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# Offenders become the victim in virtual reality: impact of changing perspective in domestic violence

## Abstract

The role of empathy and perspective-taking in preventing aggressive behaviors has been highlighted in several theoretical models. In this study, we used immersive virtual reality to induce a full body ownership illusion that allows offenders to be in the body of a victim of domestic abuse. A group of male domestic violence offenders and a control group without a history of violence experienced a virtual scene of abuse in first-person perspective. During the virtual encounter, the participants’ real bodies were replaced with a life-sized virtual female body that moved synchronously with their own real movements. Participants' emotion recognition skills were assessed before and after the virtual experience. Our results revealed that offenders have a significantly lower ability to recognize fear in female faces compared to controls, with a bias towards classifying fearful faces as happy. After being embodied in a female victim, offenders improved their ability to recognize fearful female faces and reduced their bias towards recognizing fearful faces as happy. For the first time, we demonstrate that changing the perspective of an aggressive population through immersive virtual reality can modify socio-perceptual processes such as emotion recognition, thought to underlie this specific form of aggressive behaviors.

## Introduction

Theoretical models of aggression have suggested that the perpetration of violent acts against others is linked to a lack of empathy or to a deficit in the ability of offenders to put themselves in the perspective of their victims1. In fact, some empirical studies have established a link between aggression and empathy2,3. For example, studies have found that offenders have difficulties in accurately recognizing emotions such as fear and anger, which has been hypothesized to hinder offenders’ compassionate responses4. However, other studies have failed to replicate these findings5. Discrepancies between studies might arise due to the multidimensional nature of empathy (ranging from emotion perception to cognitive perspective-taking), the differences in the methodologies used to assess empathy, the use of heterogeneous populations, and constraints found when trying to induce empathic responses in a laboratory setting6. In this sense, standard experimental studies aiming to understand core aspects of empathy often lack ecological validity due to ethical concerns. It is considered unethical to recreate in a laboratory a situation capable of inducing intense feelings of fear or distress in participants7. Additionally, research linking empathy and aggression has been frequently carried out in non-clinical populations, casting doubt on the transferability of these results to violent populations.

Here we introduce a new paradigm to study empathy and aggression in violent populations, which includes and goes beyond perspective-taking, since it allows participants to vividly experience a violent virtual situation from the perspective of the victim. This work is based on recent findings that have used immersive virtual reality (VR) to induce full body ownership illusions. In these studies, participants experience the perceptual illusion of ownership over a life-sized virtual body that visually substitutes their own body. Despite the fact that these virtual bodies may look drastically different to the participants’ own bodies in terms of size, height, skin tone, or age, individuals can still experience a strong subjective feeling of ownership8,9. Interestingly, such embodiment illusions can also influence social cognition by altering participants’ perceptions, attitudes and behaviors10,11,12,13. Furthermore, VR scenarios afford ecologically valid experimental setups while maintaining a high degree of experimental control. In these virtual scenarios, individuals have the illusion of being in a real environment and behave accordingly14. Therefore, VR provides a valuable tool to simulate and study violent behaviors without exposing participants to any real danger, hence overcoming the ethical issues that arise in non-virtual experiments15. This has been previously shown in several studies in which VR was used to assess sexually deviant behaviors of child molesters16, replicate Stanley Milgram’s obedience experiments17, and to study bystander responses to a violent incident to better understand helping and aggressive behaviors18.

The main goal of this study was to investigate some of the mechanisms of a type of violent behavior: domestic violence. Specifically, we have studied the impact of perspective taking (through virtual embodiment) and empathy on emotion recognition in participants including domestic violence offenders and controls. To this end, male participants entered a virtual environment where their body was substituted by that of a virtual female and they went through a process of “embodiment” (Fig. 1a–c). From this perspective, they saw a virtual male entering the scene and exhibiting abusive speech and gestures along with a progressive invasion of the victim’s (i.e., the participant’s) personal space (Fig. 1d–f; Video S1). The environment was interactive as the virtual abuser gazed towards the face of the participant shouting “Shut up!” if they spoke, or “Look at me!” if they looked away. We assessed emotion recognition skills before and after the VR experience, the degree of body ownership illusion12, and the subjective opinion regarding the virtual reality experience.

## Results

The experimental design was between-groups with one factor that we refer to as Condition, consisting of two groups of males: Offenders (n = 20) and Controls (n = 19; see Table S1 for demographic information). Participants completed the Face-Body Compound emotion recognition test before and after the VR exposure. In this test, faces expressing fear, anger or happiness were displayed for 100 ms on a computer screen. Each face was shown in isolation or on top of a body. The body expressed either the same emotional state as the face (congruent) or not (incongruent)19,20 (Fig. 2). Participants were given a forced-choice option (happiness vs. fear or happiness vs. anger) to categorize the facial expression of the different stimuli, and they were instructed to ignore the body. The test consisted of two independent blocks presented in a counterbalanced order. In one of the blocks, faces expressing either fear or happiness were displayed, and the other block followed the same methodology as the first one, but this time fearful faces were substituted by angry ones. Signal detection analysis was used to calculate the sensitivity index (d′) and the response criterion (c) of the recognition of angry and fearful facial expressions in male and female stimuli21,22,23. A higher value of d′ means greater sensitivity for detecting angry or fearful faces compared with happy faces. The c criterion indicates the overall response bias. A negative c reflects a bias to respond with either angry or fearful, while a positive c indicates a bias to respond with happy to the emotional expressions. A c around zero suggests no bias in indicating the emotion of the presented expressions.

These scores resulted in 8 response variables as shown in Table 1. An additional response variable was a ‘Social Desirability Scale’ which measures the extent to which participants tend to respond truthfully or misrepresent themselves in order to put forward a positive self-representation.

A Bayesian analysis was carried out that consisted of 9 simultaneous linear models24,25. Each model has as its response variables the post-VR exposure variables. The independent factor was the Condition, and the pre-VR exposure was an Offset variable. Details are given in the Materials and Methods section.

The equations model the difference between post-VR and pre-VR as a linear model of Condition (equivalent to ANOVA) such that a positive value of the coefficient of the independent factor (Condition) indicates that the difference between post-VR and pre-VR is greater for Offenders than Controls. There is a similar model for the pre-VR exposure variable, solely as a function of Condition, in order to check any differences in the response variables prior to the VR exposure.

Table 2 summarizes the results obtained and below we consider each in turn. It is important to note that this methodology specifies one overall model of the posterior distributions of all the parameters simultaneously.

### Sensitivity to recognize facial emotions (d′)

We first investigated the baseline (pre-VR) differences between domestic violence Offenders and Non-Offenders in their sensitivity (d′) to detect angry and fearful facial expressions against happy ones. The Control group recognized female fearful expressions more accurately (B_d′FearFemale, posterior probability = 0.97) than the Offenders group. They also recognized fearful (B_d′FearMale) and angry (B_ d′AngerMale) male facial expressions more accurately (both probabilities >0.84), but not angry female facial expressions (B_d′AngerFemale, posterior probability approximately 0.5) (see Tables 2 and 3).

After the VR exposure, the Offenders group substantially improved their recognition of fearful female facial expressions compared to the Control group (d′FearFemale, posterior probability = 0.99). However, the Offenders remained worse than the Control group in recognizing angry female faces (d′AngerFemale, posterior probability = 0.90). There were no evident differences in the recognition of angry and fearful male expressions after the VR experience in either group (see Fig. 3a and Table 2). This reveals that male domestic violence Offenders have difficulties recognizing female fearful facial expressions compared to male Controls. However, this deficit improved after the VR exposure in Offenders, while the detection of fear in female faces in the Control group remained constant.

### Bias towards classifying faces as depicting a particular emotion (c)

Baseline, pre-VR, showed a bias towards classifying fearful female faces as happy in the Offenders with respect to the Control group (B_cFearFemale, posterior probability = 0.80). Moreover, Offenders have a greater bias towards classifying fearful male faces as happy compared to Controls (posterior probability = 0.99). However, there does not seem to be a difference in the bias (c) used to classify angry or happy stimuli between groups (see Tables 2 and 3).

Regarding the impact of the VR experience on the bias (c), our results suggest that the bias of the Offenders towards incorrectly classifying fearful male faces as expressing happiness was reduced after the VR compared to the Control group (posterior probability = 0.96). This was also found for fearful female faces (posterior probability = 0.83). There was no evident difference between groups in the change in bias after the VR experience towards classifying angry faces as angry or happy (see Fig. 3b and Tables 1 and 2).

