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# Heart Rate Variability reveals the fight between racially biased and politically correct behaviour

## Abstract

In this study, we explored vagally-mediated heart rate variability (vmHRV) responses, a psychophysiological index of cognitive self-regulatory control, to map the dynamics associated with empathic responses for pain towards an out-group member. Accordingly, Caucasian participants were asked to judge the experience of African and Caucasian actors touched with either a neutral or a harmful stimulus. Results showed that (1) explicit judgment of pain intensity in African actors yielded higher rating score and (2) took longer time compared to Caucasian actors, (3) these behavioural outcomes were associated with a significant increment of RMSSD, Log-HF-HRV and HF-HRV n.u., (4) resting HF-HRV n.u. predicted the participants’ lag-time to judge painful stimulations delivered to African actors. Interestingly, these dynamics were associated with a measure of implicit racial attitudes and were, in part, abolished when participants performed a concurrent task during videos presentation. Taken together our results support the idea that a cognitive effort is needed to self-regulate our implicit attitude as predicted by the ‘Contrasting Forces Model’.

## Introduction

Empathy represents a cornerstone of human life1 as the ability to understand and share others’ emotions and mental states2,3 is at the basis of adaptive social interactions. For this reason, over the last decades clinical and cognitive psychologists, as well as neuroscientists, deeply investigated its behavioral, cognitive and neural correlates. One of the most accepted models in the literature suggests that empathy may be divided at least into two dissociable, although interacting, components: ‘affective empathy’ and ‘cognitive empathy’4,5. The former is regarded as a phylogenetically ancient automatic response that allows humans to vicariously experience the affective state of others by embodied simulation6. The latter process is closely tied to the mentalizing mechanism necessary to understand other’s perspective; this latter component may be affected by personal and contextual factors. From the theoretical point of view, cognitive empathy is strictly related to the concept of ‘perspective taking’7 and with the theory-of-mind1, as a consequence, while affective empathy may manifest itself also in the absence of ‘cognitive control’, cognitive empathy entails a certain degree of awareness1,2. In keeping with this model, various self-report measures (such as the Empathy Quotient8), as well as behavioral paradigms (such as the ‘Reading the Mind in the Eyes’ Test9) have supported the existence of the dichotomy between bottom-up affective and top-down cognitive processes during the unfolding of empathic responses. In line with the results of the self-report and behavioral studies, a number of neuroimaging evidence suggests that these two components can dissociate also at the neurofunctional level. Accordingly, affective empathy, as being responsible for the initial decoding phase of empathic processing, engages brain regions typically associated with motivational-affective dimensions, such as the insular and the anterior cingulate cortices, thalamus, amygdala, fusiform gyrus, somatosensory and motor cortices (see2,3 for details reviews). On the contrary, cognitive empathy engages brain regions typically associated with executive, higher-order functions, such as ventromedial, dorsolateral and dorsomedial prefrontal cortices, superior temporal gyrus, temporo-parietal junction, superior and inferior parietal lobules (see1,2 for details reviews).

## Results

### Participants and behavioral scales

Male and female participants were matched both for age [t(46) = 0.218, p = 0.8283; see Table 1] and, for the Body Mass Index [t(46) = 1.893, p = 0.0646; see Table 1]. A series of further Student’s t-test also revealed no differences between male and female participants in Resting Root Mean square of Successive Differences - RMMSD [t(46) = 0.261, p = 0.7949; see Table 1], Resting natural Log transformed High-Frequency power - Log-HF-HRV [t(46) = 0.377, p = 0.7079; see Table 1], Resting High-Frequency power normalized units - HF-HRV n.u [t(46) = 0.276, p = 0.7837; see Table 1], Resting peak High-Frequency values – pHF-HRV [t(46) = 1.144, p = 0.2585; see Table 1] and Symphatovagal balance Index – SVI [t(46) = 0.223, p = 0.8227; see Table 1]. On the other side, results also revealed no differences between male and female participants in Resting natural Log-transformed Low-Frequency power and Resting Low-Frequency power normalized units (results not shown, for details see Supplementary Information; S-Table 4). Regarding the implicit association measures (IAT24), overall our sample obtained a mean IAT D score of 0.80 (standard error of mean = 0.04; one sample Student’s t-test (H0: μ = 0): t(47) = 19.84, p < 0.001), which indicates a significant implicit association between negative words and Africans pictures. The descriptive statistics of the behavioral scales, Internal Motivation Scale and External Motivation Scale score, Scale for Ethnocultural Empathy score, Subtle and Blatant Prejudice score and Trait Empathy Scale score are reported in Table 2, it is worth noting that the sample included in this study behaves, on average, similarly to the participants included in previous studies16,18,20.

### Behavioral results

Based on previous findings by Berlingeri et al.20, we expected that (i) participants would judge the African actors’ harmful stimuli as painful as those administered to the Caucasian actors (see Methods and Fig. 1), (ii) they would need more time to produce an explicit response in line with social norms. To test this hypothesis, the explicit empathic responses and the RTs collected during both single and dual experimental task were analyzed by means of general linear mixed-effects model (GLMM; see Supplementary Information for further details).

Explicit behavioural scores (Scores; measured by a Likert scale from 1 to 9): in the single-task condition, results showed a significant main effect of type of stimuli (χ2 (1,1078) = 205.84; p < 0.001), a significant main effect of race (χ2 (1,1078) = 17.34; p < 0.001) and a significant type of stimuli-by-race interaction effect (χ2 (1,1078) = 8.50; p = 0.003). We further explored this interaction, and we observed that participants attributed higher explicit scores to the Africans’ experience with harmful stimuli when compared with scores elicited by Caucasian actors (χ2 (1,1078) = 24.90; pFDR-corrected < 0.001; see Fig. 2b).

