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07-11-2018 | Uitgave 5/2019 Open Access

# Development of Aggression Subtypes from Childhood to Adolescence: a Group-Based Multi-Trajectory Modelling Perspective

Tijdschrift:
Research on Child and Adolescent Psychopathology > Uitgave 5/2019
Auteurs:
Lisa-Christine Girard, Richard E. Tremblay, Daniel Nagin, Sylvana M. Côté
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## Electronic supplementary material

The online version of this article (https://doi.org/10.1007/s10802-018-0488-5) contains supplementary material, which is available to authorized users.

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##

Much attention has been paid to the onset and developmental course of different forms and functions of aggression throughout the lifespan. Rightly so given the individual, societal, and economic burden of aggression that persists beyond the normative period during early childhood. Research suggests that chronic aggression of one type or another is associated with many negative life outcomes. For example, chronic physical aggression during childhood has been linked to violent delinquency, criminality, and stable unemployment into adulthood (Broidy et al. 2003; Kokko and Pulkkinen 2000; Nagin and Tremblay 1999). Indirect aggression, which is characterised by social behaviours that are often covert in nature and used to manipulate others within a social context, has been associated with positive ‘reinforcing’ shorter-term outcomes such as being ‘ perceived popular’ (Cillessen and Rose 2005; Hawley 2003) and negative outcomes such as internalizing symptoms, depression, suicide ideation, and substance abuse (Herrenkohl et al. 2009; Murray-Close et al. 2007; Van der Wal et al. 2003).
Within the early childhood perspective of aggression, research has shown that physical aggression commences as early as infancy, when the capacity to inflict harm onto others, such as hitting, kicking, and biting develops (e.g., Tremblay et al. 1999; Vitaro et al. 2006; Tremblay et al. 2018), coinciding with the development of infant’s early motor ability. The early childhood perspective of aggression emerged with the increasing use of longitudinal cohort studies, which shed a different developmental perspective to the previously held traditional theoretical framework of learned aggression resulting in the increased use of physical aggression as children age (e.g., Bandura 1973; Loeber and Stouthamer-Loeber 1998). Direct observations from longitudinal cohort studies of early development supported the perspective that engagement in physical aggression is instead at its peak during infancy and early childhood, in particular during the toddler years (Hay et al. 2014), with less frequent engagement in physical aggression as children develop (Broidy et al. 2003; Nagin and Tremblay 1999; Tremblay et al. 1996); rather than onset in late childhood or adolescence. This suggested the unlearning of an instinctual behaviour rather than a learned behaviour via observation and modeling. These observed decreases in the use of physical aggression following early childhood are argued to coincide with the increasing ability to self-regulate, coupled with children’s increasing cognitive and language development, and social information processing (Dodge and Frame 1982; Dodge and Schwartz 1997; Dionne et al. 2003; Séguin et al. 2009; Girard et al. 2014).
On the other hand, engagement in indirect forms of aggression, also observed in early childhood, has been found to increase in frequency as children develop from late childhood and into adolescence (Cairns et al. 1989; Vaillancourt et al. 2007). Börkqvist and colleagues’ developmental perspective suggested that these increases in indirect aggression coincide with the development of cognitive and language skills, alongside a better understanding of social norms and expectations (Björkqvist et al. 1992a; b; Björkqvist 1994). That is, the development of higher-order cognitive and language ability, coupled with a better understanding of social norms which would not support continual engagement in overt forms of aggression, would necessarily be expected to precede the use of a more sophisticated form of aggression such as indirect aggression. Thus, a developmental model of physical and indirect aggression would suggest early onset and higher frequency of physically aggressive behaviours in early childhood, which are then replaced by more covert forms of aggression, i.e., indirect aggression, as children develop in later childhood and adolescence (Björkqvist et al. 1992a; b; Tremblay et al. 1996). Taken together, independent engagement in either form of aggression has been shown to differ in both etiology and consequence, as well as present differently with respect to their developmental trajectories, despite the moderate-strong associations found between them (e.g., Crick et al. 2006; Kaukiainen et al. 1999; Vaillancourt et al. 2003; Côté et al. 2007).

### Functions of Aggression

While theoretical perspectives differentiate forms of aggression, functions of aggression are similarly differentiated. For example, proactive aggression, which is calculated, instrumental, and predatory in nature, has been linked to gang membership, substance abuse, delinquency, anti-sociality and psychopathic features in adulthood (e.g., Barker et al. 2006; Fite et al. 2010; Vitaro et al. 1998). Proactive aggression has been argued to coincide with a social learning model of aggression (Bandura 1973; Dodge and Coie 1987), in so much as the aggression can be operationalized as a learned behaviour that is goal driven. That is, the aggression is used to obtain an instrumental goal or reward (e.g., a desired object or social status within the peer group), and reinforced via operant conditioning (i.e., goal attainment). Within this framework, proactive aggression ought to either remain stable, or increase overtime.
Reactive aggression on the other hand, often provoked by anger in reaction to a perceived threat, is defensive in nature and has been associated with internalizing difficulties such as negative affect, depression, anxiety and additionally problems with self-regulation (Vitaro and Brendgen 2011). Reactive aggression has predominantly been operationalized within Berkowitz’s frustration model of aggression (Berkowitz 1988, 1989) given that the aggression is reactive rather than instrumental. It is a consequence of perceived provocation resulting in anger and retaliatory responses. Deficits in self-control, emotional regulation, and impulsivity are characteristics of high levels of reactive aggression (Denson et al. 2012; Marsee and Frick 2007). As a result, reactive aggression is likely to decrease with brain maturation across development and as children become better able to self-regulate (Tremblay 2000). While there was large debate surrounding the utility of distinguishing between reactive and proactive aggression, particularly given challenges of correct identification, studies continue to confirm their discriminative validity (e.g., Dodge and Coie 1987; Poulin and Boivin 2000; Kempes et al. 2005), along with distinct etiologies and consequences (e.g., Dodge 1991; Paquin et al. 2017; Vitaro et al. 2002).

## Trajectories of Forms and Functions of Aggression

Longitudinal studies from early childhood to adolescence have examined both single and joint trajectories of either forms, (i.e., physical and indirect; e.g., Cleverley et al. 2012; Côté et al. 2007) or functions of aggression (i.e., reactive and proactive; e.g., Barker et al. 2006), revealing heterogeneity in trajectories. Yet, to the best of our knowledge, no study to date has combined both forms and functions of aggression within a developmental perspective from childhood to adolescence, whilst examining their given co-occurring trajectories using a person-centered approach. This is an important next step given the distinction between forms and functions of aggression highlighted first by Pitkänen in Pitkänen 1969, and more recently by Little et al. ( 2003); coupled with the suggestion that even though functions are largely dependent on the form taken, proactive, reactive, physical and indirect aggression are nevertheless recognizably distinguishable constructs. This may help to better understand the prevalence of distinct yet co-occurring trajectories of multiple forms and functions of aggression from childhood to adolescence. This developmental period is particularly salient given social-cognitive-behavioural development that is occurring, which can facilitate either engagement or desistance in both forms and functions of aggression.
By taking into account the form of the aggressive behaviours (physical vs. indirect) as well as their function (proactive vs. reactive) within longitudinal studies, we are more likely to understand to what extent there are groups of children who specialise in given forms and functions of aggressive behaviours, either independently or simultaneously, as well as to what extent these specialisations change during the course of development. For example, in one study using a variable-centered approach to model both forms and functions of aggression trajectories in adolescence (i.e., from 12 to 14 years of age), Ojanen and Kiefer ( 2013) found that on average, there were increases in instrumental (i.e., proactive) relational aggression and decreases in reactive overt aggression across time. These findings are in line with previous studies of the individual and joint trajectories of aggression and shed important insights into a more holistic view of continuity and discontinuity, of form and functions of aggression in adolescence specifically. However, an important methodological consideration in interpretation of these findings was the inability to examine non-linear trajectories. As the authors acknowledge, linear models may not be best suited to modelling trajectories of forms and functions of aggression. Moreover, given the social-cognitive-behavioural changes occurring at entry into formal schooling, it would be highly informative to start examining these co-occurring trajectories of forms and functions of aggression at earlier stages, prior to adolescence. Particularly so given this important developmental period when the opportunities for social interactions with peers have substantially increased. Thus, one aim of our study was to extend upon the above findings by examining a longer developmental period, using a person-centered approach, whilst not restricting the shape of co-occurring forms and functions of aggression trajectories.

