""
from itertools import izip_longest
width_col1 = max(map(len, col1))
it = izip_longest(col1,col2, fillvalue='')
L = [a.ljust(width_col1 + space) + b for (a,b) in it]
return '\n'.join(L)

### comment:28 in reply to: ↑ 24 ; follow-up: ↓ 29 Changed 2 years ago by jhpalmieri

Thanks for clarifying the documentation of html(_) and for making html calling _html_ if it exists.

I have two questions:

1. Doc of html now says:
In any case, *print* the resulting html string. This method
always *returns* an empty string.

What if I want to print the output in a file and not in the shell. What should I do? This is something I came across in the past.

This is just documentation; I haven't changed the code. If we want to change the behavior of html (maybe just adding a str method or something like that), let's do it on another ticket and keep this one focused on tables.

1. I really like the table function. I am defining such a function in a patch I posted just yesterday at #13069. In my case, the function takes columns as input and not rows. So, if I want to use your code, would you propose an interface that takes columns as input:
sage: columns = [('a', 100, 4), ('b', 2, 5), ('c', 3, 60)]
sage: table(columns=columns)

I can add this, but what should happen if you do table(array, column=columns)? Raise an error? Or try to combine the two sets of data somehow? I think I prefer to raise an error.

or the user should transpose columns into rows himself?

The easiest thing to do with the code as it stands is table(columns).transpose().

### comment:29 in reply to: ↑ 28 Changed 2 years ago by slabbe

This is just documentation; I haven't changed the code. If we want to change the behavior of html (maybe just adding a str method or something like that), let's do it on another ticket and keep this one focused on tables.

I agree. I just can't close my mouth sometimes!

I can add this, but what should happen if you do table(array, column=columns)? Raise an error? Or try to combine the two sets of data somehow? I think I prefer to raise an error.

or maybe the interface could be :

sage: table(data=rows)
...
sage: table(data=columns, orientation='columns')
...

where orientation would be an optional argument taking 'rows' as default value? This way there can't be inconsistent inputs.

The easiest thing to do with the code as it stands is table(columns).transpose().

ok

### comment:30 Changed 2 years ago by jhpalmieri

I think I like

sage: table(rows=...)
...
sage: table(columns=...)
...

because it puts the two formats on equal footing. I've attached a new patch which incorporates this. It also expands on the documentation somewhat.

### comment:31 Changed 2 years ago by jhpalmieri

• Description modified (diff)

### comment:32 follow-up: ↓ 34 Changed 2 years ago by slabbe

• Description modified (diff)
• Reviewers changed from Jason Grout, Karl-Dieter Crisman to Jason Grout, Karl-Dieter Crisman, Sébastien Labbé

Just added a patch which does nothing but removing trailing spaces (on 3 lines) and added the r to the """ to make doc be raw strings. ...sometimes I get doctest errors because I do not put this r. Maybe I am afraid of witches...

Jason, Karl-Dieter, to me this is positive review. Did you have anything else to say?

Sébastien

### Changed 2 years ago by slabbe

review: Spaces and r""" fixes

### comment:33 Changed 2 years ago by slabbe

Apply trac_13131-tables-with-columns.patch, trac_13131_review-sl.patch

### comment:34 in reply to: ↑ 32 Changed 2 years ago by kcrisman

Jason, Karl-Dieter, to me this is positive review. Did you have anything else to say?

If you looked at the latest code carefully, nothing would please me more than for this to get in. I love all the improvements since the very first version. It is probably the number 1 or 2 most-requested basic feature in Sage available at the top level, in my experience, and will be very well received.

### comment:35 Changed 2 years ago by jason

I haven't tested it, but I don't see any big red flags in the code either. Thanks!