### Social desirability, empathy, and subjective perception of the VR

A questionnaire on social desirability revealed that the Offenders group (Mean = 20.30, Standard Error = 0.88) had higher scores than Controls (Mean = 17.42, Standard Error = 1.09). This suggests a tendency of Offenders to answer in a manner that would be viewed favorably by others, which calls into question the reliability of their self-reported questionnaires in general. The posterior probability that Offenders have higher scores than Controls on the Social Desirability Scale is 0.98 (Table 2). These results are in line with those of Andrews & Meyer26, who found that a large sample of males sentenced for domestic violence had a mean score of 18.77 (SD = 5.97) on the social desirability scale, and that such scores were significantly higher than those of male Non-Offenders. Therefore, following standard strategies used to deal with socially desirable responses27, the Interpersonal Reactivity Index28,29 and VR questionnaires data were not further analyzed, but descriptive results are presented in Supplementary Information (SI). The scores given by the Offenders cannot be regarded as reliable. This unreliability does not apply to the implicit measures such as the emotion recognition test, in which participants are presented with stimuli very briefly (100 ms) and have to give a very fast response that cannot therefore be based on a conscious decision.

## Discussion

The main goal of this study was to investigate the impact of perspective taking during violent behavior and the empathy that it engenders on emotion recognition in participants including domestic offenders and controls. A VR paradigm for embodiment of males in a female virtual body was carried out (Fig. 1a–c)30,31 for participants to experience the virtual scene as if they were the target of domestic violence. Our findings indicate that, at baseline levels, male domestic violence Offenders do not well-recognize fearful female facial expressions when compared to Control males (Non-Offenders). Furthermore, we found that domestic violence Offenders have a greater bias towards classifying both male and female fearful faces as happy, rather than fearful, compared to Non-Offenders. However, being embodied in a female victim who suffers verbal abuse and intimidation by a male character using VR resulted in an improvement of the ability of Offenders to recognize fear in female faces, and reduced their response bias towards wrongly attributing happy emotional states to fearful facial expressions, independently of gender. This effect occurred overall in Offenders and not in Controls. Our findings indicate that changing the perspective of an aggressive population by means of virtual embodiment in the victim impacts emotion recognition, which is a socio-perceptual process thought to underlie this specific form of aggressive behavior.

The deficit in fear recognition found in the Offenders group is in line with several studies that have shown a deficit in recognizing negative emotional states in violent populations32. Our data strongly support the notion that men who are violent towards their female partners are less able to recognize fear in female faces than non-violent men. Based on our findings, we hypothesize that this deficit can partially explain why domestic violence Offenders may behave abusively towards their female partners. This is in line with a previous study showing that the inability of these violent men to recognize fearful expressions in their wives or unfamiliar women correlates with intimate partner violence perpetration33.

Blair’s Violence Inhibition Mechanism model34 proposes that poor recognition of the distress or fear experienced by the other can prevent a compassionate and empathic response. Similarly, Simulation Theory suggests that deficits in the production of a specific emotion, and deficits in the recognition of that emotion, co-occur35. In fact, poor ability to recognize fear can also encompass problems in other socially relevant skills, such as theory of mind, and these abnormalities have been linked to impairments in the neural functioning of brain regions that underpin social information processing36,37. For example, structures such as the amygdala might be involved in these deficits, since it has been previously found that this region is crucial for the processing of threating signals under ambiguous or incongruent conditions38,39.

However, our study shows that this deficit can be reduced through the type of VR exposure that we employed here. Our results indicate that a virtual domestic abuse scene experienced from the perspective of a female victim can positively increase Offenders’ sensitivity to recognize fear in fearful female facial expressions, and reduce Offenders’ bias towards misclassifying facial expressions as expressing happiness rather than fear. We also found that there was an enhancement in the recognition of angry female faces in the Control group after the VR experience when compared to the Offenders group.

These results are in line with other studies that found that non-virtual body ownership illusions can specifically enhance the recognition of fear. Such an example is the “enfacement” illusion, in which the illusory feeling of ownership over a different face can be produced by seeing a face being touched while feeling touched synchronously and correspondingly on one’s own face. This illusion can facilitate the recognition of fearful expressions but not of other types of emotions40. It is hypothesized that shared multisensory experiences, such as embodiment over a different body, can facilitate simulation mechanisms in the somatosensory cortex that are specifically related to fear perception in others. In accordance with this, it has also been shown that the Visual Remapping of Touch effect (a phenomenon in which seeing a face being touched can enhance the observer’s accuracy in detecting tactile perception on their face) is modulated to a greater degree by fearful expressions than by other emotional expressions41. Additionally, individuals experiencing mirror-touch synesthesia, where subjects report feeling touch on their own body when they see someone else being touched, also better recognize emotional states presumably because of their ability to simulate others’ somatosensory experiences42,43.

Importantly, the enhancement of fear recognition after embodiment in a female victim only occurred in the domestic violence Offenders group, and not in the Control group. A possible explanation is that the emotion recognition test used in the present study was sensitive enough to detect changes in the Offender group, who had a deficit, and thus, larger room for improvement, but not in the Control group, who had better recognition of fearful faces to start with. This difference could also be explained in terms of how perception and cognition might be affected by previous personal history and exposure to violence19. Several studies have found that violent and non-violent populations differ in their perceptual-biases and allocation of attentional resources when confronted with threatening stimuli44,45. In this sense, it is possible that the subjective interpretations of the embodiment experience were different between the groups. In the case of offenders, the VR scene reverses their usual experienced perspective - they become a victim instead of a perpetrator. This could have resulted in a profound shift of perspective while still identifying with the male avatar, which only impacted their processing of fear related emotions. No such drastic change of perspective occurred in the control group participants, who might have allocated more attentional resources to the processing of threatening cues, which might in turn have enhanced their recognition of anger in female facial expressions after the IVR experience.

In the present study, we also found that Offenders seem to have lower sensitivity to accurately recognize anger in male facial stimuli compared with Controls, although we did not find this effect for female stimuli. Additionally, only small between-group differences were found in the bias towards classifying angry faces. These results are compatible with a study showing that generally-violent/antisocial men who also committed acts of intimate partner violence, taken from a community sample, tended to misidentify angry expressions as disgust or happiness46. However, the authors observed that such a pattern of anger recognition did not occur in family-only offenders or in borderline personality disorder offenders. Although we cannot directly compare our results with these findings since we did not follow the Holtzworth-Munroe & Stuart47 typology classification for male batterers, these studies highlight the need to distinguish between subtypes of intimate partner violence male abusers based on the severity of their violent behaviors. Future studies should aim to elucidate whether such patterns in facial emotion recognition ability can be replicated in a sample such as that of the present study, i.e. in men who are sentenced and enrolled in a rehabilitation program for domestic violence.

The results of the present study further support the notion that experiencing embodiment over a different body to one’s own can influence perceptions, attitudes and behaviors. This and other studies indicate that altering the perception of the self, from a purely bodily aspect, seems to also modify self-related socio-cognitive processes, which could be exploited in order to tackle real life problems10. For example, it has been demonstrated that embodiment of light-skinned people in a dark-skinned body can reduce implicit racial biases against that outgroup11,48, an effect that seems to be long-lasting49. Similarly, evidence also indicates that embodied perspective-taking in VR is a promising tool for facilitating the resolution of intrapersonal conflicts13,50. In the present study, for the first time, we provide evidence demonstrating that embodied perspective-taking through VR also has the potential to improve current rehabilitation programs for offenders by enhancing emotional recognition skills. Finally, our results also partially support the theoretical framework proposed by Yee and Bailenson51 (known as the Proteus effect), which states that a virtual self-representation can have a significant impact on behavior. These authors demonstrated that participants assigned to more attractive or taller avatars tended to exhibit more intimate and confident behaviors towards a confederate, compared to conditions in which participants were assigned to less attractive or shorter avatars. However, in the present study we did not test the effects of embodiment over a virtual female victim on subsequent inter-personal interactions nor the effect of self-stereotyping.