## Antecedent Risk Factors

Studies of individual and joint subtypes of aggressive behaviour have shown that persisting childhood aggression carries high social burden and economic cost to societies. To prevent these chronic trajectories of aggression, we need to understand to what extent different types of aggression feed into one another during development. Particularly so given that chronic engagement in multiple forms and functions of aggression likely carries even higher risk for consequent negative life outcomes. Conversely, if we can identify groups of children who predominantly exhibit one type of aggression only, then it becomes important to examine whether individuals exhibiting distinct types of aggression have different antecedent characteristics and risk factors associated with group membership. Given that this is the first study to the best of our knowledge, looking at multi-trajectories of both forms and functions of aggression within a developmental framework, whilst using a person-centered approach, we start by examining whether common antecedent factors previously identified in single trajectories of aggression, are also predictive of potential co-occurring trajectories. Indeed, different intervention strategies may be required for the prevention of different types of aggression and especially for different mixtures of aggression types. Thus, understanding whether there are distinct patterns of group membership for co-morbid engagement, and the antecedent characteristics associated with each group, is an important first step in the development of effective prevention strategies.
In a review of the literature, some key antecedent family-, maternal-, and child-level risk factors of both individual forms and functions of aggression were identified. For example, at the family level, lower level socio-economic status and family status (i.e., single mothers) have conferred increased risk for elevated physical and indirect aggression in offspring (Côté et al. 2006, 2007; Tremblay et al. 2004; Vaillancourt et al. 2007). Additionally, in line with the theoretical model proposed by Dodge ( 1991), the origins of reactive and proactive aggression stem from early social experiences and in particular parenting behaviours such as harsh and coercive parenting (Vitaro et al. 2006). Harsh and coercive types of parenting behaviours have also been implicated in engagement in both physical and indirect aggression (Campbell et al. 2010; Hentges et al. 2018; Orri et al. 2018; Girard et al. 2014; Tremblay et al. 2004; Côté et al. 2007; Vaillancourt et al. 2007). At the maternal level, risk factors such as education and IQ, age at birth of first child, smoking during pregnancy, previous antisocial history during adolescence, and depression, have all been implicated as risk factors for children who display with higher levels of aggression (e.g., Huijbregts et al. 2008; Tremblay et al. 2004; Côté et al. 2007; Hay et al. 2003). For a comprehensive overview of implicated family and maternal level risk factors related to chronic physical aggression specifically, see Tremblay et al. ( 2018). With respect to child-level factors, both receptive and expressive language (e.g., Dionne 2005), preterm birth (e.g., Potijk et al. 2012), low birth weight (e.g., Pharoah et al. 1994), sibling status (e.g., Stauffacher and DeHart 2006; Goodwin and Roscoe 1990), and sex (e.g., Tremblay et al. 2004), have all been found to increase the risk of higher engagement in subtypes of aggressive behaviours. Given previous identification of the above-mentioned risk factors for individual and/or joint forms and functions of aggression, we examined whether these same risk factors would predict membership in trajectories of combined forms and functions of aggression. Additionally, we also examined breastfeeding. There has been increasing interest in examining the association between breastfeeding and externalising behaviours such as conduct problems, which has yielded mixed results (e.g., Jackson 2016; Girard et al. 2017; Girard et al. 2018). Proposed mechanisms for the association include psychological (e.g., via attachment), brain development (e.g., via nutrients and white growth matter), and genetic risk. It may then be possible that breastfeeding is implicated in forms or functions of aggression.

## Objectives

The first objective was to examine the heterogeneity in developmental trajectories of combined physical, indirect, proactive, and reactive aggression across childhood and into adolescence. Rooted within a developmental model and the early childhood perspective of aggression, we expected that a majority of children would follow moderate to low decreasing physical aggression trajectories with variation in stable to increasing indirect aggression over time. We also predicted that these same children would likely decrease in reactive aggression over time, with potentially increasing proactive aggression in children who were following increasing trajectories of indirect aggression. Additionally, and in line with previous findings, we expected to find a small group of children who engaged in high and chronic forms and functions of aggression over time. Finally, we expected to see a group of children who did not engage in either forms or functions or aggression over time. Taken together, we expected to find a three or four group model to best fit the data.
The second objective was to better understand which antecedent characteristics, at the child, maternal, and family level, were associated with group membership. We made no predictions regarding the specific antecedent characteristics that would distinguish between differing trajectories given that this is the first study to examine multi-trajectories of both forms and functions of aggression across childhood and into adolescence. We did however expect to find that the previously identified characteristics outlined above would similarly be implicated in the multi-trajectory groups identified in the current study. Finally, we examined the risk associated with these antecedent characteristics for group membership, which distinguished the chronic groups from the others. Here we expected to find that children, particularly boys from lower SES backgrounds, who were from single parent families, with younger mothers who had a history of engagement in anti-social behaviour along with lower IQ or educational background, who also engaged in high-risk behaviours during pregnancy such as smoking, and who used harsh and coercive parenting practices, would be at higher risk of membership in the chronic group.

## Method

### Participants

This study uses data collected from children enrolled in the Quebec Longitudinal Study of Child Development (QLSCD). The QLSCD is a nationally representative cohort of singletons born in Quebec, Canada, between 1997 and 1998 that were selected from the Quebec birth registry. Sample selection and stratification procedures have been extensively documented (Jette and Des Groseilliers 2000). The initial sample was comprised of 2223 children and their families. Inclusion criterion in the current study was having a minimum of three assessments for each subtype of aggression, thus resulting in a final sample of 787 children. The inclusion criterion was required given that three data points are necessary to properly fit a quadratic polynomial term in the group-based multi-trajectory approach. The sample characteristics of the entire cohort as compared to those included in the current study can be found in Table 1. Informed written parental consent and children’s assent were collected prior to each wave of data collection. Ethical approval was obtained by the Québec Institute of Statistics’ Ethics Committee.
Table 1
Demographic characteristics of entire sample and families included in the current study

Excluded Sample
( N = 1436)
Current Sample
( N = 787)
p
Highest Maternal Diploma:

<0.001
No Diploma
288 (20%)
113 (14%)

High School Diploma
384 (27%)
200 (25%)

College Diploma
413 (29%)
231 (29%)

University Diploma
348 (24%)
243 (31%)

Maternal Age:

0.742
Less than 20 years
45 (3%)
19 (2%)

20–24 years
285 (20%)
145 (18%)

25–29 years
454 (32%)
248 (32%)

30–34 years
458 (32%)
271 (34%)

35–39 years
164 (11%)
86 (11%)

More than 40 years
29 (2%)
18 (2%)

Maternal Working Status (Unemployed):
451 (32%)
203 (26%)
0.004
Maternal Smoking during Pregnancy (Yes):
373 (26%)
182 (23%)
0.332
566 (40%)
245 (31%)
<0.001
Maternal Antisocial Behaviour
0.9 (1.0)
0.7 (0.8)
0.002
Single Parent Family (Yes):
133 (9%)
41 (5%)
0.003
Family Income Less than 19,999 (Yes):
275 (20%)
83 (11%)
<0.001
Child Sex (Boys):
800 (56%)
338 (43%)
<0.001
Means and (standard deviations) are presented for maternal adolescent antisocial behaviour

### Outcome Measures

Teacher-reports were used to collect information on aggressive behaviours at six, seven, eight, 10, 12 and 13 years of age. The teacher-report used in the QLSCD cohort was taken from the Canadian National Longitudinal Survey of Children and Youth (NLSCY), which used items from the Child Behaviour Checklist (i.e., comprised of118 items assessing problem behaviours), the Children’s Behaviour Questionnaire (i.e., comprised of 195 items assessing temperament), and the Social Behaviour Questionnaire (i.e., comprised of 38 items assessing both problem and prosocial behaviours). All scales have previously been well validated in the literature (Achenbach and Edelbrock 1983; Rutter 1967; Tremblay et al. 1991). In the current study, we used only items measuring forms and functions of aggression. That is, teachers rated the frequency of children’s engagement in physical, indirect, proactive, and reactive aggression on a scale from 0 ( never) to 3 ( always). The numbers of items differed across subtype of aggression and thus all subscales were rescaled to range between 0 and 10. Physical aggression (PA) included three items (i.e., physically attacks others; fights often with others; hits, bites, kicks others). Cronbach’s alpha in the current sample at 6, 7, 8, 10, 12, and 13 years was 0.88, 0.86, 0.87, 0.89, 0.87, and 0.85, respectively. Indirect aggression (IA) included three items (i.e., when angry with someone, tries to get others to dislike that person; when angry with someone, became friends with another as revenge; when angry with someone, says bad things behind the other’s back). Cronbach’s alpha at 6, 7, 8, 10, 12, and 13 years was 0.85, 0.88, 0.87, 0.88, 0.86, and 0.90, respectively. Proactive aggression (PAA) also included three items (i.e., intimidates others to get what he/she wants; tries to dominate other children; encourages children to pick on a particular child). Cronbach’s alpha at 6, 7, 8, 10, 12, and 13 years was 0.74, 0.77, 0.72, 0.77, 0.83, and 0.84, respectively. Finally, reactive aggression (RA) included four items (i.e., reacts in an aggressive manner when something is taken away from him/her; reacts in an aggressive manner when contradicted; reacts in an aggressive manner when teased; when hurt by another child, gets angry and reacts by fighting). Cronbach’s alpha at 6, 7, 8, 10, 12, and 13 years was 0.89, 0.87, 0.86, 0.89, 0.85, and 0.92, respectively.