### comment:36 Changed 2 years ago by slabbe

• Status changed from needs_review to positive_review

### comment:37 Changed 2 years ago by jdemeyer

• Milestone changed from sage-5.9 to sage-5.10

### comment:38 Changed 2 years ago by jdemeyer

• Merged in set to sage-5.10.beta0
• Resolution set to fixed
• Status changed from positive_review to closed

### comment:39 Changed 2 years ago by slabbe

see #14601 for a small follow up bug

Note: See TracTickets for help on using tickets.


text:
Partial derivative of a term should give zero

Posted 2 months ago
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 Hello! There is an equation. If I take the partial derivative from it, the part with ∂ appears in the output. This part should be automatically equal to 0. How can I fix this? Ek = (m1*Dt[y, t]*Dt[y, t])/2 + (m2*(l*Dt[\[Phi], t]*Sin[\[Phi]])^2)/ 2 + (m2*(l*Dt[\[Phi], t]*Cos[\[Phi]] + Dt[y, t])^2)/2 Simplify[D[Ek, y]] Simplify[D[Ek, \[Phi]]] Thank you in advance! Attachments:
 When you take partial derivatives with respect to y,\[Phi], these must be independent variables, not themselves functions (of what? of themselves?).


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Question Paper: Theoretical Computer Science : Question Paper Dec 2016 - Computer Engineering (Semester 4) | Mumbai University (MU)
0

## Theoretical Computer Science - Dec 2016

### Computer Engineering (Semester 4)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks.
1(a) Design a DFA over an alphabet ∑={a, b} to recognize a language in which every 'a' is followed by 'b'.(5 marks) 1(b) Give formal definition of a Push Down Automata.(5 marks) 1(c) State and explain the power and limitations of a Turing machine.(5 marks) 1(d) Design a mealy machine to determine the residue mod 3 of a binary number.(5 marks) 2(a) Convert the following NFA to an equivalent DFA

 State a b c →q0 {q0,q1} q1 {} q1 {q2} {q1,q2} {} q2 {q0} {q2} {q1}
(10 marks) 2(b) State and explain pumping lemma for regular languages. Using pumping lemma prove that the language $L=\left \{ o^n1^n|n\geq 0 \right \}$/ is not regular.(10 marks) 3(a) Design a Turing machine that computers a function f(m,n)=m+n i.e. addition of two integers.(10 marks) 3(b) Design a Turing machine to accept the language 0n1nn(10 marks) 4(a) Draw a state diagram and construct a regular expression corresponding to the following state transition table.
 State →q1 0 q1 1 q2 q2 q3 q2 q3 q1 q2
(10 marks)
4(b) State and explain decision properties of regular languages(10 marks) 5(a) i) Convert the following CFG to GNF
S→AA|a
A→SS|b
(10 marks)
5(b) Design a PDA correponding to the to the grammar S→aSA |ε
A→bB
B→b
(10 marks)

### Write a short note any two Q6.(a,b,c,d)

6(a) Recursive and Recursively Enumerable Languages.(10 marks) 6(b) Chomsky Hierarchy(10 marks) 6(c) Rice's Theorem(10 marks) 6(d) Halting problem(10 marks)


text:
Question

# The value of $\underset{x\to \infty }{\mathop{\lim }}\,{{\left( \dfrac{{{20}^{x}}-1}{19({{5}^{x}})} \right)}^{\dfrac{1}{x}}}$ is_________.

Hint: You can solve this problem by taking logarithm and after that you will come to know that you have to use L-Hospital’s Rule