Similarly, in the dual-task condition, results showed a significant main effect of type of stimuli (χ2 (1,976) = 164.82; p < 0.001), a significant main effect of race (χ2 (1,976) = 9.22; p = 0.002) and a significant type of stimuli-by-race interaction effect (χ2 (1,976) = 8.22; p = 0.004). This interaction was further explored, and we observed that, even with a concurrent task during videos presentation, higher explicit scores were attributed to the Africans’ experience with harmful stimuli (χ2 (1,976) = 17.37; pFDR-corrected < 0.001; see Fig. 2d).

RTs: in the single-task condition results showed a significant main effect of type of stimuli (χ2 (1,1047) = 7.23; p = 0.007) and a significant type of stimuli-by-race interaction effect (χ2 (1,1047) = 5.18; p = 0.022). Accordingly, there was a significant difference between the RTs collected for Caucasian and African actors in the painful condition (χ2 (1,1047) = 6.78; pFDR-corrected = 0.018) and a significant type of stimuli effect for Africans actors only (χ2 (1,1047) = 12.31; pFDR-corrected < 0.001; see Fig. 2a). On the other hand, in the dual-task condition, only a significant main effect of type of stimuli (χ2 (1,970) = 4.12; p = 0.04) was observed (see Fig. 2c).

### Psychophysiological results

In accordance with previous findings43,45,47, we expected that the greater self-regulatory effort needed to explicitly judge the painful experience of African actors should be associated with a higher vmHRV. To test this hypothesis, all the vmHRV measures calculated in both single and dual experimental task, namely, RMSSD (as time-domain measure of vmHRV), HF-HRV and HF-HRV n.u. (as frequency-domain measures of vmHRV), were analysed by means of a GLMM (see Supplementary Information for further details about the model’s syntax). RMSSD: in the single-task condition, results showed a significant main effect of race (χ2 (1,192) = 14.30; p < 0.001) and a significant type of stimuli-by-race interaction effect (χ2 (1,192) = 3.70; p = 0.050). This interaction was further explored, and we observed a higher RMSSD referred to Africans painful stimulation (χ2 (1,192) = 16.44; pFDR-corrected < 0.001; see Fig. 3a). Prior to analyse the frequency-domain measures of vmHRV, the peak of HF-HRV (pHF-HRV) a measure of respiratory frequency, was explored in both single and dual-task to control for potential bias induced by respiratory frequency during the experimental tasks33,53. In both single and dual-task experimental conditions, pHF-HRV average values were in the range of 0.15–0.23 Hz and no significant main effects were observed (results not shown, for details see Supplementary Information; S-Fig. 4), thus showing that respiratory frequency did not affect the frequency-domain measures of vmHRV54.

Frequency-domain measures of vmHRV: similarly, to the pattern of results that emerged from RMSSD, even with HF-HRV in the single-task condition we found a significant main effect of race (χ2 (1,192) = 3.70; p = 0.05), as well as, a significant type of stimuli-by-race interaction effect (χ2 (1,192) = 4.44; p = 0.035). In particular, participants showed higher HF-HRV referred to Africans painful stimulation (χ2 (1,192) = 8.12; pFDR-corrected = 0.008) and a significant type of stimuli effect for African actors (χ2 (1,192) = 5.74; pFDR-corrected = 0.032; see Fig. 3b). HF-HRV n.u.: in the single-task condition, we found a significant type of stimuli-by-race interaction effect (χ2 (1,192) = 5.70; p = 0.016) with a higher HF-HRV n.u. when African actors were touched with a harmful stimulus as compared to the Caucasians (χ2 (1,192) = 5.15; pFDR-corrected = 0.046) and a significant type of stimuli effect specific for Africans actors (χ2 (1,192) = 7.43; pFDR-corrected = 0.012; see Fig. 3c). Notwithstanding, the results of the post-hoc comparisons survived the false discovery rate correction, here it is worth noting that the lack of a complete cross-over interaction suggests that this latter result should be taken with a grain of salt. On the other hand, in the dual-task condition, no significant main effects were observed in all vmHRV measures (see Fig. 3d–f); the pattern of results emerged from Low Frequency-HRV (LF-HRV) analyses mirrored the one obtained with HF-HRV (for details on LF-HRV measures, see Supplementary Information S-Fig. 5).

### Resting HRV and reaction times

According to recent works34,38,39,42,43,45,51,52, the higher the level of vmHRV at rest, the higher the capacity to self-regulate our inner emotional state flexibly adapting our behavioural responses to social norms. A self-regulating effort is reflected by lag-time, according to the ‘Contrasting Forces Model’20, to explicitly judge a painful experience of an out-group member compared to an in-group member, and this lag-time is associated with an increase of the activity of the prefrontal cortex: an area typically associated with self-regulation26, strategic and controlled behavior27. Collectively these findings prompted us to hypothesise that the delay in rating the painful experience of out-group actors could be likely related to the involvement of prefrontal-subcortical inhibitory circuits implicated in the self-regulatory effort of emotional and cognitive processes37,39,51. To explicitly test this hypothesis, the relationship between resting vmHRV and Δ-RT-AC was explored by means of a General Linear Model (GLM) in the single-task only. The results of the GLM showed that the resting HF-HRV n.u. is a significant predictor of Δ-RT-AC (b = 20.92, t(43) = 2.79, p = 0.005; for more details see Supplementary Information S-Fig. 2), thus with higher HF-HRV n.u., we are expecting a longer delay in judging the painful experience for Africans as compared with Caucasians (see Fig. 4). RMSSD and Log-HF-HRV did not show any significant linear relationship to Δ-RT-AC.

### HRV changes and implicit racial bias

Based on the findings by Forgiarini et al.16, we expected that in a single-task condition vmHRV differences between the out-group vs. in-group painful stimulations should correlate with the subjective level of implicit racial bias (IAT). To test this hypothesis, the association between IAT D scores and Δ-vmHRV-AC values in single-task condition was explored by means of a GLM. The results showed that the IAT D score is a significant predictor of both Δ-RMSSD-AC (b = 9.34, t(44) = 3.91, p = 0.04; see Fig. 5a;) and Δ-HF-HRV n.u.-AC (b = 14.72, t(44) = 4.00, p = 0.04; see Fig. 5b; for more details see Supplementary Information S-Fig. 3), thus for every increase in the IAT D score, we are expecting an increase in the Δ-RMSSD- and Δ-HF-HRV n.u.-AC measures (see Fig. 5).