From a mechanistic point of view, we hypothesize that the novel multisensory experience of facing violence from a victim’s perspective can facilitate the identification of the potential distress and fear-related emotions experienced by victims of domestic violence. Such enhancement in emotion recognition could be at the basis of a behavioral change towards the victim through the reduction of future re-offenses, which is ultimately the desired outcome of the present study. Furthermore, improvement in emotion recognition could also be explained in terms of the impact that embodiment experiences can have on brain regions involved in affective face encoding, such as the somatosensory cortex, amygdala, cingulate cortex, ventromedial prefrontal cortex, and fusiform face area. The use of brain imaging techniques in future research will be crucial in order to understand the neural mechanisms implicated in the effects that embodiment can have on emotion recognition. Finally, the critical studies will be those which explore how these virtual reality experiences can impact recidivism of violent behavior.

Our findings also highlight the need to use measures that cannot be consciously manipulated when working with these types of populations, such as the emotion recognition test employed here or the recording of physiological reactions. In the present study we quantified the impact of the experience using an implicit measure that is independent of the questionnaire results and not affected by the increased social desirability detected in the Offenders group. Hence, the Face-Body Compound test was used, since a previous event-related potentials study showed that the congruency of such stimuli is processed within 86–128 ms, a timing that precludes conscious decision making19,20. Our study illustrates that implicit and explicit measures do not seamlessly map onto each other, as shown by the results obtained in the questionnaire-based responses that reflect the impact of enhanced social desirability amongst the Offenders. We found that domestic violence Offenders record higher scores in the Social Desirability Scale in comparison to Non-Offenders, suggesting that Offenders tend to respond in a way that they presume is socially desirable. This is in line with studies showing that violent populations try to give a compliant and positive image of themselves when enrolled in court rehabilitation programs52.

Furthermore, this study also contributes to the development of new strategies aimed to improve the efficacy of rehabilitation programs, which include empathy training or perspective-taking techniques based on watching movies, reading testimonies or carrying out role-playing activities. These activities aim to help Offenders better understand the feelings of their victims by putting them in the female’s shoes53. A major drawback of such techniques is that they are explicit in their message, providing verbal information rather than actual experience, and therefore may not impact the Offenders’ behaviors towards their victims. Although Offenders might obtain abstract knowledge that their behavior is wrong, this may not actually result in behavioral changes. Furthermore, the fact that such methodologies rely on the motivation and capacity of Offenders to take the perspective of the victim in a violent situation can be problematic for those with a lower capacity of imagination, or amongst those with a low level of motivation towards the treatment. The virtual embodiment paradigm used in this study overcomes such limitations by making the Offenders actually experience an abuse situation from the perspective of the victim, independently of motivational factors and of the participants’ facility to imagine hypothetical situations.

In this study we were interested in whether the shift in embodied perspective taking would result in observable changes when comparing the Offenders to the Controls. The methodology used in the present study was grounded on past research which showed that providing congruent multisensory information is crucial to induce a body ownership illusion8. However, in future work we aim to manipulate different parameters of the embodiment experience in order to explore the most relevant factors contributing to the changes in emotion recognition. For example, future studies should explore the impact on emotion recognition of conditions where participants are provided with incongruent multisensory information concerning their virtual body in order to test whether the observed effects are due to the body ownership illusions per se54. Furthermore, the role that the aggressive social interaction established by the male and female virtual characters and the stereotypes associated with the virtual female body played in the observed effects also remains to be answered55. In particular, it has been shown that domestic violence offenders frequently present stereotypes and cognitive distortions concerning females56. Future studies using a similar methodology should assess whether such changes in emotion recognition occur after experiencing embodiment over a male virtual character that is being abused or by simply experiencing embodiment over a virtual female body in a neutral scenario. Finally, future research should control for inter-individual differences through validated scales (e.g. personality traits, psychopathology, and dispositional emotional contagion) in order to analyze how such variables can influence the results.

In conclusion, accurately recognizing emotions is a key component of empathic responses and several studies have demonstrated that better recognition of fearful expressions predicts prosocial behavior57. The results of this study suggest that experiencing a virtual situation of domestic abuse from the perspective of the victim can result in an enhancement of fear recognition by the offenders. Although the implicit measures of emotion recognition are valid objective measures of the impact of this experience in violent perpetrators, this does not imply that all the effects are restricted to those currently measured. Our study is a first demonstration of the potential of immersive VR to induce positive changes in actual offenders. However, subjective, physiological, and explicit changes should be investigated in the future. Furthermore, the final objective should be to search for behavioral changes that contribute to reducing future re-offenses, thus contributing to the well-being of the victims, the offenders themselves, and society in general.

## Materials and Methods

### Sample

The Offenders group (N = 20) was recruited in collaboration with the Catalan Justice Department. This group consisted of men convicted by the Spanish legal system for an aggression against a woman and sentenced to attend a domestic violence intervention program.

The Control group (N = 19) consisted of men without a history of domestic violence. They were recruited via advertisement in the community and also from the maintenance staff of the University. The Control group matched the Offenders group in age, educational level, nationality and occupational status. The original sample had 44 participants, 23 Offenders and 21 Controls, but 3 were discarded as outliers and 2 due to missing critical information (for further demographic information and exclusion criteria, see SI Sample section).

This study followed ethical standards as per the Declaration of Helsinki and was approved by the Ethical Committee for Clinical Research at the Hospital Clinic of Barcelona. All included participants gave written informed consent (for more details see SI Further Ethical Considerations section).

### Measures

Emotion recognition was assessed before and after the VR experience with the Face-Body Compound test19,58. Face-body compound images were created by assembling emotionally congruent or incongruent body expressions with emotional faces of five male and five female actors (Fig. 2). These stimuli have been previously validated in a study that found that violent offenders have difficulties in processing emotional incongruence and have a possible bias towards perceiving emotional expressions as aggressive19.

We used an adaptation of the Face-Body compound test that consisted of two independent blocks presented in a counterbalanced order. In one of the blocks, faces expressing either fear or happiness were displayed in isolation or on top of a body that expressed the same emotional state (congruent) or not (incongruent). The other block followed the same methodology as the first one, but this time fearful emotional expressions were substituted by angry ones. When faces were displayed in isolation, no outlines of the bodies were visible and the space of the body was filled with a grey background color. However, the same size and position on the screen was preserved when compared to stimuli which included bodies. Subjects were given a forced double-choice option (happiness vs. fear or happiness vs. anger) to categorize the facial expression of the different stimuli and they were instructed to ignore the body. The stimuli were briefly presented (100 ms). After each response, a fixation cross appeared for 750 to 1250 ms. Each block of the test included 60 pictures (2 facial expressions * 3 body postures * 10 actors) that were presented twice in a random order. All participants completed this test before and immediately after the VR experience. This test was analyzed using a signal detection analysis (details in SI Signal Detection Analysis).

Furthermore, the Spanish version59 of the Social Desirability Scale was used to assess potential response bias in the other self-reported questionnaires since it measures a subject’s tendency to respond in a socially desirable way. The scale comprises 33 true-false items concerning everyday behaviors. Higher scores in this scale indicate a stronger tendency to portray a positive image of the self and a tendency to respond in what is presumed to be desired by society.

We also administered the Interpersonal Reactivity Index60 and a self-developed VR questionnaire, which were not included in the final study due to socially desirable responding (see more details on the analysis of these variables in SI IRI and VR questionnaire).

### Procedure and Virtual Scene

Upon arrival, all participants signed an informed consent form, and completed a demographics form, the IRI, and the social desirability scale. Subsequently, we administered the Face Body Compound Test (pre-VR). Participants sat 55 cm from the computer display.

Before the VR experience, participants were fitted with the head mounted display (HMD) and a pair of headphones. The VR scenario depicted a room with a long hallway where the participant’s own body was replaced with that of a virtual female. All participants saw the virtual female from a first-person perspective. The virtual body moved in real-time in accordance with the actual movements of the participants, producing visuomotor synchrony. The virtual female character stood at one end of the hallway facing a full-length mirror, which had a table with a radio and a phone on its side. Towards the other end of the hallway was an open door. A 1 × 1 m rug on the ground in front of the full-length mirror, below the virtual female, was used as a reference for the area in which the participant could stand and navigate.