Step by step solution:
We will rewrite the given equation first,
$\underset{x\to \infty }{\mathop{\lim }}\,{{\left( \dfrac{{{20}^{x}}-1}{19({{5}^{x}})} \right)}^{\dfrac{1}{x}}}$
Consider it as L,
$\therefore L=\underset{x\to \infty }{\mathop{\lim }}\,{{\left( \dfrac{{{20}^{x}}-1}{19({{5}^{x}})} \right)}^{\dfrac{1}{x}}}$
We will put limits directly,
$\therefore L={{\left( \dfrac{{{20}^{\infty }}-1}{19({{5}^{\infty }})} \right)}^{\dfrac{1}{\infty }}}$
$\therefore L={{\left( \dfrac{\infty }{\infty } \right)}^{0}}$
As it is an indeterminate form therefore we should solve it by using different method,
$\therefore L=\underset{x\to \infty }{\mathop{\lim }}\,{{\left( \dfrac{{{20}^{x}}-1}{19({{5}^{x}})} \right)}^{\dfrac{1}{x}}}$
Taking log on both sides, we will get
$\therefore \log L=\log \left[ \underset{x\to \infty }{\mathop{\lim }}\,{{\left( \dfrac{{{20}^{x}}-1}{19({{5}^{x}})} \right)}^{\dfrac{1}{x}}} \right]$
As we all know that ‘log’ can be inserted in limits,
$\therefore \log L=\underset{x\to \infty }{\mathop{\lim }}\,\left[ \log {{\left( \dfrac{{{20}^{x}}-1}{19({{5}^{x}})} \right)}^{\dfrac{1}{x}}} \right]$
To proceed further we should know some rules of logarithm which are given below,
Formulae:
1.$\log ({{m}^{n}})=n\times \log m$
2.$\log \left( \dfrac{m}{n} \right)=\log m-\log n$
3.$\log \left( m\times n \right)=\log m+\log n$
By using formula 1 we can write log L as shown below,
$\therefore \log L=\underset{x\to \infty }{\mathop{\lim }}\,\left[ \dfrac{1}{x}\times \log \left( \dfrac{{{20}^{x}}-1}{19({{5}^{x}})} \right) \right]$
By using formula 2 we can write log L as shown below,
$\therefore \log L=\underset{x\to \infty }{\mathop{\lim }}\,\dfrac{1}{x}\times \left[ \log \left( {{20}^{x}}-1 \right)-\log \left( 19\times {{5}^{x}} \right) \right]$
By using formula 3 we can write log L as shown below,
$\therefore \log L=\underset{x\to \infty }{\mathop{\lim }}\,\dfrac{1}{x}\times \left[ \log \left( {{20}^{x}}-1 \right)-\left[ \log 19+\log {{5}^{x}} \right] \right]$
By using formula 1 again we can write log L as shown below,
$\therefore \log L=\underset{x\to \infty }{\mathop{\lim }}\,\dfrac{1}{x}\times \left[ \log \left( {{20}^{x}}-1 \right)-\left[ \log 19+x\times \log 5 \right] \right]$
Now we will give negative sign inside the bracket,
$\therefore \log L=\underset{x\to \infty }{\mathop{\lim }}\,\dfrac{\left[ \log \left( {{20}^{x}}-1 \right)-\log 19-x\times \log 5 \right]}{x}$
We will divide each term by x which is in the denominator to simplify the expression,
$\therefore \log L=\underset{x\to \infty }{\mathop{\lim }}\,\left[ \dfrac{\log \left( {{20}^{x}}-1 \right)}{x}-\dfrac{\log 19}{x}-\dfrac{x\times \log 5}{x} \right]$
$\therefore \log L=\underset{x\to \infty }{\mathop{\lim }}\,\left[ \dfrac{\log \left( {{20}^{x}}-1 \right)}{x}-\dfrac{\log 19}{x}-\log 5 \right]$
As we all know limits can be given separately for each term, therefore we can write above equation as,
$\therefore \log L=\underset{x\to \infty }{\mathop{\lim }}\,\dfrac{\log \left( {{20}^{x}}-1 \right)}{x}-\underset{x\to \infty }{\mathop{\lim }}\,\dfrac{\log 19}{x}-\underset{x\to \infty }{\mathop{\lim }}\,\log 5$
If we put the limits in the last term it won’t change as it’s a constant therefore, it can be written as,
$\therefore \log L=\underset{x\to \infty }{\mathop{\lim }}\,\dfrac{\log \left( {{20}^{x}}-1 \right)}{x}-\underset{x\to \infty }{\mathop{\lim }}\,\dfrac{\log 19}{x}-\log 5$………………………………………………. (a)
(i) (ii)
We should solve (i) and (ii) separately,
Consider,
$L1=\underset{x\to \infty }{\mathop{\lim }}\,\dfrac{\log \left( {{20}^{x}}-1 \right)}{x}$
If we put limits directly we will get,
$\therefore L1=\dfrac{\log \left( {{20}^{\infty }}-1 \right)}{\infty }$
$\therefore L1=\dfrac{\infty }{\infty }$
As it is giving a $\dfrac{\infty }{\infty }$ form which is an indeterminate form therefore we should L-Hospital’s Rule which is given below,
L-Hospital’s Rule:
If a limit of a function is giving an indeterminate form then,
$\underset{x\to a}{\mathop{\lim }}\,\dfrac{f(x)}{g(x)}=\underset{x\to a}{\mathop{\lim }}\,\dfrac{\dfrac{d}{dx}f(x)}{\dfrac{d}{dx}g(x)}$,
And we can take the derivatives till the denominator is not becoming zero if we put the limits.