## Discussion

The purpose of the present study was to investigate whether self-regulatory effort needed to explicitly judge the pain experienced by an out-group member (out-group DEAR effect) is associated with specific vagally-mediated HRV (vmHRV) changes and whether these changes can be affected by the subjective level of implicit racial attitude.

From a behavioral point of view, our data confirm and extend previous findings by Berlingeri et al.20. In particular, we replicate that, on average, significant longer RTs (approximately 100 ms) are needed to explicitly judge the painful experience stimulation of African than Caucasian actors (see Fig. 2a). Moreover, by using a nine-digit-Likert scale providing a finer grain characterization (thus higher resolution) of the out-group DEAR effect, we extend previous findings showing that significant higher explicit scores were attributed to harmful stimuli experienced by Africans than Caucasians actors (see Fig. 2b). Thus, consistent with previous findings20 our data confirm that in the single-task condition (i.e. in absence of additional cognitive load) participants needed more time to explicitly judge the painful experience of an out-group actor, and for the first time we show that participants, in order to adapt their behavior to social norms, judge the level of pain experienced by out-group actor higher than the one experiences by in-group actors (see Fig. 2a,b). This latter finding is particularly intriguing from a social point of view since it seems to suggest that the social norms and social desirability can elicit such a pressure to promote something more than an egalitarian response, i.e. an overcorrection of our explicit empathic response that, paradoxically, may become unfair toward in-group members, something, in a way, similar to the neurocognitive processes described in the ‘altruistic behavior’55.

From a psychophysiological point of view, our data show that the out-group DEAR effect is associated with specific vmHRV changes measured in both time and frequency domains. In particular, we found a significant, on average, enhancement of RMSSD, HF-HRV and HF-HRV n.u. associated with the task of giving an explicit judgment of pain experienced by African actors (see Fig. 3a–c). According to the ‘Neurovisceral Integration Model’ proposed by Thayer and Lane30,31, vmHRV enhancement would reflect the engagement of prefrontal-subcortical inhibitory circuits, the core circuitry of the ‘central autonomic network’, which play a critical role in self-regulatory function by dynamically coupling central neural activity with peripheral autonomic drives in the service of behavioral flexibility56. In this regard, vmHRV enhancement was observed in several behavioral tasks (e.g. food temptation task43, emotional suppression or emotion reappraisal task45) which demanded self-regulatory effort. Our data extend these previous findings by showing for the first time, in a model of empathic reaction for pain stimuli, that vmHRV enhancement can be regarded as a reliable psychophysiological marker reflecting the cognitive self-regulatory effort needed to mitigate our negative attitudes and to adapt our behavior to social norms in an empathy for pain task. Present findings are in line with previous neurofunctional findings revealing that the out-group DEAR effect is associated with the activity of a cortical area involved in self-regulation and cognitive reappraisal process as the dorsolateral prefrontal cortex20.

Interestingly, the administration of a concurrent task, imposing high cognitive load on the executive prefrontal system (i.e. dual-task condition), was able to abolish the differences observed in single-task condition in RTs (see Fig. 2c) and psychophysiological outcomes (see Fig. 3d–f). These findings, in accordance with Yzerbty et al.28, further support that a prefrontal top-down self-regulatory drive is needed to mitigate our negative attitudes and to manifest an overt behavioral response toward out-group members in line with social rules. However, it worth noting that, even in dual-task condition, our participants showed an overcorrection toward out-group members (see Fig. 2d). This finding supports the assumption that social norms and education can shape our neurocognitive system to such an extent that the inner racial-bias may be counterbalanced also in a relatively stressful condition, and that a higher cognitive load, more stressful than that used in the present study, is needed to possibly unveil negative attitudes, something that in real-words conditions can actually happen.

Previous psychophysiological studies showed that individuals with high resting vmHRV are more efficient in regulating both emotional and cognitive processes during explicit emotion regulation tasks38,39,51. Our findings are partially in line with this evidence. In fact, we showed that resting HF-HRV n.u. is a predictor of the out-group DEAR effect, i.e. the higher was participant’s resting HF-HRV n.u., the longer was the lag-time to judge the pain experienced by African vs. Caucasian actors explicitly. However, resting RMSSD and HF-HRV failed to predict the out-group DEAR effect. Besides differences in the experimental task used in the present study (empathic responses for pain task vs explicit emotional regulation task), this apparent discrepancy may be related to the integrated nature of the HF-HRV n.u., representing the HF-HRV values in proportion to the total power (thus including LF-HRV) minus the VLF components54. Collectively, present evidence suggests, that resting HF-HRV n.u. may be adopted as a reliable predictor of the attitude to cognitive reappraisal in the serve of counterbalancing the inner biases during empathic responses for pain toward out-group members (Fig. 4).

Finally, we analyzed the relationship between vmHRV signatures of the out-group DEAR effect (an explicit explanation of how this effect was computed can be found in the methods section) and the subjective level of implicit racial bias (IAT). Our participants showed, overall, a significant implicit association between negative words and Africans’ pictures (evaluated by means of IAT D score) to an extent that was comparable with previous reports16,20. Interestingly, we found that IAT D scores were positively associated with RMSSD and HF-HRV n.u. enhancement when judging out-group harmful stimuli (with HF-HRV showing a positive trend but not reaching the significance see Fig. 5). Therefore, we can conclude that the IAT D scores of the participants significantly predict the vagally-mediated racially-driven empathic responses for pain. This latter finding may be seen as a complement of previous findings by Forgiarini et al.16 showing that, when watching harmful stimuli inflicted to out-group members, IAT D scores were negatively correlated with the amplitude of skin conductance response, a sympathetic psychophysiological index of the autonomic response in empathic reactions for pain.