First, we asked participants to become familiar with their virtual female body and the surroundings by looking around and moving their new virtual body (Fig. 1a). For this, the virtual female character faced the full-length mirror. We also asked them to look down towards their own body so that they could inspect their virtual body from a first-person perspective (Fig. 1b). Next, participants were asked to describe their new virtual body and the room. At this moment, participants heard some pre-recorded instructions while 4 virtual balls appeared in front of them. In order to enhance the body ownership illusion through visuomotor synchrony and agency, participants were instructed by audio to touch the virtual balls. Two virtual balls appeared high in the air to be touched with their hands and two at floor level to be touched with their feet. When a ball was touched it disappeared until all four were touched, then a new iteration of the exercise started where the balls appeared again in similar positions. Participants did three iterations of this exercise, which lasted approximately 2 min (Fig. 1c).

After doing the exercises, the virtual female body faced the far end of the hallway instead of the mirror. A male virtual character entered the room and began to verbally abuse the female virtual character following a pre-defined script (Fig. 1d). The dialogue of the abuser contained various insults against the female character. More specifically, the male character constantly derided the physical appearance and defenselessness of the female virtual character. He also made frequent references and threats regarding the fact that the female was financially dependent on him.

The male virtual character was programmed to constantly look into the eyes of the female character. The male character approached the female character and at a certain point he hit the telephone, which fell off the table in the direction of the virtual female body (Fig. 1f). This was done in order to include an act of physical violence. On three occasions during the verbal abuse, the male virtual character walked towards the female character, and by the end of the scene he was right in front of her (Fig. 1e). Additionally, if participants interrupted the monologue of the male virtual character at the start of the experiment, the experimenter could make the male character say “Shut up!”. The verbal abuse lasted for approximately 3 min 20 s (see more details in SI Procedure, Technical setup, and Movie S1).

Once the VR scene finished, all equipment was removed from the participants and they were asked to immediately complete the emotion recognition test (post-VR). Subsequently, participants filled in the post VR experience questionnaire, we carried out a semi-structured interview and we conducted the debriefing about the virtual experience.

### Statistical Analysis

For the d′ and c variables shown in Table 1, each equation was of the form (1, 2):

$${\mu }_{post,i}={y}_{pre,i}+{\alpha }_{post}+{\beta }_{post}{C}_{i}$$
(1)
$${\mu }_{pre,i}={\alpha }_{pre}+{\beta }_{pre}{C}_{i}$$
(2)
$${y}_{post,i} \sim N({\mu }_{post,i},{\sigma }_{post}^{2})$$
(3)
$${y}_{pre,i} \sim N({\mu }_{pre,i},{\sigma }_{pre}^{2})$$
(4)

where $${y}_{post,i}$$ and $${y}_{pre,i}$$ are the ith observations of the variables in Table 1 as measured post and prior to the IVR exposure. $${y}_{post,i}$$ is normally distributed with mean equal to offset $${y}_{pre,i}$$ plus the linear model involving the unknown parameters $${\sigma }_{post}$$ and $${\beta }_{post}$$ where C i is the Condition (Control = 0, Offender = 1) of the ith participant. The prior distributions of the α′s and β′s were taken as bivariate normal with mean vector (0,0), variances 1600 and covariance 160, thus highly non-informative. The prior distribution of the variances $${\sigma }^{2}$$ were also taken as a highly non-informative Gamma distribution with parameters $$r=\lambda =0.001$$, where the probability density $$f(y)\propto {\lambda }^{r}{y}^{r-1}exp(-\lambda y)$$.

For the Social Desirability scale the same model was used, except that this was only measured prior to the VR exposure so that there is no offset term in the equation.

Analysis was carried out using the JAGS system25, together with MATLAB via MATJAGS (http://psiexp.ss.uci.edu/research/programs_data/jags/), and some graphs were produced using Stata 14 (see SI Bayesian Analysis).

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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## Acknowledgements

We thank the Department of Alternative Penal Measures of the Catalan Justice Department, in particular the director, Marc Cerón, and Fundació AGI, for their help in conducting the project with a population of violent offenders. We would also like to thank Cristina Gonzalez-Liencres for her comments on the manuscript. This project was supported by the Virtual Embodiment and Robotic Re-Embodiment Integrated Project funded under the European Seventh Framework Programme, Future and Emerging Technologies, Grant Agreement 257695. CERCA Programme, Generalitat de Catalunya to IDIBAPS. RH and BdG were supported by funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement number 295673.

## Author information

Authors

### Contributions

Conceived and designed the experiments: M.V.S.V., M.S., S.S., J.A.P., R.H., B.d.G. Performed the experiments: S.S., J.A.P., G.I., L.Z. Supervision of the experiments: M.V.S.V., M.S. Analyzed the data: M.S. Wrote the paper: S.S., M.V.S.V., M.S., with the help of rest of authors. Programming of the virtual reality system: D.B., G.I., and J.A.P.

### Corresponding author

Correspondence to M. V. Sanchez-Vives.

## Ethics declarations

### Competing Interests

Authors MS and MVSV are founders of Virtual Bodyworks Inc. which currently licenses to the Catalan Justice Department a virtual environment for domestic violence offenders used in the frame of its rehabilitation training programme. The research in this manuscript was carried out according to scientific methods and was not influenced at all by the company.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Seinfeld, S., Arroyo-Palacios, J., Iruretagoyena, G. et al. Offenders become the victim in virtual reality: impact of changing perspective in domestic violence. Sci Rep 8, 2692 (2018). https://doi.org/10.1038/s41598-018-19987-7

• Accepted:

• Published:

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text:
<meta http-equiv="refresh" content="1; url=/nojavascript/"> Reflections | CK-12 Foundation

# 12.5: Reflections

Difficulty Level: At Grade Created by: CK-12
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Progress
Practice Reflections
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What if you noticed that a lake can act like a mirror in nature? Describe the line of reflection in the photo below. If this image were on the coordinate plane, what could the equation of the line of reflection be? (There could be more than one correct answer, depending on where you place the origin.) After completing this Concept, you'll be able to answer this question.

### Guidance

A transformation is an operation that moves, flips, or changes a figure to create a new figure. A rigid transformation is a transformation that preserves size and shape. The rigid transformations are: translations, reflections (discussed here), and rotations. The new figure created by a transformation is called the image . The original figure is called the preimage . Another word for a rigid transformation is an isometry . Rigid transformations are also called congruence transformations . If the preimage is $A$ , then the image would be labeled $A'$ , said “a prime.” If there is an image of $A'$ , that would be labeled $A''$ , said “a double prime.”

A reflection is a transformation that turns a figure into its mirror image by flipping it over a line. Another way to describe a reflection is a “flip.” The line of reflection is the line that a figure is reflected over. If a point is on the line of reflection then the image is the same as the original point.

##### Common Reflections
• Reflection over the $y-$ axis: If $(x,y)$ is reflected over the $y-$ axis, then the image is $(-x,y)$ .
• Reflection over the $x-$ axis: If $(x,y)$ is reflected over the $x-$ axis, then the image is $(x,-y)$ .
• Reflection over $x = a:$ If $(x, y)$ is reflected over the vertical line $x = a$ , then the image is $(2a - x, y)$ .
• Reflection over $y = b:$ If $(x, y)$ is reflected over the horizontal line $y = b$ , then the image is $(x, 2b - y)$ .
• Reflection over $y = x$ : If $(x, y)$ is reflected over the line $y = x$ , then the image is $(y, x)$ .
• Reflection over $y = -x$ : If $(x, y)$ is reflected over the line $y = -x$ , then the image is $(-y, -x)$ .

#### Example A

Reflect the letter ‘‘F’’ over the $x-$ axis.

To reflect the letter $F$ over the $x-$ axis, now the $x-$ coordinates will remain the same and the $y-$ coordinates will be the same distance away from the $x-$ axis on the other side.

#### Example B

Reflect $\triangle ABC$ over the $y-$ axis. Find the coordinates of the image.

To reflect $\triangle ABC$ over the $y-$ axis the $y-$ coordinates will remain the same. The $x-$ coordinates will be the same distance away from the $y-$ axis, but on the other side of the $y-$ axis.

$A(4,3) & \rightarrow A'(-4,3)\\B(7,-1) & \rightarrow B'(-7,-1)\\C(2,-2) & \rightarrow C'(-2,-2)$

#### Example C

Reflect the triangle $\triangle ABC$ with vertices $A(4, 5), B(7, 1)$ and $C(9, 6)$ over the line $x = 5$ .

Notice that this vertical line is through our preimage. Therefore, the image’s vertices are the same distance away from $x = 5$ as the preimage. As with reflecting over the $y-$ axis (or $x = 0$ ), the $y-$ coordinates will stay the same.