To better understand the etiology and specific risk factors associated with concomitant aggressive trajectories, maternal reports and standardised measures were used. When children were five months, mothers reported on whether having obtained a high school diploma (yes/no), maternal age at the birth of her first child (21 years or less/22 years or over), annual household income during the past 12 months (dichotomized into less than $19,999, over 20,000 Canadian dollars, representing low income families; Statistics Canada 2011), family status at birth (single parent/dual parent), maternal smoking during pregnancy (yes/no), maternal depression (a score of ≥16 using the Centre for Epidemiological Study of Depression, short version, CES-D; Radloff 1977), which has been previously well validated (Lewinsohn et al. 1997), type of delivery (vaginal/caesarean), preterm birth derived from gestational age (delivered prior to the start of the 37th week, yes/no), the child’s birth weight (≥2500 g, yes/no), and the child’s birth order. Maternal adolescent antisocial behaviour was assessed at five months using seven items previously validated in the literature (Zoccolillo et al. 2004). Examples of the items include: before you completed your secondary school studies had you already been involved with the direction of youth protection or with the police, or had you been stopped by the police because of your bad behavior. Cronbach’s alpha is 0.92. The Cumulative Score for Neonatal Risk (CSNR) comprised of the infant’s APGAR score, birth weight, gestational length, congenital abnormalities, retardation of cranial perimeter growth, retardation of intrauterine growth and neonatal complications was taken from medical records at birth. At one and a half years, mothers were asked about breastfeeding engagement (scored as never breastfed, up to six months, more than six months). A proxy of maternal IQ (i.e., crystallized intelligence) was assessed when children were age five years, with 14 items measuring receptive ability. For each item, mothers were asked to fill in the missing word that most correctly completed the idea of the sentence from a list of five potential options. Positive and coercive parenting was collected using the Parent Practices Scale (Strayhorn and Weidman 1988) when children were one and a half (positive only), two and a half, three and a half, four and a half, and five years. Cronbach alpha for positive parenting was 0.64, 0.62, 0.62, 0.63, and 0.65, respectively, and 0.68, 0.71, 0.67, and 0.74, respectively for coercive parenting. A mean composite score over time was then created. The scale’s reliability and validity have been well documented (Strayhorn and Weidman 1988). The Peabody Picture Vocabulary Test (PPVT; Dunn and Dunn 1997), a standardized measure of children’s receptive language was also assessed when children were six years; Cronbach’s alpha, 0.93. The PPVT has demonstrated good reliability and validity (e.g., Campbell 1998). ### Statistical Analysis There remains ongoing debate surrounding the use of residuals in modelling forms and functions of aggression given the challenges in conceptualising the residualised construct in a meaningful way, in addition to challenges with its validity (e.g., Miller and Lynam, 2006). Despite this, research in aggression must take into consideration the potential measurement and analytical challenges in examining both forms and functions as distinct subtypes of aggression (Little et al. 2003). Thus, we first ran a one, two, and four factor confirmatory factor analysis using the Mplus software, version 7.4. Model fit indices used to assess the best model fit, in addition to the Chi-square which is commonly significant in larger sample sizes, include the root mean square error of approximation (RMSEA), the comparative fit index (CFI), and the root mean square residual (SRMR). Good model fit can be represented by a RMSEA of equal to or less than 0.08 (MacCallum et al. 1996), a CFI of greater than 0.95 (Hu and Bentler 1999), and a SRMR of equal to or less than 0.05 (Diamantopoulos and Siguaw 2000). The model fits revealed that across all ages, the four-factor model continually provided the best fit to the data, suggesting discriminant validity. Model fit comparisons can be found in the online Supplement 1. Additionally, a correlation table between forms and functions of aggression across time is presented in Supplement 2. Group-based modelling was conducted next. Unlike the traditional growth curve modelling approach (i.e., variable-centered) which focuses on population means, individual variability around the population mean, and the contributing factors for that variability, the group-based modelling approach (i.e., person-centered) aims to model distinctive trajectory groups within the population and to identify factors that differentiate the groups (Nagin 2005). It is a semi-parametric approach to modelling the heterogeneity of developmental trajectories within the targeted population. A recent extension of the group-based trajectory model, called the multi-trajectory approach, was used in the current study to model multiple subtypes of aggression jointly from childhood to adolescence (Nagin et al. 2016). The advantage of this extension is its ability to conjointly model multiple subtypes of behaviours, thus providing an overall behavioural ‘profile’ of, in our case, aggression across multiple dimensions. A two-stage approach was used to identify the best fitting model. First, two, three, four, five and six group models were run to compare the Bayesian information criteria (BIC) of each model (Supplement 3). A larger BIC is indicative of a better fitting model (D’Unger et al. 1998; Nagin 2005). Both four, five, and six group models provided the best BIC fits. Next, polynomial terms were fitted in the four, five, and six-group model. Based on these two criteria, a five group multi-trajectory model provided the best fit to the data and was selected. Further, criteria for judging the adequacy of the selected, suggested in Nagin ( 2005) such as the average posterior probabilities of group membership by assigned trajectory group and odds of correct classification, supported the adequacy of the five-group model. Model fit criteria are presented in Table 2. Table 2 Model fit criterion of the multi-trajectories of subtypes of aggression Trajectory Group N Average Posterior Probability of Group Membership Odds of Correct Classification 1 203 (26.6%) 81.7% 12.8 2 272 (33.7%) 90.5% 17.9 3 56 (7.4%) 87.0% 87.1 4 218 (27.4%) 86.8% 17.1 5 38 (4.9%) 95.3% 402.8 Membership probability greater than 70 and OCC greater than 5 represent good model fit Chi-square tests and analysis of variance were conducted to first understand which characteristics were significantly associated with group-membership. Next, multinomial logistic regression models were examined to better understand risk factors for belonging to the moderate-desisting, high-desisting, and chronic group as compared to the low to non-aggressive groups. Only factors that were statistically significant in bivariate analysis were examined in the multivariate model based on the principle of parsimony. The statistical threshold was set at p = 0.050, two-tailed. These analyses were performed using Stata 14.0 software. We use the term significant to denote statistical significance henceforth. ## Results Five groups of children emerged, displaying distinct developmental patterns of concomitant PA-IA-PAA-RA. The first group, which was estimated to compose 26.6% of the sampled population and labelled as the ‘low-stable’ group, were low engagers in PA, PAA, & RA. Levels of IA were slightly higher as compared to the other subtypes of aggression yet were still low across the developmental period from six to 13 years of age. This group was entirely stable in their engagement in each subtype of aggression across time. The second, and largest group was estimated to compose of 33.7% of the sampled population. This group, labelled as the ‘non-aggressors’, were rated as not having engaged in any subtype of aggression between six and 13 years of age. The third group, the ‘moderate-engaging’ group, were rated as having moderate engagement in each subtype of aggression at age six, followed by a linear decreasing trajectory of PA, and RA thereafter. The frequency of IA engagement in this group revealed a quadratic rather than linear shape, whereby IA was moderate at age six, increased between seven to 10 years, and gradually returned to moderate engagement between 10 and 13 years. The shape of engagement in PAA in this group however did not decrease over time, but rather it remained stable. The moderate-engaging group was estimated to account for 27.4% of the sampled population. Group four, the ‘high-desisting’ group revealed similar shapes in their trajectories of PA, IA, and RA as the moderate-engaging group, albeit at higher levels of initial aggression and throughout development. Further, PAA in this group followed a linear decreasing trajectory rather than remaining stable as in the case of the moderate-engaging group. This group was estimated to compose 7.4% of the sampled population. The fifth and final group, consisted of an estimated 4.9% of the sampled population, and were labelled as the ‘high-chronic’ engagers. This group consistently engaged in elevated and stable levels of all forms aggression from age six to 13. Multi-trajectory groups are displayed in Fig. 1. Based on chi-square tests and analysis of variance, Table 3 displays antecedent child, maternal, and family characteristics associated with group membership. At the child level, sex was the only significant characteristic. At the maternal level, highest diploma obtained, maternal IQ, maternal adolescent antisocial behaviour, and postnatal depression, were all significant. Family characteristics were not found to distinguish between group-membership although parenting factors such as breastfeeding and coercive parenting were. Table 3 Proportions and mean level of risk factors for the five distinct multi-aggression trajectory groups: bivariate analysis Low Stable ( n = 203) Non-Aggressors ( n = 272) Moderate Engagers ( n = 218) High Desisting ( n = 56) High Chronic ( n = 38) p Family Factors Low income (less than$19,999 annually):
6.5%
11.0%
11.1%
14.3%
21.1%
0.063
Single parent family:
5.4%
3.3%
5.5%
7.1%
13.2%
0.120
Maternal Factors
Less than high school diploma:
11.8%
11.4%
15.1%
23.2%
31.6%
0.003
Less than 21 years at birth of 1st child:
18.0%
16.1%
21.8%
20.0%
31.6%
0.163
Smoking during pregnancy:
22.2%
20.6%
24.8%
23.2%
36.8%
0.103
Postnatal depression:
9.9%
8.1%
12.8%
26.8%
23.7%
<0.001
Child Factors:
Born preterm:
6.4%
2.6%
7.3%
3.6%
2.6%
0.113
Low birth weight:
5.9%
2.6%
3.7%
1.8%
0.0%
0.202
Caesarean birth:
14.8%
12.9%
18.1%
10.7%
10.5%
0.416
First born child:
42.9%
49.3%
44.0%
37.5%
50.0%
0.397
Sex: Boy
24.1%
37.1%
61.0%
41.1%
84.2%
<0.001
Parenting Factors:
Never breastfed:
22.2%
21.5%
32.1%
33.9%
43.2%
<0.001
Means
0.8 (0.9)
0.6 (0.8)
0.8 (0.8)
1.0 (0.9)
0.8 (0.8)
0.037
Maternal IQ:
8.3 (0.9)
8.2 (1.0)
8.2 (0.9)
7.9 (1.0)
7.5 (1.2)
<0.001
Positive Parenting:
6.8 (0.7)
6.8 (0.9)
6.8 (0.9)
6.7 (0.8)
6.7 (0.6)
0.972
Coercive Parenting:
2.3 (0.7)
2.2 (0.7)
2.5 (0.8)
2.5 (0.6)
2.7 (0.8)
<0.001
Receptive Language (child at 6 years):
117.2(15.0)
117.7(17.1)
117.2(15.3)
113.5(14.5)
113.4(17.2)
0.333
Risk at birth (CSNR):
0.9 (1.1)
0.9 (1.1)
1.0 (1.3)
0.8 (1.2)
0.9 (1.2)
0.209
Maternal IQ is scored on a scale from 0 to 10 with higher scores indicative of higher IQ. Percentages of prevalence of antecedent characteristic per group displayed for chi-square analysis; means and (standard deviations) presented in analysis of variance
Finally, to better understand which characteristics were associated with risk for belonging to trajectory groups, a multinomial logistic regression was conducted by grouping the low-stable and non-aggressors together to form the reference group, and then comparing the moderate-engaging, high-desisting, and high-chronic groups; Model x 2 = 151.62, p = <0.001, pseudo R 2 = 0.12. For the moderate-engaging group, coercive parenting, never being breastfed, and being male were all risk factors that increased the relative risk ratio of membership in this group. Maternal and parental risk factors that increased the relative risk ratio of membership in the high-desisting group included lower maternal IQ, and postnatal depression. Finally, maternal risk factors for membership in the high-chronic group included maternal education and IQ. Coercive parenting also increased the relative risk ratio for high-chronic aggression. The strongest risk factor for group membership in the high-chronic group was being male. Please see Table 4.
Table 4
Risk factors associated with multi-aggression trajectories: multivariate analysis