Of course, the present study is not free from limitations. Firstly, future studies should better characterise participants empathy for pain and vicarious pain by adding, for example, a pain-specific empathy questionnaire like the Empathy for Pain Scale57 to complement the results obtained with the Trait Empathy Scale. In addition, we are aware that gender-related differences in vmHRV parameters have been reported in literature58; in a recent meta-analysis, Koenig and Thayer58 reported that females, as compared to males, showed greater resting HF-HRV power. However, in our sample, we failed to observe any major differences in resting HF-HRV between males and females (see Table 1). Sex-related differences tend to increase with advancing age58; therefore, we can hypothesize that the similarity of resting vmHRV values between female and male participants observed in our study might be due, at least in part, to the relatively young age of the participants. Concerning the abolishment of vmHRV differences in the dual-task setting, it cannot be ruled out, at least in part, that this could reflect a cardiac involvement in a stress response. Indeed, it is worth mentioning that a mental arithmetic task could evoke a stress response, as reported by several studies (for a detailed review see59, where a variety of cognitive tasks, including mental arithmetical tests, especially when performed under social evaluation, evoke changes in neuro-vegetative balance typical of acute stress responses. Finally, future studies should also consider a possible correlation between IAT D scores and measures of participants’ executive function.

In conclusion, here we show for the first time that vmHRV, a well-known psychophysiological index of prefrontal top-down executive/cognitive control, is a reliable index of the self-regulatory effort needed to adapt our behavior to social norms when we are required to empathize with the pain experienced by an out-group member. Accordingly, we gave the first psychophysiological picture of the contrasting forces model in action to suggest that despite the well-documented existence of implicit racially-biased empathic responses for pain16,17,18,19,20,21,22, we are able to exhibit controlled behaviors toward out-group members by cognitive self-regulating our-selves19,20,21. Interestingly, our findings could be considered also like the ‘other side of the coin’ of the results reported in the study by Tamir and Mitchell60 in which it has been showed that the anchoring and adjustment process is specifically engaged for in-group members, and not for out-group members where, as shown by our findings, a self-regulation process seems to be implicated.

This is an important empirical finding with a promising transferable value in the field of applied psychology and in supporting social and educational programs capable of promoting racial integration in modern societies.

## Methods

### Participants

Participants were recruited according to the following criteria: aged between 18 and 35 years, native Italian speakers and Caucasians. According to established recommendations for using vmHRV in psychophysiological research54, an ad-hoc created questionnaire was administered to all participants at the beginning of the experimental session. The ad-hoc created questionnaire included items associated with the following variables: gender, smoking, habitual levels of alcohol consumption, weight and height, cardioactive and psychotropic medications, oral contraceptive intake for female participants, follow a normal sleep routine the day before the experiment, no intensive physical training the day before the experiment, no meal the last 2 h before the experiment, no coffee – or caffeinated drinks or tea in the 2 h before the experiment and no alcohol consumption for 24 h prior the experiment (see Supplementary Information for further details about the ad-hoc created questionnaire; Fig. S-1).

In keeping with an a priori analysis performed with G*Power361, we estimated to recruit 40 participants to be able to detect large effects size (f = 0.72, 1-β = 0.95, α = 0.01) about the association between specific vmHRV changes and self-regulatory effort while explicitly judging the severity of pain inflicted to an out-group member, namely, out-group DEAR effect. This analysis was performed by taking into account the results reported in the paper by Forgiarini et al.16, i.e. by considering the study that is closer to ours in terms of experimental paradigm, stimuli, and collected measures. Thus, taking into account potential outliers in the baseline condition, fifty-two young and healthy Caucasian participants were recruited among undergraduate university students and young workers. Of the fifty-two participants that were included in the study, we excluded data from four (females) who displayed high resting stress level values (assessed as Sympathovagal Balance Index62,63,64) during the 5-min resting HR recordings (for more details see the Behavioral and physiological analyses section). Consequently, the data of forty-eight participants, 27 males and 21 females, were considered in the analyses (see Table 1). All participants provided their written informed consent to take part in the study that was approved by the Ethics Board of the University of Urbino – Carlo Bo, and carried out in accordance with the Declaration of Helsinki65.

### Behavioural scales

Once arriving at the Neuropsychology and Psychometrics laboratory of the University of Urbino, all participants were individually tested in a quiet and dimly-lit room (temperature 22–23 °C). Before experimental task and vmHRV recording, participants completed the Internal Motivation to Respond Without Prejudice Scale (IMS)66, the Scale for Ethnocultural Empathy (EES)67, the Subtle and Blatant Prejudice Scales68,69 and the Trait Empathy Scale (TES)70. Participants also completed a race (Caucasian and African) Implicit Association Test (IAT) to assess the implicit racial bias24. In particular, in each trial of the race IAT, participants categorized a stimulus from one of the following four categories: a picture of Caucasian man, a picture of an African man, a positive word (e.g. joy, peace, love, good), or negative word (e.g. agony, war, pain, evil). The stimuli were organized into seven blocks. The critical blocks consisted of 24 trials. The IAT scores were calculated using an ad-hoc-created R routine to obtain the D score as described in Greenwald et al.71. The D scores were coded in the direction of the association between positive words and Caucasian targets; as a consequence, the higher the score, the higher the association between positive concepts and the Caucasian race (as well as the stronger the association between negative concepts and African actors).

Stimuli were 12-second-long video clips depicting male or female actors (six Caucasian, 3 males and 3 females; six African, 3 males and 3 females) touched on their hand with either painful or neutral stimulus. Each video clip began with a frame showing the actor’s face holding a neutral expression. By the 3th second, the camera zoomed in on the actor’s hand, which was then touched by a Caucasian experimenter by either a neutral stimulus (eraser, the top of a pencil, cotton bud), or a harmful stimulus (needle, dagger, nail), followed by a 4-second still image of the hand/tool interaction. Stimuli were presented on a 60-Hz Samsung S22F350FH, with a screen diagonal of 22 in. connected to a DELL Precision Tower 5810 pc running the Windows 10 operating system.