By using L-Hospital’s Rule we can write L1 as,
$\therefore L1=\underset{x\to \infty }{\mathop{\lim }}\,\left[ \dfrac{\dfrac{d}{dx}\left[ \log \left( {{20}^{x}}-1 \right) \right]}{\dfrac{d}{dx}\left( x \right)} \right]$
Before proceeding further we should know the formulae of derivatives given below,

Formulae:

4. $\dfrac{d}{dx}\left( \log x \right)=\dfrac{1}{x}$

5. $\dfrac{d}{dx}\left( {{a}^{x}} \right)={{a}^{x}}\times \log a$

6. $\dfrac{d}{dx}\left( x \right)=1$

By using formula 4 and 6 we can write L1 as,
$\therefore L1=\underset{x\to \infty }{\mathop{\lim }}\,\left[ \dfrac{\dfrac{1}{\left( {{20}^{x}}-1 \right)}\times \dfrac{d}{dx}\left( {{20}^{x}}-1 \right)}{1} \right]$
By using formula 5 we can write above equation as,
$\therefore L1=\underset{x\to \infty }{\mathop{\lim }}\,\left[ \dfrac{1}{\left( {{20}^{x}}-1 \right)}\times {{20}^{x}}\times \log 20 \right]$
As we observe above equation we can easily see that the denominator is still not vanished and still
maintaining the $\dfrac{\infty }{\infty }$ form so we have to use L-Hospitals rule again. But it will be
lengthy as solved above.
To solve it shortly and to save time we can just take ${{20}^{x}}$common from denominator, therefore
we can write above equation as,
$\therefore L1=\underset{x\to \infty }{\mathop{\lim }}\,\left[ \dfrac{{{20}^{x}}\times \log 20}{{{20}^{x}}\left( 1-\dfrac{1}{{{20}^{x}}} \right)} \right]$
$\therefore L1=\underset{x\to \infty }{\mathop{\lim }}\,\left[ \dfrac{\log 20}{\left( 1-\dfrac{1}{{{20}^{x}}} \right)} \right]$
Now put the limits directly to get the answer,
$\therefore L1=\dfrac{\log 20}{\left( 1-\dfrac{1}{{{20}^{\infty }}} \right)}$
$\therefore L1=\dfrac{\log 20}{\left( 1-\dfrac{1}{\infty } \right)}$
As we all know that the value of $\dfrac{1}{\infty }$ is 0,
$\therefore L1=\dfrac{\log 20}{\left( 1-0 \right)}$
$\therefore L1=\log 20$……………………………………………………………….. (b)
Consider,
$L2=\underset{x\to \infty }{\mathop{\lim }}\,\dfrac{\log 19}{x}$
If we put the limits directly we will get,
$\therefore L2=\dfrac{\log 19}{\infty }$
As we all know that the value of $\dfrac{1}{\infty }$ is 0, therefore above equation will become,
$\therefore L2=0\times \log 19$
$\therefore L2=0$………………………………………………………………….. (c)
Now put the value of equation (b) and (c) in equation (a) we will get,
$\therefore \log L=\underset{x\to \infty }{\mathop{\lim }}\,\dfrac{\log \left( {{20}^{x}}-1 \right)}{x}-\underset{x\to \infty }{\mathop{\lim }}\,\dfrac{\log 19}{x}-\log 5$
$\therefore \log L=\log 20-0-\log 5$
$\therefore \log L=\log 20-\log 5$
If we use the formula 2 in above equation we will get,
$\therefore \log L=\log \dfrac{20}{5}$
$\therefore \log L=\log 4$
We will take antilog on both sides to get the final answer,
$\therefore L=4$
$\therefore \underset{x\to \infty }{\mathop{\lim }}\,{{\left( \dfrac{{{20}^{x}}-1}{19({{5}^{x}})} \right)}^{\dfrac{1}{x}}}=4$
Therefore the value of $\underset{x\to \infty }{\mathop{\lim }}\,{{\left( \dfrac{{{20}^{x}}-1}{19({{5}^{x}})} \right)}^{\dfrac{1}{x}}}$ is 4.
Note: You can commit a mistake which I have shown below, but do remember that x is tending to infinity
and not tending to Zero and therefore the formula is not applicable in the above case.
$\underset{x\to 0}{\mathop{\lim }}\,{{\left( {{a}^{x}}-1 \right)}^{\dfrac{1}{x}}}=e$ but $\underset{x\to \infty }{\mathop{\lim }}\,{{\left( {{20}^{x}}-1 \right)}^{\dfrac{1}{x}}}\ne e$