$A(4,5) & \rightarrow A'(6,5)\\B(7,1) & \rightarrow B'(3,1)\\C(9,6) & \rightarrow C'(1,6)$

#### Example D

Reflect square $ABCD$ over the line $y = x$ .

The purple line is $y = x$ . To reflect an image over a line that is not vertical or horizontal, you can fold the graph on the line of reflection.

$A(-1, 5) & \rightarrow A'(5, -1)\\B(0, 2) & \rightarrow B'(2, 0)\\C(-3, 1) & \rightarrow C'(1, -3)\\D(-4, 4) & \rightarrow D'(4, -4)$

Watch this video for help with the Examples above.

#### Concept Problem Revisited

The white line in the picture is the line of reflection. This line coincides with the water’s edge. If we were to place this picture on the coordinate plane, the line of reflection would be any horizontal line. One example could be the $x-$ axis.

### Vocabulary

A transformation is an operation that moves, flips, or otherwise changes a figure to create a new figure. A rigid transformation (also known as an isometry or congruence transformation ) is a transformation that does not change the size or shape of a figure. The new figure created by a transformation is called the image . The original figure is called the preimage . A reflection is a transformation that turns a figure into its mirror image by flipping it over a line. The line of reflection is the line that a figure is reflected over.

### Guided Practice

1. Reflect the line segment $\overline{PQ}$ with endpoints $P(-1, 5)$ and $Q(7, 8)$ over the line $y = 5$ .

2. A triangle $\triangle LMN$ and its reflection, $\triangle L'M'N'$ are to the left. What is the line of reflection?

3. Reflect the trapezoid TRAP over the line $y = -x$ .

1. Here, the line of reflection is on $P$ , which means $P'$ has the same coordinates. $Q'$ has the same $x-$ coordinate as $Q$ and is the same distance away from $y = 5$ , but on the other side.

$P(-1,5) & \rightarrow P'(-1,5)\\Q(7,8) & \rightarrow Q'(7,2)$

2. Looking at the graph, we see that the preimage and image intersect when $y = 1$ . Therefore, this is the line of reflection.

3. The purple line is $y = -x$ . You can reflect the trapezoid over this line just like we did in Example D.

$T(2,2) & \rightarrow T'(-2, -2)\\R(4, 3) & \rightarrow R'(-3,-4)\\A(5, 1) & \rightarrow A'(-1,-5)\\P(1, -1) & \rightarrow P'(1,-1)$

### Explore More

1. Which letter is a reflection over a vertical line of the letter $b$ ?
2. Which letter is a reflection over a horizontal line of the letter $b$ ?

Reflect each shape over the given line.

1. $y-$ axis
2. $x-$ axis
3. $y = 3$
4. $x = -1$

Find the line of reflection of the blue triangle (preimage) and the red triangle (image).

Two Reflections The vertices of $\triangle ABC$ are $A(-5, 1), B(-3, 6)$ , and $C(2, 3)$ . Use this information to answer questions 10-13.

1. Plot $\triangle ABC$ on the coordinate plane.
2. Reflect $\triangle ABC$ over $y = 1$ . Find the coordinates of $\triangle A'B'C'$ .
3. Reflect $\triangle A'B'C'$ over $y = -3$ . Find the coordinates of $\triangle A''B''C''$ .
4. What one transformation would be the same as this double reflection?

Two Reflections The vertices of $\triangle DEF$ are $D(6, -2), E(8, -4)$ , and $F(3, -7)$ . Use this information to answer questions 14-17.

1. Plot $\triangle DEF$ on the coordinate plane.
2. Reflect $\triangle DEF$ over $x = 2$ . Find the coordinates of $\triangle D'E'F'$ .
3. Reflect $\triangle D'E'F'$ over $x = -4$ . Find the coordinates of $\triangle D''E''F''$ .
4. What one transformation would be the same as this double reflection?

Two Reflections The vertices of $\triangle GHI$ are $G(1, 1), H(5, 1)$ , and $I(5, 4)$ . Use this information to answer questions 18-21.

1. Plot $\triangle GHI$ on the coordinate plane.
2. Reflect $\triangle GHI$ over the $x-$ axis. Find the coordinates of $\triangle G'H'I'$ .
3. Reflect $\triangle G'H'I'$ over the $y-$ axis. Find the coordinates of $\triangle G''H''I''$ .
4. What one transformation would be the same as this double reflection?

## Date Created:

Jul 17, 2012

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text:
# find derivative

• August 14th 2009, 08:56 AM
skeske1234
find derivative
1. If f(x)=log_2[log_3(x)], find f'(4)

can someone check to see if my answers match yours?

f'(x)=1/[(xln3)]/[log_3(x)ln2]
=1/[xln3ln2log_3(x)]
f'(4)=1/[4ln3ln2log_3(4)]

2. Find the equation of the tangent to the curve defined by y=lnx-1 that is parallel to the straight line with equation 3x-6y-1=0.

M=0.5

y=lnx-1
y'=1/x
M_t=0.5
0.5=1/x
x=2

at x=2

y=lnx-1
y=ln2-1

y=mx+b
ln2-1=0.5(2)+b
ln2-1=b

y=0.5x+ln2-1

• August 14th 2009, 09:56 AM
Amer
Quote:

Originally Posted by skeske1234
1. If f(x)=log_2[log_3(x)], find f'(4)

can someone check to see if my answers match yours?

f'(x)=1/[(xln3)]/[log_3(x)ln2]
=1/[xln3ln2log_3(x)]
f'(4)=1/[4ln3ln2log_3(4)]

2. Find the equation of the tangent to the curve defined by y=lnx-1 that is parallel to the straight line with equation 3x-6y-1=0.

M=0.5

y=lnx-1
y'=1/x
M_t=0.5
0.5=1/x
x=2

at x=2

y=lnx-1
y=ln2-1

y=mx+b
ln2-1=0.5(2)+b
ln2-1=b

y=0.5x+ln2-1

first one is correct

$f'(x)=\frac{\frac{1}{x\ln (3)}}{\ln (2)\log _3 (x)}$

$f'(x)=\frac{1}{x\ln (2)\ln (3) \log _3 (x)}$
• August 14th 2009, 10:14 AM
skeske1234
Quote:

Originally Posted by Amer
first one is correct

$f'(x)=\frac{\frac{1}{x\ln (3)}}{\ln (2)\log _3 (x)}$

$f'(x)=\frac{1}{x\ln (2)\ln (3) \log _3 (x)}$

First one is correct.

Is the second one correct?
If not, could you explain. ty
• August 14th 2009, 10:23 AM
Amer
the second one

$y-y_o = Slope(x-x_o)$

$x_o=2 ,y_o=ln(2)-1$

slope = 1/2

$y-(ln(2)-1)=.5(x-2)$

$y=.5(x) -1 +ln(2) -1$

$y=.5(x) +ln(2) -2$

you mistake when you find b value

b=ln(2)-2


text:
4
Q:

In a row of thirty boys, R is fourth from the right end and W is tenth from the left end . How many boys are there between R and W?

 A) 15 B) 16 C) 17 D) can't be determined

Explanation:

W = 10th from the left

R = 4th from the right = (30 - 4 + 1= ) 27th from left

$\inline \therefore$ No of boys between R and W = (27 - 10) - 1 = 16

Q:

Manoj and Satya are ranked seventh and eleventh respectively from the top in a class of 31 students. What will be their respective ranks from the bottom in the class ?

 A) 25th and 21st B) 26th and 22nd C) 20th and 24th D) 24th and 20th

Explanation:

Number of students behind Manoj in rank = (31 - 7) = 24.

So, Manoj is 25th from the bottom.

Number of students behind Satya in rank = (31 - 11) = 20.

So, Satya is 21st from the bottom.

2 287
Q:

If the positions of the first and the third digit within each number are interchanged, which of the following will be the third digit of the second lowest number ?

987, 514, 658, 487, 404, 269

 A) 8 B) 9 C) 2 D) 4

Explanation:

According to the question, after the position of the first and third digit interchanged new numbers are
789, 415, 856, 784, 404, 962

962, 856, 789, 784, 404, 415
Here the second lowest number is 404 and the last digit of it is '4'.

4 384
Q:

Some boys are sitting in a row. P is sitting fourteenth from the left and Q is seventh from the right. If there are four boys between P and Q, how many boys are there in the row ?