### Heart rate variability

HR was continuously recorded using the V800 HR monitor (Polar Electro Italia Srl, Bologna, Italy), a mobile HR monitor that has been shown to record changes in consecutive heart beats as accurate as conventional HR monitors74. The V800 comprises a two-lead chest belt system (HRM strap – Polar H7) for data recording (sample rate of 1000 Hz) and a wristwatch for data storage. Device specific software (Polar FlowSync; flow.polar.com/start) was used to synchronise the recorded data from the wristwatch to Polar Flow web platform (flow.polar.com) from which HR recordings were downloaded to a computer for data processing with Kubios HRV analysis package 3.1 (www.kubios.com)75. According to established guidelines76, the HR recordings were firstly detrended (smooth priors: λ = 500), visually inspected and, when necessary, artefact corrected using adaptive filtering (cubic spline interpolation). Following artefact correction, the HR recordings were subjected to both temporal and spectral analysis to estimate different vmHRV measures. As for the temporal analysis, the root mean square of successive differences (RMSSD), measured in milliseconds, that is considered a stable77 and valid time-domain measure of vmHRV54,78, was extracted for each participant. As for the spectral analysis, the Burg autoregressive modeling based method75 (AR model-order = 16; for details see79,80) was used to estimate high-frequency HRV (HF-HRV, 0.15–0.4 Hz) power, low-frequency HRV (LF-HRV, 0.04–0.15 Hz) power, as well as high and low frequency normalized powers (HF-HRV n.u., LF-HRV n.u.) for each participant. In particular, the normalized units represent the relative value of HF- and LF-HRV power components in proportion to the total power minus the very low frequency (VLF-HRV, 0.003–0.04 Hz) component54 and were obtained from the absolute values according to the formulae:

$$HF \mbox{-} HRV\,n.\,u.\,=\,HF[m{s}^{2}]/(total\,power[m{s}^{2}]-VLF[m{s}^{2})$$
$$LF \mbox{-} HRV\,n.\,u.\,=\,LF[m{s}^{2}]/(total\,power[m{s}^{2}]-VLF[m{s}^{2})$$

HF-HRV and LF-HRV values were natural log transformed (namely, Log-HF-HRV and Log-LF-HRV) to fit assumptions of linear analyses81. Among all the frequency-domain measures obtained the HF-HRV and HF-HRV n.u. powers are considered to represent vmHRV and were used as frequency-domain measures of vmHRV. Additionally, a measure of respiration frequency, named peak high-frequency heart rate variability (pHF-HRV) values were obtained from autoregressive analyses to control for potential respiratory-induced bias32,53. Thus, taking into account the above considerations, in this study, the RMSSD, the HF-HRV and HF-HRV n.u. were considered as the primary indices of the self-regulatory effort needed for the explicit judgment of the video-clips.

### Behavioral and physiological analyses

RTs smaller than 300 ms were considered as outliers and discharged from the following analyses, the overall distribution of the RTs was then compared with a standard normal distribution according to an ad-hoc created routine in R (the R script can be obtained by emailing MB). RTs and scores per each block were considered in behavioral analyses. To identify vmHRV outliers based on resting stress level values, after artifact correction, the 5-min resting HR recordings were analysed to assess the Sympathovagal Balance Index (SVI)62,63,64. The SVI was computed according to the formula62,63,64:

$$SVI=\frac{LF-HRV\,power}{HF-HRV\,power}$$

After the calculation of participants’ SVI, the oulier labeling rule82,83 was used to identify and exclude possible outliers in our population based on resting SVI values (see Supplementary Information for further details about SVI outliers detection). Taking into account that several works showed that vmHRV changes can be reliably computed using ultra-short-term recordings (UST; less than 5-min of duration)84,85,86, we relied on an UST time-window of 110 s, on average, to obtain a fine-grained vmHRV measure capable of detecting the rapid changes in autonomic responses associated with DEAR effects. Thus, the HR recordings were divided into 8 sample-windows of 110 s each to detect the physiological response associated with each experimental condition. As a consequence, for each subject, we obtained 8 values for each vmHRV measures considered in the analyses, namely, RMSSD, HF-HRV and HF-HRV n.u.

### Statistics

All the analyses were run in the R-studio (version: 1.1.442) environment using ad-hoc created routines (the R script can be obtained by emailing MB) based on the standard libraries available online. Preliminary analyses were performed to investigate the characteristics of male and female participants. To this end, a series of Student’s t-test for independent samples were used to compute differences in age (years), Body Mass Index (BMI, kg/m2), Resting RMSSD, Resting Log-HF-HRV, and Resting HF-HRV n.u., Resting pHF-HRV and Resting SVI between male and female participants. The same analyses were performed to investigate Resting LF-HRV, as well as Resting LF-HRV n.u. (data not shown for details see Supplementary Information, see Table S-4). A further student’s t-test was used to explore the IAT D score used to asses the implicit racial bias of participants, while a series of one-sample Wilcoxon test were used to explore other behavioural scales (namely IMS, EMS, EES, Subtle and Blatant Prejudice Scales and TES). In order to assess stimuli- and race-related differences in behavioural data (namely, Scores and RTs) collected during both single and dual-task experimental conditions, we estimated a mixed model (‘lme4’ R package87, version: 1.1–17) with Type of stimuli (Harmful vs. Neutral) and Race (Africans vs. Caucasians) as fixed predictors while the Subject was considered as clustering factor to model random intercept and the type of stimuli were used to model random slope (see Supplementary Information for further details about the model’s syntax). The model with the best fit to the data was selected on the basis of likelihood ratio tests and goodness of fit indexes88, moreover, if significant, the Type of stimuli-by-Race interaction effect was further explored by means of pairwise comparisons while adopting an FDR correction of multiple comparisons.