Moderate-Engagers
High-Desisting
High-Chronic
Maternal Diploma (Less than High School):
1.5(0.4) [0.8–2.8]
1.9(0.8) [0.8–4.3]
4.3(2.3) [1.5–12.4]
Maternal IQ: (High)
1.0(0.1) [0.8–1.2]
0.7(0.1) [0.5–0.9]
0.6(0.1) [0.4–0.9]
Maternal Postnatal Depression (Yes):
1.6(0.5) [0.8–3.1]
3.2(1.3) [1.4–7.4]
2.9(1.5) [0.9–8.3]
1.1(0.1) [0.8–1.3]
1.3(0.2) [0.8–1.7]
1.1(0.2) [0.6–1.7]
Coercive Parenting:
1.7(0.2) [1.2–2.1]
1.5(0.3) [0.9–2.2]
2.5(0.6) [1.4–4.2]
Breastfeeding
Never:
2.2(0.5) [1.2–3.6]
2.2(1.0) [0.8–5.7]
2.3(1.3) [0.7–7.2]
Up to 6 months:
1.2(0.2) [0.7–1.9]
2.1(0.8) [0.8–4.7]
0.9(0.5) [0.3–2.9]
Child Sex (Boy):
3.9(0.7) [2.6–5.8]
1.8(0.5) [0.9–3.4]
13.5(7.7) [4.4–41.3]
The comparison group combined the low-stable and non-aggressors. The reference category for breastfeeding was six months or more. The relative risk ratio is presented along with the (standard error) and the [95% Confidence Intervals]

## Discussion

This is the first study which has used the multi-trajectory approach to be able to model both forms and functions of aggression simultaneously across development. In so doing, allowing us to visually represent distinct groups of children who exhibit unique patterns of both change and continuity over time, in and across, concurrent forms and functions of aggression. Our overarching aims, were to identify any heterogeneity in the co-occurrence of PA, IA, PAA, and RA using this group-based multi-trajectory approach, and to identify risk factors, specific to the unique trajectories identified, that may be good targets for early prevention and/or intervention efforts. These questions are of particular importance given the multitude of emergent maladaptive outcomes experienced over the life course for persistent aggressors.
The literature has typically placed the examination of subtypes of aggression within a single or dual-dimensional context only, whereby a focus on either form (PA-IA) or functions (PAA-RA) are studied in combination, with two-, three- and four group trajectory models often being identified (e.g., Barker et al. 2006; Côté et al. 2007; Nagin and Tremblay 1999). Our findings, while in line with previous developmental patterns found in these studies (e.g., groups of non-engagers, high-desisting, chronic engagers), offer new insights on observable patterns of four co-occurring subtypes of aggression from six to 13 years of age. Our results suggest that both subtype and severity models are necessary for early programming efforts given the differing trajectories. Moreover, the differences in antecedent factors identified between these five trajectory groups has important implications for developmental models of aggression as the field moves forward, given that there was little overlap between the risk factors implicated across group membership in groups displaying with moderate to chronic levels of aggression.

### Heterogeneity in Multi-Trajectories of Aggression and Theoretical Perspectives

Our first hypothesis, based on previous work of single and joint trajectories, was that a three- or four-group model would best fit the data. Our data however best supported a five-group model: low-stable, non-aggressors, moderate-engaging, high-desisting, and high-chronic. We further hypothesised that a majority of children would engage in moderate to low decreasing PA with variation in stable to increasing IA over time, coupled with decreases in RA over time, and potentially increasing PAA for children who were following increasing trajectories of IA. Additionally, we expected to find a small group of children who engaged in high-chronic forms and functions of aggression over time along with another group of children not engaging in either forms or functions of aggression over time. Our data partially supported this hypothesis. From the age of six to 13 years, 60.3% of children were rarely engaging in any subtype of aggression, 34.8% engaged in some form of mostly desisting trajectory and only 4.9% remained stable in their use of elevated subtypes of aggressive behaviours. However, given the host of associated maladaptive outcomes across the life course for chronic engagers, the 4.9% who persist in both form and function of aggression, is not negligible.
From a theoretical perspective, the findings in our study (for children in trajectory groups who were engaging in aggressive behaviours), appear most strongly aligned with the early childhood perspective of aggression. For example, most children who engage in aggressive behaviours are learning how not to aggress as they grow older (Nagin and Tremblay 1999; Tremblay et al. 1999; Tremblay 2010), rather than learning to aggress with time via modelling and conditioning, as is suggested in the social learning theory of aggression (Bandura 1973). Support for this early childhood perspective is demonstrated through the decreases found in aggressive trajectories from childhood to adolescence in the multi-trajectory groups identified in our study. The findings not only support this perspective for physically aggressive behaviours, but it appears that when modelling concomitant forms and functions of aggression, this perspective also fits with reactive, and to some extent, proactive aggression (i.e., group 4). Indirect aggression on the other hand revealed slight digressions from this early unlearning perspective, in particular during middle childhood.
That is, both the moderate-engaging and high-desisting groups revealed an interesting pattern with respect to their engagement in indirect forms of aggression. While PA and RA were linearly decreasing, IA increased between seven and 10 years before eventually also decreasing. This suggests that while these children are learning how not to engage in overt and reactive displays of aggression, they are also learning more sophisticated forms of covert aggression, perhaps to replace the less socially accepted forms of PA and RA as they grow older, at least in the interim and prior to adolescence (Côté et al. 2007). Indirect aggression has been argued to coincide with the development of linguistic competence and social intelligence (Björkqvist et al. 1992b, 2000; Garandeau and Cillessen 2006), which may approximate the reason for the observed peak at a later developmental stage. However, the steady decline of IA in both groups between 10 years of age and adolescence suggest that a developmental trajectory of inhibiting the frequency of engagement in aggressive behaviours over time may also apply to indirect forms of aggression just prior to adolescence. The period between 10 and 13 years of age for this sample marks the transition between elementary school and high-school. Thus, an alternative explanation for the observed decrease in IA after age 10 may be reflective of this transition period. While children are starting to increase their use of indirect aggression, the transition may help to suppress its continuation with shifts in positions within school-based social hierarchies.
Regarding the high-chronic group of aggressors, (i.e., mainly boys), our results revealed that there were no preferences regarding either forms or functions of aggression used. Given the stability of engagement in multiple subtypes of elevated aggression from kindergarten onwards, this group of children appear at greater risk for continued aggression, mental health problems and maladaptive functioning in later adolescence and adulthood.

### Antecedent Characteristics Associated with Group-Membership

As this is the first study to the best of our knowledge to examine multi-trajectories of both forms and functions of aggression trajectories across an eight-year developmental period, we made no hypotheses with respect to the specific risk factors that would predict group membership across the individual trajectories identified. We did however expect to find that for any chronic group identified, risk factors would include being male, from young single-mother families, with their own history of engagement in anti-social behaviour, lower IQ, educational, and SES backgrounds, and who engaged in high-risk prenatal behaviours (i.e., smoking). Further, we expected higher levels of coercive parenting behaviours to predict membership in the chronic group. Partial support for this hypothesis was found and is discussed below.
We identified specific risk and protective factors that were both homogeneous and heterogeneous between the moderate-engagers, high-desisting and high-chronic groups as compared to the combined low-stable and non-aggressor groups, although the majority of factors in combination were specific to individual trajectory groups. One factor, being male, was a risk factor common to two trajectory groups (i.e., the moderate-engagers and high-chronic), although this risk was largest for the high-chronic group (i.e., a staggering relative risk ratio of 13.5). This finding is line with previous studies examining either forms or functions of aggression and suggests that young males engaging in high levels of early aggression may need additional programming support to provide them with alternative strategies to replace their use of aggressive behaviours.
Another common risk factor was maternal IQ. That is, lower maternal IQ was a a risk factor associated with membership in both the high-desisting and high-chronic groups in particular. More specifically, for every decreasing unit (i.e., point scored), the relative risk ratio increased by 30 and 40%, respectively. While another factor (i.e., coercive parenting), was also common to two groups (i.e., the moderate-engagers and the high-chronic groups), albeit with differing magnitudes. For example, coercive parenting was associated with higher probabilities of membership in the high-chronic group as compared to the moderate-engagers, which would be expected. Coercive parenting, has repeatedly been implicated in children’s aggressive behaviour, which is likely attributable to both genetic transmission of risk and negative early child-rearing environments (reflected in poor quality interactions between the mother and child; e.g., Tremblay et al. 2004). Thus, the implication of coercive parenting for membership in the moderate-engagers and high-chronic trajectories would suggest that inadequate parenting skills are a good target for early prevention efforts. It was unclear however why coercive parenting was not associated with higher probabilities of membership in the high-desisting group, warranting future investigation.
Heterogeneous risk factors were also identified. For example, mothers not having received a high school diploma increased the relative risk ratio for membership in the high-chronic group by 4.3. This finding is in line with Nagin and Tremblay ( 2001) who also found low maternal education to increase risk for being on a chronic trajectory of PA. Moreover, maternal postnatal depression had an impact on membership in the high-desisting group: a relative risk ratio of 3.2. Maternal depression has previously been implicated in children’s aggressive behaviour (e.g., Hay et al. 2003), and is likely implicated via similar pathways as that of coercive parenting, particularly regarding early poor-quality dyadic interaction (e.g., Kim-Cohen et al. 2005). Interestingly, not being breastfed appeared to increase the risk for membership in the moderate-engagers trajectory group. This finding provides some support to the growing body of work examining the association between breastfeeding and externalising problems. While it is not entirely clear why not being breastfed would only be implicated for the moderate-engaging group, this finding is consistent with the suggestion that breastfeeding may only be associated with externalising problems at non-clinical levels (Girard et al. 2017).
Taken together, these findings would suggest that early intervention efforts really ought to be tailored dependent on group membership, as risk factors, and the combination of risk factors, are not uniform across concomitant trajectories of form and function of aggression engagement. Moreover, our results highlight the need for additional studies that use person-centered, rather than variable-centered approaches to modelling aggression across development. While the popular variable-centered approach in developmental studies has critically advanced theory in developmental aggression, it may have also partially inhibited our understanding of the degree of heterogeneity among groups of children engaging in concomitant forms and functions of aggression, along with the antecedent characteristics that increase risk for specific group membership.