text:
# How To Test & Roll

Elea McDonnell Feit, Drexel University
Ron Berman, The Wharton School

Prepared for the July 2020 R Ladies Philly Meetup

## Workshop Description

Marketers often use A/B testing as a tool to compare marketing treatments in a test stage and then deploy the better-performing treatment to the remainder of the consumer population. These tests have traditionally been analyzed using hypothesis testing. In this workshop, Elea and Ron will explain a new approach to A/B testing that they developed called “Test & Roll” that focuses on the profit earned during and after the test. It is based on a Bayesian decision theory and the (very technical) Test & Roll paper explains why marketers should use this approach. This workshop will focus on how to Test & Roll using hands-on examples in R. Elea and Ron will cover:

• When you should use “Test & Roll” versus traditional hypothesis test
• How to compute the Test & Roll sample size that maximizes profits
• How to estimate priors from your own data to help plan your Test & Roll
• How to analyze a Test & Roll experiment (It’s easy!)

## Materials

If you want follow along, you can view the slides in your browser.

If you want to run the code as we go through the workshop, You will need to have R, R Studio and the rstan package installed. All the code is in the R Markdown Code for the slides and you will also need the file nn_functions.R.

If you use git, you can also clone the entire workshop repository. The howto folder contains the R Markdown files for the workshop. (There is other good stuff in the repo, including a copy of the paper and R scripts for replicating the analysis in the paper.)

## Expectations

• We will assume you have a general idea of what a probability distribution (like the normal distribution) is.
• We expect you are comfortable reading R. We will use R for mathematical calculations, for loops, and plotting.
• You do not need to know how to plan and analyze an A/B test using hypothesis testing, but see Elea’s tutorial on Advanced A/B Testing which was recorded in a previous R Ladies Philly workshop, if you are interested.


text:
# American Institute of Mathematical Sciences

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July  2019, 18(4): 1711-1734. doi: 10.3934/cpaa.2019081

## Scattering results for Dirac Hartree-type equations with small initial data

 1 Korea Institute for Advanced Study, Seoul 20455, Korea 2 Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 54896, Korea

Received  May 2018 Revised  September 2018 Published  January 2019

Fund Project: This work was partially supported by NRF (NRF-2015R1D1A1A09057795) and by German Science Foundation (IRTG 2235)

We consider the Dirac equations with cubic Hartree-type nonlinearity which are derived by uncoupling the Dirac-Klein-Gordon systems. We prove small data global well-posedness and scattering results in the full scaling subcritical regularity regime. The strategy of the proof relies on the localized Strichartz estimates and bilinear estimates in $V^2$ spaces, together with the use of the null structure that the nonlinear term exhibits. This result is shown to be almost optimal in the sense that the iteration method based on Duhamel's formula fails over the supercritical range.