To assess the existence of vmHRV differences during the explicit judgment of painful experience between African and Caucasian actors, the RMSSD, HF-HRV and HF-HRV n.u. values obtained in both single and dual-task experimental conditions were entered as dependent variables into a mixed model with Type of stimuli (Painful vs. Neutral) and Race (Africans vs. Caucasians) as fixed predictors with random intercept (grouped by subject) and random slope (grouped by Type of stimuli). If significant, the Type of stimuli-by-Race interaction effect was further explored by means of pairwise comparisons while adopting an FDR correction of multiple comparisons (see Supplementary Information for further details about the model’s syntax).

We then explored the relationship between resting vmHRV measures and RTs in the single-task condition. Differences between RTs elicited by African vs. Caucasian painful stimulations (i.e. the behavioural cost associated with judging the painful experience of African actors), were computed according to the following formula:

$${\rm{\Delta }} \mbox{-} RT \mbox{-} AC=RTs\,African\,painful\,stimulation\,-\,RTs\,Caucasian\,painful\,stimulation$$

A simple regression analysis was used to test the association between Δ-RT-AC and resting RMSSD, HF-HRV and HF-HRV n.u. values (see Supplementary Information for further details about the model’s syntax).

A similar approach was adopted to explore the relationship between vmHRV mesaures, calculated during the single-task condition, with the subjective level of implicit racial bias (IAT). In particular, at first, differences between RMSSD, HF-HRV and HF-HRV n.u. values for painful vs. neutral conditions were calculated both in Africans (vmHRV-A) and Caucasians (vmHRV-C) using the following formulae:

$$vmHRV \mbox{-} A=vmHRV \mbox{-} mesaure \mbox{-} African\,painful\,-\,vmHRV \mbox{-} mesaure \mbox{-} African\,neutral;$$
$$vmHRV \mbox{-} C=vmHRV \mbox{-} measure \mbox{-} Caucasian\,painful\,-\,vmHRV \mbox{-} measure \mbox{-} Caucasian\,neutral$$

Subsequently, the vmHRV-A (namely, RMSSD-A, HF-HRV-A and HF-HRV n.u.-A) were subtracted to vmHRV-C (by obtaining: Δ-RMSSD-AC, Δ-HF-HRV-AC and Δ-HF-HRV n.u.-AC) using the following formula,

$${\Delta } \mbox{-} vmHRV \mbox{-} AC=vmHRV \mbox{-} A\,-\,vmHRV \mbox{-} C$$

The Δ-RMSSD-AC, Δ-HF-HRV-AC and Δ-HF-HRV n.u.-AC values, that according to our theoretical model should reflect the physiological counterpart of the out-group DEAR effect, were then included in a simple regression analysis to test their association with the IAT D score (see Supplementary Information for further details about the model’s syntax).

## Data Availability

Data reported in this manuscript are available on request from the authors.

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## Author information

M.B., M.G., A.M. and M.D.P. designed the experimental paradigm, A.G. designed the stimuli, M.D.P., M.B., E.A. and D.L. analyzed the data, M.B. and M.D.P. drafted the introduction and the discussion, M.D.P. prepared all the figures, all the authors revised the results and the manuscript.

Correspondence to A. Minelli or M. Berlingeri.

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Appendix 8

The Power Spectral Density

Let x(t) be a real-valued signal derived from the behavior of a system in time. For a vehicle–driver system, x(t) could be the steering angle or the vertical position of the vehicle body. The energy, related to x(t), is proportional to x(t)2. For the vehicle body, the square of the vertical position is proportional to the energy stored in the suspension, due to road disturbances. For a driver, the square of the steering angle corresponds to the energy that is being delivered by the driver to steer the vehicle. The corresponding average power P(x;T) over a time interval with length T can be expressed as the summation of the supplied energy per unit of time:

$P\left(x;T\right)=\frac{1}{T}{\int }_{0}^{T}x{\left(t\right)}^{2}\cdot \mathrm{d}t$ (A8.1)

(A8.1)

For the example ...

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## The calm after the storm

After the weather cleared, we spent more time in Rab's old town and the nearby Komrcar Park.  Visited the spot where a nun shared peaches with the bell tower attendant, and me, yesterday evening.   In the pm, we took a very unpleasant walk to a nearby Franciscan monastery, only to discover that the opening hours [...]

## A complaints procedure can be found at the bar

We left Rovinj this morning and took a break at the Opatija Riviera. It was deserted. There must be some Brussels regulation, as we keep noticing that restaurants include a notice about "complaint procedures" at the bottom of their menus. Later, we boarded the ferry to Rab, where we came across some old-style coaches which [...]


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# Math Help - Prove that if xy is odd, then x and y are odd. Proof by cases

1. ## Prove that if xy is odd, then x and y are odd. Proof by cases

Course: Foundations of Higher Mathematics

Let x and y be integers. Prove that if x*y is odd, then x and y are odd.

I assume that x and y are odd, i.e. $x=2a + 1$ and $y=2b + 1$.

Then, $xy = (2a + 1)(2b + 1)$

= $4ab + 2a + 2b + 1$

= $2(2ab + a + b) +1$

= $2c + 1$, for some integer c

There's only one case for x and y to be odd, so is another case needed?

2. ## Re: Prove that if xy is odd, then x and y are odd. Proof by cases

There's only one case for x and y to be odd, so is another case needed?
No, this is fine. I would add that at this level of detail, you should note that $a$ and $b$ are integer.

You could also prove this by contradiction. Then you need to consider two cases: $a$ is even and $b$ is even (each case includes the possibility that both are even).

3. ## Re: Prove that if xy is odd, then x and y are odd. Proof by cases

Originally Posted by emakarov
You could also prove this by contradiction. Then you need to consider two cases: $a$ is even and $b$ is even (each case includes the possibility that both are even).
I want to give that a try. I'm assuming that you mean that case 1 would be that a is even and case 2 would be that b is even, right?