### Strengths and Limitations

As this is the first study to simultaneously model both forms and functions of aggression and to estimate the ways in which they feed into one another using a developmental multi-trajectory person-centered approach, there are notable strengths. These include the use of a semi-parametric approach for modelling heterogeneity in four distinct subtypes of aggression simultaneously, the relatively large sample size, the use of repeated measures across time and assessment of maternal IQ, along with the use of a multi-informant approach whereby limiting the potential of shared method variance. Despite these strengthens, the study has limitations that must be acknowledged. Good practice for properly fitting a polynomial term requires a minimum of three data points per child, which significantly reduced the sample size. Thus, our results may be specific to the subsample used in this study, despite being largely consistent with previous epidemiological findings. Further, statistically significant differences were found between the entire cohort and those included in the current study, two of which were found to be significant risk factors for group membership in the high-chronic group (i.e., maternal education and child sex). Thus, it is possible that the prevalence rate of those belonging to the high-chronic group may in fact be underrepresented, warranting replication with larger sample sizes. Additional studies are also needed to examine a longer period of development (i.e., from infancy into adulthood), and which evaluate the long-term outcomes associated with specific group membership. The internal consistency for parenting variables were also low, which may have resulted in underestimation of the associations, thus warranting replication in future studies. Moreover, as with any study examining aggression, concerns may be raised with either construct validity or reliability across forms and functions of aggression variables. As a result of the challenges in providing a meaningful conceptualisation of residualised constructs when modelling forms and functions of aggression, we did not use residualised constructs in the current study. Thus, the possibility of confounding remains, particularly with respect to items for indirect aggression which may have also captured elements of reactive aggression. The results of the confirmatory factor analysis however do provide support for the construct validity of aggression variables used. Finally, additional items measuring both forms and functions of aggression would have been desirable. The high Cronbach’s alphas provided in this study however, provide support for the reliability of aggression variables used.

### Conclusions

Our results are in line with previous studies suggesting that programmes for the prevention of aggression should be offered to mothers with lower levels education or cognitive capacities (Eckenrode et al. 2017; Enoch et al. 2016). Further, children of mothers with a history of depression or who may have a higher disposition for postnatal depression may benefit from preventative efforts. These mothers in particular may need additional supports for strategies to provide their children with rich and stimulating environments from pregnancy throughout childhood (Enoch et al. 2016). Such programming efforts ought to focus on teaching parents the strategies needed for promoting positive quality dyadic interactions along with disciplinary tactics that are not coercive in nature. Additional support should also be given for mothers of young boys.

## Compliance with Ethical Standards

### Conflict of Interest

The authors have no conflicts of interest relevant to this article to disclose.

### Ethical Approval

Ethical approval was obtained by the Québec Institute of Statistics’ Ethics Committee.

### Informed Consent

Informed written parental consent and children’s assent were collected prior to each wave of data collection.

## Publisher’s Note

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

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text:
$SnO_2$ forms a positively charged colloidal Sol in the basic medium why ?
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text:
Math Help - logarithmic word problem

1. logarithmic word problem

Hello, here is my problem:

Alex invests $50,000 at an interest rate of 0.7% compounded monthly. Laura invests$40,000 at 9.5% compounded annually. After how many years will the two investments be equal in value?

okay so this is what I have so far

50,000 x (.07x12)^n=x

40,000 x .095^n=x

I am not really sure how to set up the equation and I'm pretty sure I started off wrong, help is much appreciated!

Thank you!
Dan

2. Re: logarithmic word problem

The 1st amount gets multiplied by 1.007 each month and the 2nd amount by 1.095 each year.

3. Re: logarithmic word problem

it should have been: 50,000 x (1.07^n)12=x and 40,000 x (1.095 ^n)=x I think im figuring it out. Biffboy, i think you ment to write 1.07

4. Re: logarithmic word problem

If the interest rate is 0.7% we multiply by 1.007. If you meant 7% we multiply by 1.07. In your question you said 0.7%
Also n years =12n months so with the 50000 the index will be 12n

5. Re: logarithmic word problem

The reason i cant figure it out is because it has two variables. In the answers n= 10.6, but if you sub that in the x variable in each equation are different when they should be the same according to the question in the first post. I am clearly missing something...

6. Re: logarithmic word problem

The first rate can't be 7% per month as amounts then would never be equal. So it must be 0.7% and you should be multiplying by 1.007^(12n)

7. Re: logarithmic word problem

I made a mistake writing the problem, sorry it is 7% not 0.7%, and 9.5%. My bad... Thank you for you help though, I'm busting out the calculator right now!

8. Re: logarithmic word problem

Originally Posted by Dwahl
Hello, here is my problem:

Alex invests $50,000 at an interest rate of 0.7% compounded monthly. Laura invests$40,000 at 9.5% compounded annually. After how many years will the two investments be equal in value?

okay so this is what I have so far

50,000 x (.07x12)^n=x
No "7% compounded monthly" does NOT mean 7% per month. It is still 7% per year which is $.07/12= 0.00583$ percent per month. In n years, there are, of course, 12n months so it will be $50000(1.00583)^{12n}$

40,000 x .095^n=x
And this should have been $40000(1.095)^n$
Set them equal and solve for n.

I am not really sure how to set up the equation and I'm pretty sure I started off wrong, help is much appreciated!

Thank you!
Dan

9. Re: logarithmic word problem

this is what i have so far, thanks for all your help guys, I really appreciate it!

50000 (1.00583)^12n = 40000 (1.095)^n

log(50000 x 1.00583^12n) = log(40000 x1.095^n)

log 50000 + log 1.00583^12n = log 40000 + log 1.095^n

log 50000 + 12n log 1.00583= log 40000+ n log 1.095

and this is sort of where i get a bit lost, below is what i did.

log 50000 - log 40000= -12n log 1.00583 + n log 1.095

Im not sure how to simplify it anymore. so i tried this

__ log 50000- log 40000____ = n+n
-12 + log 1.00583 + log 1.095

Can I get a little more help simplifying it, because I know I did it wrong :/ thanks again!

10. Re: logarithmic word problem

Last line wrong. In the previous line write the right hand side as n(log1.095-12log1.00583)


text:
# Euler phi at a product

If the positive greatest common divisor of the integers $a$ and $b$ is $d$, then

 $\varphi(ab)\;=\;\frac{\varphi(a)\,\varphi(b)\,d}{\varphi(d)}.$

Proof.  Using the positive prime factors $p$, the right hand side of the asserted equation is

 $\displaystyle\frac{d\cdot a\prod_{p\mid a}\frac{p-1}{p}\cdot b\prod_{p\mid b}% \frac{p-1}{p}}{d\prod_{p\mid a,\,p\mid b}\frac{p-1}{p}}$ $\displaystyle\;=\;\frac{ab\prod_{p\mid a,\,p\nmid b}\frac{p-1}{p}\cdot\prod_{p% \mid a,\,p\mid b}\frac{p-1}{p}\cdot\prod_{p\mid b,\,p\nmid a}\frac{p-1}{p}% \cdot\prod_{p\mid b,\,p\mid a}\frac{p-1}{p}}{\prod_{p\mid a,\,p\mid b}\frac{p-% 1}{p}}$ $\displaystyle\;=\;ab\prod_{p\mid a\;\lor\;p\mid b}\frac{p\!-\!1}{p}\;=\;ab% \prod_{p\mid ab}\frac{p\!-\!1}{p}\;=\;\varphi(ab),$

Q.E.D.

Title Euler phi at a product EulerPhiAtAProduct 2014-02-18 14:02:24 2014-02-18 14:02:24 pahio (2872) pahio (2872) 6 pahio (2872) Theorem msc 11A25 msc 11-00 EulerPhifunction DivisibilityByPrimeNumber


text:
# Space vectors 3D

The vectors u = (1; 3; -4), v = (0; 1; 1) are given. Find the size of these vectors, calculate the angle of the vectors, the distance between the vectors.

a =  5.099
b =  1.4142
A =  13.8979 °
h =  5.4772

### Step-by-step explanation:

$b=\mathrm{\mid }v\mathrm{\mid }=\sqrt{{v}_{x}^{2}+{v}_{y}^{2}+{v}_{z}^{2}}=\sqrt{{0}^{2}+{1}^{2}+{1}^{2}}=\sqrt{2}=1.4142$

Did you find an error or inaccuracy? Feel free to write us. Thank you!

Tips to related online calculators
Our vector sum calculator can add two vectors given by their magnitudes and by included angle.
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#### You need to know the following knowledge to solve this word math problem:

We encourage you to watch this tutorial video on this math problem:

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text:
# Measuring but not looking at the result

Once a state is measured, but we don't look at the result, is the state now written as a density matrix, that is, the probability that it could land on a measurement operator multiplied by the operator applied on the state, this summed up for every measurement operator that it could land on contained in the measurement?