 A) 26 B) 24 C) 25 D) 22

Explanation:

Number of boys in the row = number of boys uptil P + number of boys between P and Q + number of boys including Q and those behind Q = 14 + 4 + 7 = 25.

2 300
Q:

In a queue, Amrita is 10th from the front while Mukul is 25th from behind and Mamta is just in the middle of the two. If there be 50 persons in the queue. What position does Mamta occupy from the front ?

 A) 14th B) 16th C) 18th D) 20th

Explanation:

Number of persons between Amrita and Mukul = 50 - (10 + 25) = 15. Since Mamta lies in middle of these 15 persons, so Mamta`s position is 8th from Amrita i.e. 18th from the front.

5 323
Q:

In a row, Kumar is at 7th place from the left and Pawan is at 9th place from the right. When they interchange the positions Kumar becomes 11th from left. How many were seated in the row ?

 A) 19 B) 20 C) 21 D) 27


text:
# zbMATH — the first resource for mathematics

Ramsey’s theorem for singletons and strong computable reducibility. (English) Zbl 1423.03159
Summary: We answer a question posed by D. R. Hirschfeldt and C. G. Jockusch jun. [J. Math. Log. 16, No. 1, Article ID 1650002, 59 p. (2016; Zbl 1373.03068)] by showing that whenever $$k > \ell$$, Ramsey’s theorem for singletons and $$k$$-colorings, $$\mathsf {RT}^1_k$$, is not strongly computably reducible to the stable Ramsey’s theorem for $$\ell$$-colorings, $$\mathsf {SRT}^2_\ell$$. Our proof actually establishes the following considerably stronger fact: given $$k > \ell$$, there is a coloring $$c : \omega \rightarrow k$$ such that for every stable coloring $$d : [\omega]^2 \rightarrow \ell$$ (computable from $$c$$ or not), there is an infinite homogeneous set $$H$$ for $$d$$ that computes no infinite homogeneous set for $$c$$. This also answers a separate question of the first author [J. Symb. Log. 81, No. 4, 1405–1431 (2016; Zbl 1368.03044)], as it follows that the cohesive principle, $$\mathsf {COH}$$, is not strongly computably reducible to the stable Ramsey’s theorem for all colorings, $$\mathsf {SRT}^2_{<\infty}$$. The latter is the strongest partial result to date in the direction of giving a negative answer to the longstanding open question of whether $$\mathsf {COH}$$ is implied by the stable Ramsey’s theorem in $$\omega$$-models of $$\mathsf {RCA}_0$$.

##### MSC:
 03D80 Applications of computability and recursion theory 03F35 Second- and higher-order arithmetic and fragments 05D10 Ramsey theory 03B30 Foundations of classical theories (including reverse mathematics) 03D30 Other degrees and reducibilities in computability and recursion theory
Full Text:
##### References:
 [1] Brattka-bib Vasco Brattka, Bibliography on Weihrauch complexity, website: \tt http://cca-net.de/publications/weibib.php · Zbl 1140.03040 [2] BR-TA Vasco Brattka and Tahina Rakotoniaina, On the uniform computational content of Ramsey’s theorem, to appear. · Zbl 1422.03132 [3] Cholak, Peter A.; Dzhafarov, Damir D.; Hirst, Jeffry L.; Slaman, Theodore A., Generics for computable Mathias forcing, Ann. Pure Appl. Logic, 165, 9, 1418-1428, (2014) · Zbl 1320.03072 [4] CDS-TA Peter A. Cholak, Damir D. Dzhafarov, and Mariya I. Soskova, Generics for Mathias forcing over general Turing ideals, Israel J. Math., to appear. [5] Cholak, Peter A.; Jockusch, Carl G.; Slaman, Theodore A., On the strength of Ramsey’s theorem for pairs, J. Symbolic Logic, 66, 1, 1-55, (2001) · Zbl 0977.03033 [6] Chong, C. T.; Lempp, Steffen; Yang, Yue, On the role of the collection principle for $$Σ ^0_2$$-formulas in second-order reverse mathematics, Proc. Amer. Math. Soc., 138, 3, 1093-1100, (2010) · Zbl 1195.03015 [7] Dorais, Fran\ccois G.; Dzhafarov, Damir D.; Hirst, Jeffry L.; Mileti, Joseph R.; Shafer, Paul, On uniform relationships between combinatorial problems, Trans. Amer. Math. Soc., 368, 2, 1321-1359, (2016) · Zbl 06560459 [8] Dzhafarov, Damir D., Cohesive avoidance and strong reductions, Proc. Amer. Math. Soc., 143, 2, 869-876, (2015) · Zbl 1386.03055 [9] Dzhafarov-TA Damir D. Dzhafarov, Strong reductions between combinatorial principles, J. Symbolic Logic, to appear. · Zbl 1368.03044 [10] GS-TA Marcia J. Groszek and Theodore A. Slaman, Moduli of computation, in preparation. [11] Hirschfeldt, Denis R., Slicing the truth, Lecture Notes Series. Institute for Mathematical Sciences. National University of Singapore 28, xvi+214 pp., (2015), World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ · Zbl 1304.03001 [12] HJ-TA Denis R. Hirschfeldt and Carl G. Jockusch, Jr., On notions of computability theoretic reduction between $$Π ^1_2$$ principles, to appear. · Zbl 1373.03068 [13] Hirschfeldt, Denis R.; Shore, Richard A., Combinatorial principles weaker than Ramsey’s theorem for pairs, J. Symbolic Logic, 72, 1, 171-206, (2007) · Zbl 1118.03055 [14] Hirst, Jeffry Lynn, COMBINATORICS IN SUBSYSTEMS OF SECOND ORDER ARITHMETIC, 153 pp., (1987), ProQuest LLC, Ann Arbor, MI [15] Jockusch, Carl; Stephan, Frank, A cohesive set which is not high, Math. Logic Quart., 39, 4, 515-530, (1993) · Zbl 0799.03048 [16] Lerman, Manuel; Solomon, Reed; Towsner, Henry, Separating principles below Ramsey’s theorem for pairs, J. Math. Log., 13, 2, 1350007, 44 pp., (2013) · Zbl 1326.03021 [17] Mileti, Joseph Roy, Partition theorems and computability theory, 76 pp., (2004), ProQuest LLC, Ann Arbor, MI [18] Patey-TA Ludovic Patey, The weakness of being cohesive, thin or free in reverse mathematics, Israel J. Math., to appear. · Zbl 1368.03018 [19] Rakotoniaina-2015 Tahina Rakotoniaina, \em The Computational Strength of Ramsey’s Theorem, PhD thesis, University of Cape Town, 2015. [20] Shore, Richard A., The Turing degrees: an introduction. Forcing, iterated ultrapowers, and Turing degrees, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 29, 39-121, (2016), World Sci. Publ., Hackensack, NJ [21] Simpson, Stephen G., Subsystems of second order arithmetic, Perspectives in Logic, xvi+444 pp., (2009), Cambridge University Press, Cambridge; Association for Symbolic Logic, Poughkeepsie, NY · Zbl 1181.03001 [22] CSY-TA Theodore A. Slaman and Yue Yang, The metamathematics of stable Ramsey’s theorem for pairs, to appear. · Zbl 1341.03015 [23] Soare-TA Robert I. Soare, \em Computability theory and applications, Theory and Applications of Computability. Springer, New York, to appear. · Zbl 1350.03001 [24] Solovay, Robert M., Hyperarithmetically encodable sets, Trans. Amer. Math. Soc., 239, 99-122, (1978) · Zbl 0411.03039 [25] Weihrauch-1992 K. Weihrauch, The degrees of discontinuity of some translators between representations of the real numbers, Technical report TR-92-050, International Computer Science Institute, Berkeley, 1992.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
# discretize.CylindricalMesh.boundary_node_vector_integral#

property CylindricalMesh.boundary_node_vector_integral#

Represents the operation of integrating a vector function dotted with the boundary normal

This matrix represents the boundary surface integral of a vector function dotted with the boundary normal and multiplied with a scalar finite volume test function on the mesh.

Returns
(n_nodes, dim * n_boundary_nodes) scipy.sparse.csr_matrix

Sparse matrix of shape.