Citation: Changhun Yang. Scattering results for Dirac Hartree-type equations with small initial data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1711-1734. doi: 10.3934/cpaa.2019081
##### References:
 [1] I. Bejenaru and S. Herr, The cubic Dirac equation: small initial data in $H^1(\Bbb R^3)$, Comm. Math. Phys., 335 (2015), 43-82. doi: 10.1007/s00220-014-2164-0. Google Scholar [2] I. Bejenaru and S. Herr, On global well-posedness and scattering for the massive Dirac-Klein-Gordon system, J. Eur. Math. Soc. (JEMS), 19 (2017), 2445-2467. doi: 10.4171/JEMS/721. Google Scholar [3] J. M. Chadam and R. T. Glassey, On the Maxwell-Dirac equations with zero magnetic field and their solution in two space dimensions, J. Math. Anal. Appl., 53 (1976), 495-507. doi: 10.1016/0022-247X(76)90087-1. Google Scholar [4] Y. Cho and T. Ozawa, On the semirelativistic Hartree-type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074. doi: 10.1137/060653688. Google Scholar [5] Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121. Google Scholar [6] P. D'Ancona and L. Fanelli, Strichartz and smoothing estimates of dispersive equations with magnetic potentials, Comm. Partial Differential Equations, 33 (2008), 1082-1112. doi: 10.1080/03605300701743749. Google Scholar [7] P. D'Ancona, D. Foschi and S. Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system, J. Eur. Math. Soc. (JEMS), 9 (2007), 877-899. doi: 10.4171/JEMS/100. Google Scholar [8] J. a.-P. Dias and M. Figueira, On the existence of weak solutions for a nonlinear time dependent Dirac equation, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 149-158. doi: 10.1017/S030821050002401X. Google Scholar [9] M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941. doi: 10.1016/j.anihpc.2008.04.002. Google Scholar [10] M. Hadac, S. Herr and H. Koch, Erratum to "Well-posedness and scattering for the KP-II equation in a critical space", [Ann. I. H. Poincaré–AN 26 (3) (2009) 917–941], Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 971–972. doi: 10.1016/j.anihpc.2010.01.006. Google Scholar [11] S. Herr and E. Lenzmann, The Boson star equation with initial data of low regularity, Nonlinear Anal., 97 (2014), 125-137. doi: 10.1016/j.na.2013.11.023. Google Scholar [12] S. Herr and A. Tesfahun, Small data scattering for semi-relativistic equations with Hartree type nonlinearity, J. Differential Equations, 259 (2015), 5510-5532. doi: 10.1016/j.jde.2015.06.037. Google Scholar [13] S. Herr and C. Yang, Critical well-posedness and scattering results for fractional Hartree-type equations, Differential Integral Equations, 31 (2018), 701-714. Google Scholar [14] H. Koch, D. Tataru and M. Vişan, Dispersive Equations and Nonlinear Waves. Generalized Korteweg-De Vries, Nonlinear SchrÖdinger, Wave and Schródinger Maps., Basel: Birkhäuser/Springer, 2014. Google Scholar [15] E. Lenzmann, Well-posedness for semi-relativistic hartree equations of critical type, Mathematical Physics, Analysis and Geometry, 10 (2007), 43-64. doi: 10.1007/s11040-007-9020-9. Google Scholar [16] S. Machihara and K. Tsutaya, Scattering theory for the Dirac equation with a non-local term, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 867-878. doi: 10.1017/S0308210507000479. Google Scholar [17] L. Molinet, J. C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal., 33 (2001), 982-988 (electronic). doi: 10.1137/S0036141001385307. Google Scholar [18] M. Nakamura and K. Tsutaya, Scattering theory for the Dirac equation of Hartree type and the semirelativistic Hartree equation, Nonlinear Anal., 75 (2012), 3531-3542. doi: 10.1016/j.na.2012.01.012. Google Scholar [19] F. Pusateri, Modified scattering for the boson star equation, Commun. Math. Phys., 332 (2014), 1203-1234. doi: 10.1007/s00220-014-2094-x. Google Scholar [20] E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, Ⅲ. Google Scholar [21] A. Tesfahun, Small data scattering for cubic Dirac equation with Hartree type nonlinearity in $\mathbb R ^{1+3}$, arXiv e-prints.Google Scholar [22] C. Yang, Small data scattering of semirelativistic hartree equation, Nonlinear Analysis, 178 (2019), 41-55. doi: 10.1016/j.na.2018.07.003. Google Scholar