Suppose you have a state $$\rho$$, and a random process that changes this to a state $$\rho_j$$ with probability $$p_j$$. If you know what the value of $$j$$ is, your knowledge of the resulting state will be described by the corresponding $$\rho_j$$. If you have no information regarding $$j$$, your knowledge will be described by

$$\sum_j ~ p_j ~ \rho_j$$

This is a general statement that holds for any random process. For the case you describe, which is measurement, the possible outcomes can often be described by a set of projectors $$\{P_j\}$$. For these

$$p_j = {\rm tr}~(~P_j~\rho~), ~~~ \rho_j = \frac{P_j \rho P_j}{p_j}.$$

Probabilities for more general measurements can be calculated by more general operators, but figuring out the post-measurement states for these is not always as easy.

In the Copenhagen interpretation, there are only two kinds of things that one can do, one is evolution and other is the measurement. Measuring but not looking is equivalent to measuring the system and hence projecting it to one of the possible eigenstates. (Or maybe you can clarify more what you meant by not looking?)

And after the system is probed in the measurement, it is no longer in a superposition and no longer in the statistical mixture anymore. It just becomes a decohered density matrix with a single element in the measurement (projector) basis. Density matrix representation then becomes trivial.

(I think in the latter part of your question you are pointing the completeness of probability in the measurement which summed over all the measurement projectors will be unity. But this has to do with the act of measurement, once that is performed, there is only one deterministic state.)


text:
# Complete valuation, norm of finite extension. Proof of Propositon.

I would like to ask for tips how to manage with proof of this proposition. How to show that $$v$$ can be uniquely extended to $$v'$$ ? Should I assume that $$v$$ can be also extended to another valuation? Also I don't know how to get equation.

This is a consequence of Hensel’s Lemma. Not the version that talks about roots of a polynomial, but the version that says that if the polynomial factors into two relatively prime factors over the residue field, then it factors over the original ring, and you can guarantee that one of the factors upstairs has the same degree as the factor downstairs.

I think that if you prove the following Lemma, you should be able to get the whole thing:

Lemma. Let $$k$$ be a field complete under the (additive) valuation $$v$$, and let $$f(X)=X^n+c_{n-1}X^{n-1}+\cdots+c_1X+c_0$$ be $$k$$-irreducible. Then:
(1) If $$v(c_0)\ge0$$, then for all $$i$$, $$v(c_i\ge0$$;
(2) If $$v(c_0)>0$$, then for all $$i$$, $$v(c_i)>0$$;
(3) If $$v(c_0)<0$$, then $$v(c_0)<\min_{0.

An important consequence is:

Proposition. Let $$k$$ be a field complete under the additive valuation $$v$$, and let $$K$$ be an algebraic extension of $$k$$, with a valuation $$V$$ extending $$v$$. Then $$\mathcal O=\{z\in K:V(z)\ge0\}$$ is the integral closure in $$K$$ of $$\mathfrak o=\{z\in k:v(z\ge0\}$$.
In other words, for $$z\in K$$, you get $$V(z)\ge0\Longleftrightarrow z\text{ is integral over }\mathfrak o$$.

I think that now it should be easy to show that if $$V$$ and $$V'$$ are extensions of $$v$$ from $$k$$ to $$K$$, then $$V=V'$$. That does it for uniqueness.

For existence, use the formula given. Of course the formula guarantees multiplicativity immediately; what you need besides is the additivity: $$V(a+b)\ge\min\bigl(V(a),V(b)\bigr)$$. You get this by showing that if $$V(z)\ge0$$, then $$V(1+z)\ge0$$ as well. Use the fact that the Norm $$N^{k(z)}_k(z)$$ is, up to sign, the constant coefficient of the minimal $$k$$-polynomial for $$z$$.


text:
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Temple (District and Circle lines): 2 minute walk. Charing Cross (Bakerloo and Northern lines): 10 minute walk, Embankment (District, Circle and Bakerloo lines): 10 minute walk, Waterloo (Jubilee, Northern, Bakerloo, Waterloo & City lines): 12 minute walk, Holborn (Central and Picadilly lines): 12 minute walk,Chancery Lane (Central line): use exit 4 - 15 minute walk.

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### 30.10.2019 (Wednesday)

#### Triangular Seminar at KCL

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### 27.03.2019 (Wednesday)

#### Bootstrapping Inflationary Fluctuations

Regular Seminar Guilherme Leite Pimentel (University of Amsterdam)

 at: 13:15 KCLroom S2.49 abstract: In flat space, four point scattering amplitudes at weak coupling can be fully determined from Lorentz symmetry, unitarity and causality. The resulting scattering amplitude depends on model details only through coupling constants and the particle content of the theory. I will show how the analogous story works in the case of inflationary fluctuations. We found explicit expressions for inflationary three and four-point functions, whose shapes depend on the field content of the theory, and do not depend on the specific inflationary model, as long as the fluctuations minimally break de Sitter symmetry. This cosmological bootstrap” is a first step towards classifying all possible shapes of primordial non-gaussianity, which can be searched for in experimental data.

### 20.03.2019 (Wednesday)

#### 3d Abelian Gauge theories at the Boundary

Regular Seminar Edoardo Lauria (University of Durham)

 at: 13:15 KCLroom S2.49 abstract: A four-dimensional abelian gauge theory can be coupled to a 3d CFT with a U(1) symmetry living on a boundary. This coupling gives rise to a continuous family of boundary conformal field theories (BCFTs) parametrized by the gauge coupling \tau and by the choice of the CFT in the decoupling limit. Upon performing an Electric-Magnetic duality in the bulk and going to the decoupling limit in the new frame, one finds a different 3d CFT on the boundary, related to the original one by Witten's SL(2, Z) action. In particular the cusps on the real \tau axis correspond to the 3d gauging of the original CFT. We study general properties of this family of BCFTs. We show how to express bulk one and two-point functions, and the hemisphere free-energy, in terms of the two-point functions of the boundary electric and magnetic currents. Finally, upon assuming particle-vortex duality (and its fermionic version), we show how to turn this machinery into a powerful computational tool to study 3d gauge theories.

### 13.03.2019 (Wednesday)

#### Anomalous supersymmetry

Regular Seminar Marika Taylor (University of Southampton)

 at: 13:15 KCLroom S-1.04 abstract: We show that supersymmetry is anomalous in N = 1 superconformal quantum field theories (SCFTs) with an anomalous R-symmetry. This anomaly was originally found in holographic theories: here we show that this anomaly is present in general and demonstrate it for the massless superconformal Wess-Zumino model via a one loop computation of four-point functions of two supercurrents with either R-currents. In fact, the Wess-Zumino consistency conditions together with the standard R-symmetry anomaly imply the existence of the anomaly. We outline the implications of this anomaly.

### 07.03.2019 (Thursday)

#### The Cosmological Constant and the Electroweak Scale

Regular Seminar Henry Tye ()

 at: 15:00 KCLroom S7.06 abstract: KCL TPPC Seminar

### 06.03.2019 (Wednesday)

#### When black holes collide: Probing the quantum state of black holes by the spectrum of emitted gravitational waves

Regular Seminar Ramy Brustein (Ben-Gurion University (Israel))

 at: 15:15 KCLroom K6.63 abstract: KCL TPPC Seminar Abstract: Black hole (BH) mergers can be viewed as cosmological “scattering experiments” resulting in an excited BH, which then decays to its ground state by emitting gravitational waves (GW). I will present general arguments, based on fundamental physics principles, as to why we should expect additional quantum ringdown modes to those predicted by general relativity. Then, I will discuss the spectrum of the predicted ringdown modes, the resulting emitted GW and the prospects for their detection in LIGO/VIRGO.

### 27.02.2019 (Wednesday)

#### Massive AdS Gravity from String Theory

Regular Seminar Costas Bachas (Ecole Normale Superieure)

 at: 13:15 KCLroom S2.49 abstract: TBA

### 21.02.2019 (Thursday)

#### L_infinity algebras, the BV formalism, and Classical Fields

Exceptional Seminar Martin Wolf (Surrey)

 at: 11:00 KCLroom Bush House S 2.01 abstract: I will first review the Batalin-Vilkovisky formalism and its mathematical foundations with an emphasis on higher algebraic structures and classical field theories. I will then move on and discuss recent developments in formulating higher gauge theory with Lie quasi-groupoids as gauge structure. Finally, I will explain how all these ideas can be combined with those of twistor theory to formulate maximally superconformal gauge theories in four and six dimensions by means of quasi-isomorphisms.

### 21.02.2019 (Thursday)

#### Twisting and localization in supergravity: equivariant cohomology of BPS black holes

Exceptional Seminar Imtak Jeon (KCL)

 at: 11:50 KCLroom Bush House S 2.01 abstract: I will talk about the formalism of supersymmetric localization in supergravity using the deformed BRST algebra defined in the presence of a supersymmetric background. The gravitational functional integral localizes onto the cohomology of a global supercharge Qeq, obeying (Qeq)2=H, where H is a global symmetry of the background. This construction naturally produces a twisted version of supergravity whenever supersymmetry can be realized off-shell. I will present the details of the twisted graviton multiplet and ghost fields for the superconformal formulation of four-dimensional N=2 supergravity. As an application of our formalism, we systematize the computation of the exact quantum entropy of supersymmetric black holes. In particular, we compute the one-loop determinant of the Qeq deformation operator for the off-shell fluctuations of the Weyl multiplet around the AdS2×S2 saddle.