Notes

The integral we are representing on the boundary of the mesh is

$\int_{\Omega} (w \vec{u}) \cdot \hat{n} \partial \Omega$

In discrete form this is:

$w^T * P @ u_b$

where w is defined on all nodes, and u_b is all three components defined on boundary nodes.


text:
# Vision Transformer

In computer vision, we usually approach the problem by using convolutional architectures due to how effective they are. From small scale to large scale, CNN-based methods remain dominant. Meanwhile, Transformer is a go-to method to work with natural language processing (NLP) because it is highly computational efficient and scalable. Nowadays we can train 500B parameters with self-attention-based architecture. Inspired from NLP success, Vision Transformer (ViT) [1] is a novel approach to tackle computer vision using Transformer encoder with minimal modifications. While small and middle-size dataset are ViT's weakness, further experiment show that ViT performs well and easily achieve state-of-the-art results. Furthermore, with pre-trained model on ImageNet-21K, ViT now can easily beats SoTA on small dataset like CIFAR10. In this blog, we will introduce the basic idea of Vision Transformer.

## #Vision Transformer (ViT)

Figure 1: Model overview. An image input is split into multiple patches, then we see them as normal token that will be embedded and feed into the encoder. To perform classification task, an extra learnable parameter is added at the beginning of the sequence, and and MLP head at its output position.

Unlike normal Transformer, which needs encoder and decoder, Vision Transformer only need the encoder part with a little modifications because our main task is image classification. The first modification is to split the input images into several patches, then feed them using multilayer perceptron or a simple CNN model to encode them. The second modification is to attach a [class] token at the beginning of the embedding input sequence, its output will behave like the image representation. Formally, we reshape the image $\mathbf{x} \in \mathbb{R}^ {H \times W \times C} \rightarrow (\mathbf{x}_{p}) \in \mathbb{R}^{N \times\left(P^{2} \cdot C\right)}$, with $\mathbf{x}_{p} \in \mathbb{R}^{\left(P^{2} \cdot C\right)}$, where:

• $(H, W)$ is the resolution of the original image,
• $C$ is the number of channels,
• $(P, P)$ is the resolution of each image patch,
• $N=H W / P^{2}$ is the resulting number of patches

Similar to BERT's [class] token, a learnable embedding $z_0^0 = \mathbf{x}_{\text{class}}$ is added at the beginning of the embedded patches sequence. And after $L$ encoder layers $z_0^0 \rightarrow \underbrace{\text{encoder} \rightarrow \text{encoder} \rightarrow \cdots}_{L} \rightarrow z_0^L$, the final $z_0^L$ is the image representation. During pretraining and fine-tuning, a MLP to classify $z_0^L$ is added at its position. Position embedding is the standard 1-D Positional Embedding. Note that all MLP in the encoder consists of 2 layers with a GELU non-linearity. Formally, the ViT encoder is:

$\mathbf{z}_{0} =\left[\mathbf{x}_{\text {class }} ; \mathbf{x}_{p}^{1} \mathbf{E} ; \mathbf{x}_{p}^{2} \mathbf{E} ; \cdots ; \mathbf{x}_{p}^{N} \mathbf{E}\right] + \mathbf{E}_{\text {pos }}, \mathbf{E} \in \mathbb{R}^{\left(P^{2} \cdot C\right) \times D}, \mathbf{E}_{\text {pos }} \in \mathbb{R}^{(N+1) \times D}$
$\mathbf{z}_{\ell}^{\prime} = \operatorname{MSA} \left(\operatorname{LN}\left(\mathbf{z}_{\ell-1}\right)\right)+\mathbf{z}_{\ell-1}, \ell=1 \ldots L$
$\mathbf{z}_{\ell} = \operatorname{MLP}\left(\operatorname{LN}\left(\mathbf{z}_{\ell}^{\prime}\right)\right)+\mathbf{z}_{\ell}^{\prime}, \ell=1 \ldots L$
$\mathbf{y} =\operatorname{LN}\left(\mathbf{z}_{L}^{0}\right)$

Where $\operatorname{LN} (\cdot)$ is LayerNorm, $\operatorname{MSA}(\cdot)$ is multihead self attention, $\mathbf{E}$ is the embedding matrix, $\mathbf{z}$ is the embedded vector and $D$ is the hidden dimension. The more visual illustration about the model is shown in Figure 1.

## #Vision Transformer's variants

There are three main variants: Base, Large, and Huge. The detail is shown in Table 1. When we are picking which backbone or variants to use, usually it is denoted as ViT - <size> / <patch size>, for example ViT-B/8 means the Base variants with $8 \times 8$ patch size.

ModelLayersHidden size $D$MLP sizeHeadsParams
ViT - Base1276830721286M
ViT - Large241024409616307M
ViT - Huge321280512016632M

Table 1: Detais of ViT model variants.

While increasing the model size usually help increasing the performance on large dataset, it is not the case for Vision Transformer. As shown in Table 2, some variants perform worse than the Base architecture. Therefore, we need to keep in mind about this characteristic when working with ViT.

ImageNetCIFAR-1098.1397.7797.8697.94-
CIFAR-10087.1386.3186.3587.04-
Oxford Flowers-10289.4985.4389.6686.36-
Oxford-IIIT-Pets93.8192.0493.6491.35-
ImageNet-21kCIFAR-1098.9598.7999.1699.1399.27
CIFAR-10091.6791.9893.4493.0493.82
Oxford Flowers-10299.3899.1199.6199.1999.51
Oxford-IIIT-Pets94.4393.0294.7393.0994.82

Table 2: Top1 accuracy (in %) of ViT variants on various datasets when pretrained on ImageNet and ImageNet-21k.

## #Reference

[1] Dosovitskiy, A., Beyer, L., Kolesnikov, A., Weissenborn, D., & Zhai, X. (2021). Thomas Unterthiner Mostafa Dehghani Matthias Minderer Georg Heigold Sylvain Gelly Jakob Uszkoreit and Neil Houlsby. An image is worth 16x16 words: Transformers for image recognition at scale. In International Conference on Learning Representations.


text:
# finding 4-torsion point on elliptic curve

Hi I have the following:

E=EllipticCurve(Qp,[0,0,0,-3267,45630])
P=E([-21,324])

Is there some coding I can use to get the 4-torsion point of this elliptic curve. I reckon it might have something to do with the nth divisional polynomial but I couldn't find it.

edit retag close merge delete

We are missing a definition of Qp to be able to play with your code.

( 2016-11-02 09:32:37 +0100 )edit
1

I think the terminology is as follows.

Points Q such that 4 Q = P are called 4-division points of P on E.

In the special case of P = E(0), they are called 4-torsion points of E.

( 2016-11-02 09:50:19 +0100 )edit

@slelievre Thank you. That definitely make sense.

( 2016-11-02 09:58:07 +0100 )edit

Sort by » oldest newest most voted

I think the method you are looking for is called division_points in Sage.

See the SageMath documentation for division points of elliptic curves which you can find in a web search engine with the query [elliptic curve torsion point sagemath].

Your example, with a guess for what Qp might be.

sage: Qp = pAdicField(17)
sage: E = EllipticCurve(Qp, [0, 0, 0, -3267, 45630])

Starting with the point of x-coordinate zero as P --- works fine.

sage: P = E(0); P
(0 : 1 + O(17^20) : 0)
sage: P.division_points(4)
[(15 + O(17^20) : O(17^10) : 1 + O(17^20)), (0 : 1 + O(17^20) : 0)]

Starting with your choice of P, I get an error --- but maybe I have the wrong Qp.

sage: P = E([-21, 324]); P
(13 + 15*17 + 16*17^2 + 16*17^3 + 16*17^4 + 16*17^5 + 16*17^6 + 16*17^7 + 16*17^8 + 16*17^9 + 16*17^10 + 16*17^11 + 16*17^12 + 16*17^13 + 16*17^14 + 16*17^15 + 16*17^16 + 16*17^17 + 16*17^18 + 16*17^19 + O(17^20) : 1 + 2*17 + 17^2 + O(17^20) : 1 + O(17^20))
sage: P.division_points(4)
Traceback (most recent call last)
...
PrecisionError: p-adic factorization not well-defined since the discriminant is zero up to the requestion p-adic precision

Different choices of Qp give different results.