4. ## Re: Prove that if xy is odd, then x and y are odd. Proof by cases

I'm assuming that you mean that case 1 would be that a is even and case 2 would be that b is even, right?
Yes. The negation of the conclusion "x is odd and y are odd" is "x is even or y is even". This negation is assumed in a proof by contradiction. When you assume a disjunction (A or B) and use it to prove some C, you do it by cases, i.e., you prove that A implies C and B implies C.

5. ## Re: Prove that if xy is odd, then x and y are odd. Proof by cases

The answer in post #1 is not correct. You are not trying to prove that if $x$ and $y$ are odd, then their product is odd. You are trying to prove that if $x\cdot y$ is odd, then both $x$ and $y$ are odd. So, if you assume that $x$ and $y$ are odd, you will always get that $x$ and $y$ are odd because that was your assumption in the first place. Instead, assume only that the product $x\cdot y$ is odd. A logically equivalent statement to the one you are trying to prove would be "If x is even or y is even, then x*y is even." This is called the contrapositive, and a conditional statement is logically equivalent with its contrapositive. So, if you prove that statement, you also prove the original statement. Assume (without loss of generality) that $x$ is even. Then $x=2k$ for some integer $k$. Hence $x\cdot y = 2ky = 2(ky)$ is even. This proves the contrapositive, and in so doing, also proves the initial statement.

6. ## Re: Prove that if xy is odd, then x and y are odd. Proof by cases

Originally Posted by SlipEternal
The answer in post #1 is not correct. You are not trying to prove that if $x$ and $y$ are odd, then their product is odd. You are trying to prove that if $x\cdot y$ is odd, then both $x$ and $y$ are odd.
Thanks. I retire for the night.

7. ## Re: Prove that if xy is odd, then x and y are odd. Proof by cases

Originally Posted by SlipEternal
The answer in post #1 is not correct. You are not trying to prove that if $x$ and $y$ are odd, then their product is odd. You are trying to prove that if $x\cdot y$ is odd, then both $x$ and $y$ are odd.
That's weird, because we've been doing things in class that way for a while. But I could be wrong. Take a look at this proof. The professor did it in class.

Let a, b, and c be integers. Prove: (a|b) ^ (b|c) => (a|c)

Assume that $(a|b)$ ^ $(b|c)$, i.e. $b=ax$ and $c=by$, for any integer x,y

Then, $c=by$
= $(ax)y$
= $a(xy)$
= $az$, for some integer z

Therefore, $c=az$, for some integer z, so $a|c$

8. ## Re: Prove that if xy is odd, then x and y are odd. Proof by cases

Originally Posted by SlipEternal
The answer in post #1 is not correct. You are not trying to prove that if $x$ and $y$ are odd, then their product is odd. You are trying to prove that if $x\cdot y$ is odd, then both $x$ and $y$ are odd. So, if you assume that $x$ and $y$ are odd, you will always get that $x$ and $y$ are odd because that was your assumption in the first place. Instead, assume only that the product $x\cdot y$ is odd. A logically equivalent statement to the one you are trying to prove would be "If x is even or y is even, then x*y is even." This is called the contrapositive, and a conditional statement is logically equivalent with its contrapositive. So, if you prove that statement, you also prove the original statement. Assume (without loss of generality) that $x$ is even. Then $x=2k$ for some integer $k$. Hence $x\cdot y = 2ky = 2(ky)$ is even. This proves the contrapositive, and in so doing, also proves the initial statement.
I am not great with logic, so please forgive me if my comment is obvious. I am not sure that the contrapositive is fully constructed here. My thought is that, yes, you can show that if the product of two numbers is even then at least one of the original numbers also must be even. But I don't feel that the possibility of one of the numbers being even and one odd (from the original problem statement) is addressed here? Or am I mistaken?

-Dan

9. ## Re: Prove that if xy is odd, then x and y are odd. Proof by cases

Course: Foundations of Higher Mathematics
Let x and y be integers. Prove that if x*y is odd,then x and y are odd.
I assume that x and y are odd, i.e. $x=2a + 1$ and $y=2b + 1$.
That's weird, because we've been doing things in class that way for a while. But I could be wrong. Take a look at this proof. The professor did it in class.
Let a, b, and c be integers. Prove: (a|b) ^ (b|c) => (a|c)
Assume that $(a|b)$ ^ $(b|c)$, i.e. $b=ax$ and $c=by$, for any integer x,y
Then, $c=by$
= $(ax)y$
= $a(xy)$
= $az$, for some integer z
Therefore, $c=az$, for some integer z, so $a|c$
The question about your OP is that you are proving the converse of what you are asked to prove.
If $xy$ is odd, then both $x~\&~y$ are odd. Not the other way around.
Personally, I would do this by contradiction. Suppose $xy$ is odd.
Then is any even number times any integer is even. SO?

As for your instructor's proof, I hope he/she said $b=ax$ and $c=by$, for some integers x & y.
Otherwise, it is the way to do it.

10. ## Re: Prove that if xy is odd, then x and y are odd. Proof by cases

That's weird, because we've been doing things in class that way for a while. But I could be wrong. Take a look at this proof. The professor did it in class.

Let a, b, and c be integers. Prove: (a|b) ^ (b|c) => (a|c)

Assume that $(a|b)$ ^ $(b|c)$, i.e. $b=ax$ and $c=by$, for any integer x,y

Then, $c=by$
= $(ax)y$
= $a(xy)$
= $az$, for some integer z

Therefore, $c=az$, for some integer z, so $a|c$
$(a|b) \wedge (b|c) \Longrightarrow (a|c)$ is the equivalent of saying, "If a divides b and b divides c, then a divides c."
Your professor assumed what came after the "if" and proved what came after the "then". In post #1, you assume what comes after the "then" and prove what comes after the "if". As Plato said, you proved the converse statement which is not logically equivalent to the original statement.