### 21.02.2019 (Thursday)

#### 3d Abelian Gauge theories at the Boundary

Exceptional Seminar Edoardo Lauria (Durham)

 at: 14:00 KCLroom Bush House S 2.01 abstract: A four-dimensional abelian gauge theory can be coupled to a 3d CFT with a U(1) symmetry living on a boundary. This coupling gives rise to a continuous family of boundary conformal field theories (BCFTs) parametrized by the gauge coupling τ and by the choice of the CFT in the decoupling limit. Upon performing an Electric-Magnetic duality in the bulk and going to the decoupling limit in the new frame, one finds a different 3d CFT on the boundary, related to the original one by Witten's SL(2, Z) action. In particular the cusps on the real τ axis correspond to the 3d gauging of the original CFT. We study general properties of this family of BCFTs. We show how to express bulk one and two-point functions, and the hemisphere free-energy, in terms of the two-point functions of the boundary electric and magnetic currents. Finally, upon assuming particle-vortex duality (and its fermionic version), we show how to turn this machinery into a powerful computational tool to study 3d gauge theories.

### 21.02.2019 (Thursday)

#### Entanglement Content of Quantum Particle Excitations

Exceptional Seminar Cecilia De Fazio (City)

 at: 14:50 KCLroom Bush House S 2.01 abstract: In the ﬁrst part of my talk I will give a brief introduction to the main features of the Entaglement Entropy of a bipartite system in Integrable Quantum Field Theories and Conformal Field Theories. Secondly I will discuss the results my collaborators and I obtained in our two papers. In particular, I will consider the Entaglement Entropy of a single connected region of a ﬁnite bipartite system in excited states described by one-dimensional massive free theories with ﬁnite numbers of particles. I will show that in the limit of large volume and large length of the region the excess of entanglement due to the presence of the particles with respect to the ground state takes a simple form and admits a "q-bit interpretation".

### 21.02.2019 (Thursday)

#### Thermo/HydroDynamics of the classical Toda system and random matrix theory

Exceptional Seminar Benjamin Doyon (KCL)

 at: 16:00 KCLroom Bush House S 2.01 abstract: This will be presenting mainly some of my recent work, and also aspects of recent work of Herbert Spohn, both developed in parallel. The classical Toda system is a one-dimensional integrable many-body system, which can be seen either as a gas of particles or as a chain of degrees of freedom. Herbert has shown how the generalised Gibbs ensembles of the Toda chain can be obtained from a certain limit of the beta-ensemble in random matrix theory. Analysing and connecting the gas and chain viewpoints, I have obtained both the generalised Gibbs ensembles and generalised hydrodynamics from a quasiparticle scattering description. Thus we make a connection between quasiparticle scattering and random matrix theory.

### 20.02.2019 (Wednesday)

#### Witten, Cardy, and the Holonomy Saddle

Polygon Seminar Piljin Yi (KIAS)

 at: 15:00 KCLroom S-1.27 abstract: This talk will explore topological invariants of susy gauge theories, with some emphasis on index-like quantities and the notion of holonomy saddles. We start with 1d refined Witten index computations where the twisted partition functions typically show rational, rather than integral, behavior. We will explain how this oddity is a blessing in disguise and propose a universal tool for extracting the truely enumerative Witten indices. In part, this finally put to the rest a two-decade-old bound state problems which had originated from the M-theory hypothesis. Along the way, we resolve an old and critical conflict between Kac+Smilga and Staudacher/Pestun, circa 1999~2002, whereby the notion of holonomy saddles emerges and plays a crucial role. More importantly, the holonomy saddle prove to be universal features of supersymmetric gauge theories when the spacetime include a small circle. We explore them further for d=4, N=1 theories, with much ramifications on recent claims on Cardy exponents of their partition functions.

### 13.02.2019 (Wednesday)

#### String theory compactifications with sources

Regular Seminar Alessandro Tomasiello (Universita di Milano-Bicocca)

 at: 13:15 KCLroom S2.49 abstract: In recent years, more and more compactifications have emerged whose existence depends crucially on the presence of internal sources to the supergravity fields, such as D-branes and orientifold planes. I will review some solutions of this type in various dimensions, and illustrate their applications to holography and potentially to the problem of finding de Sitter solutions.

### 04.02.2019 (Monday)

#### Single trace TTbar-deformations and AdS/CFT

Informal Seminar Gaston Giribet (UBA)

 at: 10:00 KCLroom K6.63 abstract: A solvable irrelevant deformation of AdS3/CFT2 correspondence leading to a theory with Hagedorn spectrum at high energy has been recently proposed by Kutasov et al. It consists of a single trace deformation of the boundary theory, which is inspired by the recent work on solvable T\bar{T}-deformations of two-dimensional CFTs. Thought of as a worldsheet sigma-model, the interpretation of the deformed theory from the bulk viewpoint is that of string theory on a background that interpolates between AdS3 in the IR and a linear dilaton vacuum of little string theory in the UV. In this talk, after giving an introduction to this class of solvable theories, I will present explicit results for their observables.

### 30.01.2019 (Wednesday)

#### Cosmic censorship violation and black hole collisions in higher dimensions

Regular Seminar Roberto Emparan (ICREA, Barcelona)

 at: 13:15 KCLroom S2.49 abstract: The cosmic censorship conjecture raises the question of whether classical gravitational dynamics can drive a low-energy configuration into an accessible regime of quantum gravity, with Planck-scale curvatures and energy densities visible by distant observers. I will present evidence that cosmic censorhip is violated in the quintessential phenomenon of General Relativity: the collision and merger of two black holes. It only requires a sufficient total angular momentum in a collision in high enough number of dimensions. Nevertheless, I will argue that even if cosmic censorship is violated in this and in some other know instances, its spirit remains unchallenged: classical relativity describes the physics seen by observers outside the black holes accurately, with only minimal quantum input that does not entail macroscopic disruptions.

### 23.01.2019 (Wednesday)

#### What spatial geometry does the (2+1)-d QFT vacuum prefer?

Regular Seminar Toby Wiseman (Imperial College London)

 at: 13:15 KCLroom S2.49 abstract: We consider the energy of a (2+1)-d relativistic QFT on a deformation of flat space in either the quantum or thermal vacuum state. Looking at both free scalars and fermions, with and without mass (and in the scalar case including a curvature coupling) we surprisingly find that any deformation of flat space is always energetically preferred to flat space itself. This is a UV finite effect, insensitive to any cut- off. We see the same behaviour for any (2+1)-holographic CFT which we compute via the gravity dual. We consider the physical application of this to membranes carrying relativistic degrees of freedom, the vacuum energy of which then induce a tendency for the membrane to crumple. An interesting case is monolayer graphene, which experimentally is observed to ripple, and on large scales can be understood as a membrane carrying free massless Dirac degrees of freedom.

### 12.12.2018 (Wednesday)

#### An Etude on 3d N=2 Conformal Manifolds

Regular Seminar Nikolay Bobev ( KU Leuven)

 at: 13:15 KCLroom S2.49 abstract: I will describe the one-dimensional complex conformal manifold that controls the infrared dynamics of a three-dimensional N=2 supersymmetric theory of three chiral superfields with a cubic superpotential. Two special points on this conformal manifold are the XYZ model and three decoupled copies of the critical Wess-Zumino model. The conformal manifold enjoys a discrete duality group and can be thought of as an orbifold of CP^1. I will discuss how to compute the spectrum of low lying operators and their OPE coefficients as a function of the coordinate on the conformal manifold using the epsilon-expansion and the numerical conformal bootstrap.

### 04.12.2018 (Tuesday)

#### Coulomb branches and mock modular forms

Informal Seminar Giorgios Korpas (Trinity College, Dublin)

 at: 10:15 KCLroom K6.63 abstract: “We revisit Donaldson-Witten theory, that is the N=2 topologically twisted super Yang-Mills theory with gauge group SU(2) or SO(3) on compact 4-manifolds. We study the effective action in the Coulomb branch of the theory and by considering a specific Q-exact deformation to the theory we find interesting connections to mock modular forms. A specific operator of this theory computes the famous Donaldson invariants and our analysis makes their computation more accessible than previously. We also extend these ideas to the case of ramified Donaldson-Witten theory, that is the theory in the presence of embedded surfaces. Our results make calculations of correlation functions of Coulomb branch operators more trackable and we hope that they can help in the search of new 4-manifold invariants.”


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# Q 'bout games with LARGE levels

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Say I have a quite large 2d game world with LOTS of entities. All at different positions of course. Now I have the camera follow one of these entities thus I want to draw other entities if they are close enough and also perform collision detection and other stuff. This would get quite tough for the CPU if it has to do this for ALL entities (even if I first check if the object is close enough I think..). So my question is: how do most games do this? Do they simply store all entities in an array or linked list and do some kind of comparision whether to check for collisions etc on the object, or do they use some kind of data tree?? I was thinking that'd be a nice solution, but there is one problem with that: if I was to store everything by its x-position, then I would end up with ALL entities with an x-position within a given range no matter what Y-POS they have. Thank you

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Most use a tree with 2^(number of dimensions) children per node. Quadtree in 2D, Octtree in 3D.
(<=,<=),(<=,>),(>,<=),(>,>) for 2d

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A tree structure is good because you only have to do actual collision testing on objects that are close together.

Quick Approximation of 2D Collision Detection

I hope that is enough information to get you into it.

Good luck.

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Quote:
 Original post by moussen15Say I have a quite large 2d game world with LOTS of entities. [...] how do most games do [collision detection]?

The keyword phrase you want to google for is "spatial partitioning".

The basic idea is to partition space into regions in some manner, with the regions being larger than the objects you have. Clearly, if two objects reside in two regions that are far from each other, they do not need to be tested for collision against each other. In other words, for a given object A you only need to test for collisions amongst the objects in the region R of A (and regions neighboring R, if objects are allowed to straddle regions).

Grids are a very effective spatial partitioning method. Trees, such as k-d trees, quadtrees, and octrees are also good spatial partitioning methods. There's lots of information available on the net about all of these.