sage: Qp = pAdicField(5)
sage: E = EllipticCurve(Qp, [0, 0, 0, -3267, 45630])
sage: P = E([-21, 324]); P
(4 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + 4*5^10 + 4*5^11 + 4*5^12 + 4*5^13 + 4*5^14 + 4*5^15 + 4*5^16 + 4*5^17 + 4*5^18 + 4*5^19 + O(5^20) : 4 + 4*5 + 2*5^2 + 2*5^3 + O(5^20) : 1 + O(5^20))
sage: P.division_points(4)

[(4 + 3*5 + 4*5^2 + 5^3 + 5^4 + 5^5 + 3*5^6 + 3*5^7 + 3*5^8 + 4*5^9 + 4*5^10 + 4*5^11 + 5^12 + 3*5^13 + 3*5^14 + 5^15 + 3*5^16 + 3*5^18 + O(5^20) : 1 + 4*5 + 4*5^2 + 4*5^4 + 2*5^5 + 4*5^7 + 2*5^8 + 3*5^9 + 5^10 + 4*5^12 + 4*5^13 + 3*5^14 + 3*5^15 + 4*5^17 + 2*5^18 + 4*5^19 + O(5^20) : 1 + O(5^20)),
(2 + 3*5 + 2*5^2 + 5^3 + 5^4 + 2*5^5 + 2*5^6 + 3*5^7 + 5^8 + 5^9 + 5^11 + 4*5^12 + 5^13 + 3*5^14 + 2*5^16 + 5^17 + 5^19 + O(5^20) : 2 + 5^4 + 2*5^5 + 3*5^6 + 4*5^8 + 5^9 + 2*5^10 + 3*5^11 + 2*5^12 + 3*5^14 + 4*5^15 + 3*5^16 + 4*5^17 + 4*5^18 + 3*5^19 + O(5^20) : 1 + O(5^20))]

sage: Qp = pAdicField(7)
sage: E = EllipticCurve(Qp, [0, 0, 0, -3267, 45630])
sage: P = E([-21, 324]); P
(4*7 + 6*7^2 + 6*7^3 + 6*7^4 + 6*7^5 + 6*7^6 + 6*7^7 + 6*7^8 + 6*7^9 + 6*7^10 + 6*7^11 + 6*7^12 + 6*7^13 + 6*7^14 + 6*7^15 + 6*7^16 + 6*7^17 + 6*7^18 + 6*7^19 + 6*7^20 + O(7^21) : 2 + 4*7 + 6*7^2 + O(7^20) : 1 + O(7^20))
sage: P.division_points(4)
[]

more

thank you.. you are right. we can't work with prime 2,3,17 as they are the bad primes on the curve. Thank you this helped a lot..

( 2016-11-02 09:45:35 +0100 )edit

( 2022-10-21 16:38:37 +0100 )edit


text:
# 1.19 Evaluate those of the following that are de?ned: (a) 5 C 3 , (b) 3 C 5 , (c) - 5 C 3 , (d) -...

1.19              Evaluate those of the following that are deﬁned: (a) 5C3, (b) 3C5, (c) −5C3, (d)

−3C5.

1.20              Use  a  binomial  expansion  to  evaluate  1/√4.2  to  ﬁve  places  of  decimals,  and

compare it with the accurate answer obtained using a calculator.

1.21              Prove by induction that

n

 2

, r = 1 n(n + 1)               and

r=1

1.22              Prove by induction that

n

 4

, r3 = 1 n2(n + 1)2.

r=1

1 − rn+1

1 + r + r2  + ·· · + rk + ·· · + rn =

.

1 − r

1.23              Prove that 32n + 7, where n is a non-negative integer, is divisible by 8.

1.24              If a sequence of terms, un, satisﬁes the recurrence relation un+1 = (1 − x)un + nx, with u1 = 0, show, by induction, that, for n ≥ 1,

1                                        n

1.25              Prove by induction that

n

un = x [nx − 1 + (1 − x) ].

, 1

2r  tan

θ

2r

1

=          cot 2n

θ

2n

− cot θ.

r=1


text:
# Temporal Disaggregation to High-Frequency

## Convert between any frequency

Since version 1.0, tempdisagg can convert between most frequencies, e.g., it can disaggregate a monthly series to daily. It is no longer restricted to regular conversions, where each low frequency period had the same number of high frequency periods. Instead, a low frequency period (e.g. month) can contain any number of high-frequency periods (e.g. 31, 28 or 29 days). Thanks to Roger Kissling and Stella Sim who have suggested this idea.

We can not only convert months to days, but also years to days, weeks to seconds, or academic years to seconds, or lunar years to hours, … The downside is that the computation time depends on the number of observations. Thus, for longer high-frequency series, the computation may take a while.

## Quarterly to daily disaggregation

In the following, we try to disaggregate quarterly GDP of Switzerland to a hypothetical daily GDP series. The example series are shipped with the package.

library(tempdisagg)
data(tempdisagg)
#>         time    value
#> 1 2005-01-01 133101.3
#> 2 2005-04-01 136320.4
#> 3 2005-07-01 137693.7
#> 4 2005-10-01 139475.9
#> 5 2006-01-01 139204.7
#> 6 2006-04-01 141112.5

Because we are dealing with daily data, we keep the data in a data.frame, rather than in a ts object. Other time series objects, such as xts and tsibble, are possible as well. For conversion and visualization, we use the tsbox package.

library(tsbox)
ts_plot(gdp.q, title = "Swiss GDP", subtitle = "real, not seasonally adjusted")

## No indicator

For models without an indicator series, method = "fast" uses the Chow-Lin method with a rho parameter value very close to 1. This is numerically close to the theoretically preferable Denton-Cholette method but much faster. Faster, as stated above, has a relative meaning; the following calculation still takes a few seconds:

m.d.noind <- td(gdp.q ~ 1, to = "daily", method = "fast")
summary(m.d.noind)
#>
#> Call:
#> td(formula = gdp.q ~ 1, to = "daily", method = "fast")
#>
#> Residuals:
#>      Min       1Q   Median       3Q      Max
#> -20275.5  -6704.0    371.7  10125.2  22872.8
#>
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)
#> (Intercept)   1704.2      517.6   3.292   0.0017 **
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> 'chow-lin-fixed' disaggregation with 'sum' conversion
#> 59 low-freq. obs. converted to 5386 high-freq. obs.
#> Adjusted R-squared: 0.09057  AR1-Parameter:     1

Because daily GDP is much lower than monthly GDP, we scale both series to the same mean and standard deviation:

gdp.d.noind <- predict(m.d.noind)
ts_plot(
ts_scale(
ts_c(gdp.d.noind, gdp.q)
),
title = "Daily disaggregated GDP",
subtitle = "no indicator"
)

Like with all disaggregation methods in tempdisagg, the resulting series fulfills the aggregation constraint:

all.equal(ts_frequency(gdp.d.noind, "quarter", aggregate = "sum"), gdp.q)
#> [1] TRUE

## Stock market data as an indicator

For a more refined disaggregation, one or several indicator series may be used. In the following, we use Swiss stock market data to disaggregate GDP. Data of the stock market index, the SMI, is also included in tempdisagg:

ts_plot(spi.d, title = "Swiss Performance Index", subtitle = "daily values, interpolated")

Weekend and holiday values have been interpolated, because td does not allow the presence of missing values. For how to interpolate the raw data, see ?spi.d.

The following uses the Chow-Lin method to disaggregate the series. A high rho parameter takes into account that the two series are unlikely to be co-integrated.

m.d.stocks <- td(gdp.q ~ spi.d, method = "chow-lin-fixed", fixed.rho = 0.9)
summary(m.d.stocks)
#>
#> Call:
#> td(formula = gdp.q ~ spi.d, method = "chow-lin-fixed", fixed.rho = 0.9)
#>
#> Residuals:
#>    Min     1Q Median     3Q    Max
#> -10656  -1760   1076   3796   8891
#>
#> Coefficients:
#>              Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 1.320e+03  2.856e+01   46.22   <2e-16 ***
#> spi.d       5.512e-02  3.735e-03   14.76   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> 'chow-lin-fixed' disaggregation with 'sum' conversion
#> 59 low-freq. obs. converted to 5493 high-freq. obs.
#> Adjusted R-squared: 0.7928   AR1-Parameter:   0.9

This results in a much more volatile estimation of daily GDP:

gdp.d.stocks <- predict(m.d.stocks)
ts_plot(
ts_scale(
ts_c(gdp.d.stocks, gdp.q)
),
title = "Daily disaggregated GDP",
subtitle = "one indicator"
)