show all references

##### References:
 [1] I. Bejenaru and S. Herr, The cubic Dirac equation: small initial data in $H^1(\Bbb R^3)$, Comm. Math. Phys., 335 (2015), 43-82. doi: 10.1007/s00220-014-2164-0. Google Scholar [2] I. Bejenaru and S. Herr, On global well-posedness and scattering for the massive Dirac-Klein-Gordon system, J. Eur. Math. Soc. (JEMS), 19 (2017), 2445-2467. doi: 10.4171/JEMS/721. Google Scholar [3] J. M. Chadam and R. T. Glassey, On the Maxwell-Dirac equations with zero magnetic field and their solution in two space dimensions, J. Math. Anal. Appl., 53 (1976), 495-507. doi: 10.1016/0022-247X(76)90087-1. Google Scholar [4] Y. Cho and T. Ozawa, On the semirelativistic Hartree-type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074. doi: 10.1137/060653688. Google Scholar [5] Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121. Google Scholar [6] P. D'Ancona and L. Fanelli, Strichartz and smoothing estimates of dispersive equations with magnetic potentials, Comm. Partial Differential Equations, 33 (2008), 1082-1112. doi: 10.1080/03605300701743749. Google Scholar [7] P. D'Ancona, D. Foschi and S. Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system, J. Eur. Math. Soc. (JEMS), 9 (2007), 877-899. doi: 10.4171/JEMS/100. Google Scholar [8] J. a.-P. Dias and M. Figueira, On the existence of weak solutions for a nonlinear time dependent Dirac equation, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 149-158. doi: 10.1017/S030821050002401X. Google Scholar [9] M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941. doi: 10.1016/j.anihpc.2008.04.002. Google Scholar [10] M. Hadac, S. Herr and H. Koch, Erratum to "Well-posedness and scattering for the KP-II equation in a critical space", [Ann. I. H. Poincaré–AN 26 (3) (2009) 917–941], Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 971–972. doi: 10.1016/j.anihpc.2010.01.006. Google Scholar [11] S. Herr and E. Lenzmann, The Boson star equation with initial data of low regularity, Nonlinear Anal., 97 (2014), 125-137. doi: 10.1016/j.na.2013.11.023. Google Scholar [12] S. Herr and A. Tesfahun, Small data scattering for semi-relativistic equations with Hartree type nonlinearity, J. Differential Equations, 259 (2015), 5510-5532. doi: 10.1016/j.jde.2015.06.037. Google Scholar [13] S. Herr and C. Yang, Critical well-posedness and scattering results for fractional Hartree-type equations, Differential Integral Equations, 31 (2018), 701-714. Google Scholar [14] H. Koch, D. Tataru and M. Vişan, Dispersive Equations and Nonlinear Waves. Generalized Korteweg-De Vries, Nonlinear SchrÖdinger, Wave and Schródinger Maps., Basel: Birkhäuser/Springer, 2014. Google Scholar [15] E. Lenzmann, Well-posedness for semi-relativistic hartree equations of critical type, Mathematical Physics, Analysis and Geometry, 10 (2007), 43-64. doi: 10.1007/s11040-007-9020-9. Google Scholar [16] S. Machihara and K. Tsutaya, Scattering theory for the Dirac equation with a non-local term, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 867-878. doi: 10.1017/S0308210507000479. Google Scholar [17] L. Molinet, J. C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal., 33 (2001), 982-988 (electronic). doi: 10.1137/S0036141001385307. Google Scholar [18] M. Nakamura and K. Tsutaya, Scattering theory for the Dirac equation of Hartree type and the semirelativistic Hartree equation, Nonlinear Anal., 75 (2012), 3531-3542. doi: 10.1016/j.na.2012.01.012. Google Scholar [19] F. Pusateri, Modified scattering for the boson star equation, Commun. Math. Phys., 332 (2014), 1203-1234. doi: 10.1007/s00220-014-2094-x. Google Scholar [20] E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, Ⅲ. Google Scholar [21] A. Tesfahun, Small data scattering for cubic Dirac equation with Hartree type nonlinearity in $\mathbb R ^{1+3}$, arXiv e-prints.Google Scholar [22] C. Yang, Small data scattering of semirelativistic hartree equation, Nonlinear Analysis, 178 (2019), 41-55. doi: 10.1016/j.na.2018.07.003. Google Scholar
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