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You may often find (it depends on the type of game you are making) that in a large world, it's the world entities that are many, not the actual moving objects (characters). Calculating the collisions etc for moving objects is actually much easier since there are usually only a few.

As for drawing too, usually most people use a quadtree or grid or some similar structure. However, i've come up with an idea for non-tilebased static world objects i call a Cell Wrapper. Because all your objects will have some kind of Draw() or Update() function, what you do is create an object with those functions whose only job is to update other objects. However a cell wrapper keeps track of the furthest X and Y coords of the objects it contains. When it gets called with Draw() or similar, it first checks if the total bounding rectangle of all the objects it contains (like a rubber band) intersects the rectangle made by the screen, then draw the objects. So we do this:

// create a cell wrapperCellWrapper cw;cw.AddObject( some_object_1 );cw.AddObject( some_object_2 );cw.AddObject( some_object_3 );// cw now understands the bounding volumes of the objects it contains// now the engine iterates over all the objects when drawing including......cw.Draw();...// Inside, the cell wrapper does this:void CellWrapper::Draw() {   if ( my_bounding_rectangle intersects screen_rectangle ) {      foreach object contained { object->Draw(); }      }   }

... and so if the cell wrapper volume intersects the screen, it then procedes to draw all of the objects contained by it. You could further reduce overdraw by putting individual bounding checks on each object if you wanted, but that might bog things down a bit.

It's not a foolproof method, but it's easy to slap together and works good very static world objects. You can change how big you want the wrappers by varying what you put in them. If you keep them small you have more Draw() called but less overdraw. If they are bigger and contain many objects, you call Draw() less times but may waste a lot of drawing.

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You might also want to consider not doing collision detection on things more than X metres away. It's common in games that entities only exist where yuor are looking and close by - for instance in a city the traffic will be generated around you rather than modelling every car in the city. If you have to have the entities as persistent for some reason, collision detection is almost certainly something you can avoid for out-of-screen entities in amost all cases.

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For each moving object which changes its velocity and/or direction.
Check every object with every other object on the same grid and calculate which one the moving object is going to collide with and when if it keeps its present cource.
Add the time of collision and the objects to the que
next

Keep a que, of items in crhonoligical order.

Per frame, check the first item to see if the objects will be colliding within the next frame.

If they are, then do collision responce,
and add any more collisions to the list. Every collision move the time forward to the collision every time your responding to it.

From,
Nice coder

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One thing I didn't see mentioned above is that most games don't have LARGE levels or LOTS of entities - if you can walk in a straight line for a long time, you're probably changing maps without knowing it (because they are loaded while you walk around instead of when you hit the seam) and entities are loaded/unloaded just like levels.
I can't give any actual numbers, though. The above combined with even the crudest of spatial partitioning algorithms changes a billion collision checks(from 10000 entities in the world) each update to maybe a few hundred.

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# Electric constant

The electric constant, or vacuum permittivity, earlier called the absolute dielectric permittivity and dielectric permittivity of vacuum, is a physical constant, a scalar quantity that:

The electric constant has the dimension of farad per meter.

## Definition

The electric constant ${\displaystyle \varepsilon _{0}}$  is determined with the help of the speed of light ${\displaystyle c}$  and the magnetic constant ${\displaystyle \mu _{0}}$  : [1]

${\displaystyle \varepsilon _{0}\,{\overset {\underset {\mathrm {def} }{}}{=}}\,{\frac {10^{7}}{4\pi \ c^{2}}}\equiv {\frac {1}{\mu _{0}c^{2}}}\approx 8,854187817620\times 10^{-12}}$  F/m.

## Application

The electric constant appears in Maxwell's equations in vacuum, which describe the properties of electric and magnetic fields, as well as electromagnetic radiation, and relate the fields with their sources.

In the matter the material equations of the electromagnetic field are used, while the electric displacement field D is expressed in terms of the electric constant, the electric field strength vector E and the polarization density vector P:

${\displaystyle \mathbf {D} =\varepsilon _{0}\ \mathbf {E} +\mathbf {P} .}$

As a rule, we can assume that ${\displaystyle \mathbf {P} =\varepsilon _{0}\chi \mathbf {E} }$ , where the quantity ${\displaystyle \chi }$  represents a tensor and is called electric polarizability. This expression means that the polarization density vector as a certain reaction of the matter is generated by the vector of the electric field strength in the matter, while the directions of these vectors may not coincide.

In the weak field, the quantity ${\displaystyle \chi }$  has the special name electric susceptibility and it is almost constant, depending on the type of matter and its state. In this case we can write:

${\displaystyle \mathbf {D} =\varepsilon _{0}\ \mathbf {E} +\varepsilon _{0}\chi \mathbf {E} =\varepsilon _{0}(1+\chi )\mathbf {E} =\varepsilon _{0}\varepsilon _{r}\mathbf {E} =\varepsilon _{a}\mathbf {E} .}$

The product of the electric constant by the relative permittivity ${\displaystyle \varepsilon _{r}}$  in this expression is called the absolute electric permittivity ${\displaystyle \varepsilon _{a}}$ .

The electric constant is included in the formulation of the Coulomb's Law, giving the expression for the force acting between two electric charges:

${\displaystyle \mathbf {F} _{12}={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}r_{12}^{2}}}{\frac {\mathbf {r} _{12}}{r_{12}}},}$

where ${\displaystyle r_{12}}$  is the distance between the charges ${\displaystyle q_{1}}$  and ${\displaystyle q_{2}}$ . If ${\displaystyle \mathbf {r} _{12}}$  is a vector directed from the charge ${\displaystyle q_{1}}$  to the charge ${\displaystyle q_{2}}$ , then the force ${\displaystyle \mathbf {F} _{12}}$  will be the force acting on the charge ${\displaystyle q_{2}}$  from the charge ${\displaystyle q_{1}}$ . From the expression for the force we can see that the electric constant in the system of physical units SI relates the electric charge with the mechanical units, such as force and distance.

## Expression in terms of the vacuum field parameters

In the concept of the force vacuum field [2] it is assumed that the electrogravitational vacuum is filled with the fluxes of particles that create gravitational and electromagnetic forces between the bodies. In particular, the fluxes of charged particles – praons, moving at relativistic velocities and transferring their momentum to the charged matter, are considered to be responsible for emergence of the Coulomb force.

In the model of cubic distribution of the praons’ fluxes, for the electric constant we obtain the following: [3]

${\displaystyle ~\varepsilon _{0}={\frac {e^{2}}{6p_{q}D_{0q}\vartheta ^{2}}}={\frac {e^{2}}{\varepsilon _{cq}\vartheta ^{2}}}.}$

Here ${\displaystyle ~p_{q}}$  is the momentum of praons, interacting with the charged matter; the fluence rate ${\displaystyle ~D_{0q}}$  denotes the number of praons dN coming per time dt through the perpendicular to the flux area dA of one face of the cube, limiting the volume under consideration; ${\displaystyle ~\vartheta =2.67\cdot 10^{-30}}$  m² is the cross-section of interaction between praons and nucleons; ${\displaystyle ~e}$  is the elementary charge; ${\displaystyle ~\varepsilon _{cq}=4\cdot 10^{32}}$  J/m³ is the energy density of the praons’ fluxes for cubic distribution.

In the model of spherical distribution of the praons’ fluxes in space:

${\displaystyle ~\varepsilon _{0}={\frac {e^{2}}{16\pi p_{q}B_{0q}\vartheta ^{2}}}={\frac {3e^{2}}{2\varepsilon _{sq}\vartheta ^{2}}},}$

where the fluence rate ${\displaystyle ~B_{0q}}$  denotes the number of praons dN coming per time dt from the unit solid angle ${\displaystyle d{\alpha }}$  inside the spherical surface dA; ${\displaystyle ~\varepsilon _{sq}=6\cdot 10^{32}}$  J/m³ is the energy density of the praons’ fluxes for spherical distribution.

Hence it follows that the electric constant is a dynamic variable, depending on the parameters of the vacuum field particles.

## References

1. CODATA Value: electric constant. The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values.
2. Fedosin S.G. The Force Vacuum Field as an Alternative to the Ether and Quantum Vacuum. WSEAS Transactions on Applied and Theoretical Mechanics, ISSN / E-ISSN: 1991-8747 / 2224-3429, Volume 10, Art. #3, pp. 31-38 (2015).
3. Fedosin S.G. The charged component of the vacuum field as the source of electric force in the modernized Le Sage’s model. Journal of Fundamental and Applied Sciences, Vol. 8, No. 3, pp. 971-1020 (2016). http://dx.doi.org/10.4314/jfas.v8i3.18, https://dx.doi.org/10.5281/zenodo.845357.


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# Finding A Large Reducible Polynomial

Algebra Level 5

Find the largest integer $$N<1000$$ such that the polynomial $f_N(x)=x^6+x^5+x^4+Nx^3+x^2+x+1$ is reducible over the integers.

Definition: A polynomial that is reducible over the integers can be expressed as a product of two non-constant polynomials with integer coefficients.

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Home » Uncategorized » Exponential matrix

# Exponential matrix

Let . We define

It is known that this series converges. Prove that

Solution

We triangulise the matrix , that is where is an invertible matrix and is an upper triangular. This is possible since our matrix is over and thus its characteristic polynomial splits. Let be its eigenvalues. Then we note that is upper triangular with in its diagonal. Hence is also upper triangular with in its diagonal. Hence

However and forall . Thus and finally

1. Since the space of all complex matrices is a finite-dimensional vector space, all norms define the same topology. So we can take a sub-multiplicative norm, that is, a norm such that . (For example, we can take to be the operator norm on E.) As a finite dimensional vector space, E is complete, so it’s enough to show normal convergence. We have that, for each integer ,

and we know that, for each real number , the series converges (it defines the exponential function). Therefore, for any , the series converges.

(We also got the additional result that . )

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