Originally Posted by topsquark
I am not great with logic, so please forgive me if my comment is obvious. I am not sure that the contrapositive is fully constructed here. My thought is that, yes, you can show that if the product of two numbers is even then at least one of the original numbers also must be even. But I don't feel that the possibility of one of the numbers being even and one odd (from the original problem statement) is addressed here? Or am I mistaken?

-Dan
You are correct, I did not explain how the statement I wrote is logically equivalent to the contrapositive. The contrapositive would be, "If not (x and y are odd), then not (x*y is odd)." By DeMorgan's Laws, the negation of a conjunction of two expressions is logically equivalent to the disjunction of the negation of each expression $\left( \neg(P\wedge Q) \equiv \neg P \vee \neg Q \right)$. So, the statement is logically equivalent to "If (x is not odd) or (y is not odd), then x*y is not odd." Since not odd is equivalent to even, "If x is even or y is even, then x*y is even."

However, the statement you mention, "If the product of two numbers is even, then at least one of the original numbers also must be even," is actually an inverse statement, not a contrapositive one. In general, the truth value of the inverse statement does not in any way depend on the truth value of the original statement.

11. ## Re: Prove that if xy is odd, then x and y are odd. Proof by cases

This is great. Thank you emakarov, SlipEternal, topsquark, and Plato!

I think what you all are saying is that I should prove that the consequent is true. Then, no matter what the truth value of the antecedent is, the implication will be true. I think I see it now. Don't assume the consequent is true, but prove that it is. I can't think of a way to prove that x and y are odd, so by finding the contrapositive I can prove that x*y is odd and would give me 2-3 cases. Case 1 being x and y are odd. Case 2 being that x is odd and y is even (and vice-versa for case 3, if it's needed).

Proof by contradiction was mentioned twice, but I don't know how to prove it that way. I can't find anything about the process in my textbook.

12. ## Re: Prove that if xy is odd, then x and y are odd. Proof by cases

For proof by contradiction: When is a conditional statement false? It is when the antecedent is true, but the consequent is false. Assume that the antecedent is true and the consequent is false and arrive at a logical contradiction. Proof by contradiction is not necessary in this case because proof by exhaustion is just as easy.

Case 1: x is odd and y is odd (you already showed that x*y is odd)
Case 2: x is even and y is odd (show that x*y is even)
Case 3: x is odd and y is even (show that x*y is even)
Case 4: x is even and y is even (show that x*y is even)

Assume that x*y is odd. Case 1 is the only possible conclusion because you showed that cases 2, 3, and 4 cannot be true if x*y is odd.


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Thread: Grid Curves View Single Post
P: 99
 Quote by HallsofIvy To find the tangent vector in the "u" direction, differentiate r(u,v)= with respect to u. To find the tangent vector in the "v" direction, differentiate r(u,v)= with respect to v. To find them at the given point, substitute $u= \pi/4$ and $v= \sqrt{2}$.
So, when r(u,v)=<vcos(u), vsin(u), v2>, the derivative with respect to u is <-vsin(u), vcos(u), 0>. And the derivative with respect to v is <cos(u), sin(u), 2v>. Now, I plug in $$\sqrt{2}$$ for v and $$\pi/4$$ for u. So I get, <-1, 1, 0> and <$$\sqrt{2}/2$$, $$\sqrt{2}/2$$, 2$$\sqrt{2}$$>. Is this correct so far?


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# Angular velocity  (Page 2/4)

 Page 2 / 4

$\begin{array}{l}{\mathbf{\omega }}_{\mathrm{avg}}=\frac{\Delta \mathbf{\theta }}{\Delta \mathbf{t}}\end{array}$

Instantaneous angular velocity is obtained by taking the limit when time interval tends to become zero. In other words, the instantaneous angular velocity (simply referred as angular velocity) is equal to the first differential of angular displacement with respect to time :

$\begin{array}{l}⇒\mathbf{\omega }=\frac{d\mathbf{\theta }}{d\mathbf{t}}\end{array}$

It is convenient to consider the point as the origin of the coordinate system. In the figure above, origin serves as the point about which angular displacement and velocity are defined. Now since angle is measured with respect to origin, the measurement of angular velocity, in turn, will depend on the coordinate system chosen for the measurement. Further note that the linear distance of the particle from the origin is not constant like in circular motion.

## Angular and linear velocity

In the case of circular motion about an axis, we have seen that linear velocity of a particle is related to angular velocity by the vector relation given by :

$\begin{array}{l}\mathbf{v}=\mathbf{\omega }\mathbf{x}\mathbf{r}\end{array}$

In the case of rotation about a fixed axis, the plane of velocity and position vector is same. This is not necessary for motion of a particle about any arbitrary point. The position vector and velocity can be oriented in any direction. It means that plane of motion can be any plane in three dimensional coordinate system and is not constrained in any manner.

Interpretation of the vector equation as given above is difficult for the motion of a particle along a path, which is not circular. We observe that angular velocity is an operand of the cross product. What we mean to say that it is not the vector, which is expressed in terms of other vectors. By looking at the vector relation, we can say that linear velocity is perpendicular to the plane formed by angular velocity and position vector. From the point of view of angular velocity, however, we can only say that it is perpendicular to linear velocity, but it can have any orientation with respect to position vector. This presents difficulty in interpreting this relation for angular velocity about a point.

Here, we consider a situation as shown in the figure. We see that velocity of the particle and the position vector form an angle, "θ" between them. For convenience, we have considered the point about which angular velocity is defined as the origin of coordinate system.

To analyze the situation, let us consider components of velocity in the radial and tangential directions.

The radial component of velocity is not a component for rotation. Its direction is along the line passing through the origin. The rotation is accounted by the component of velocity perpendicular to position vector. Now, tangential velocity is perpendicular to position vector. Thus, we can use the fact that instantaneous angular velocity is equal to ratio of tangential velocity and linear distance from the origin (as defined for circular motion about an axis). Hence, the magnitude of angular velocity is given as :

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