**10. Sinusoidal Responses of Three Classes of Fractional Oscillators**

When the excitation force takes the sinusoidal one in the form of cos *ωt* or sin *ωt*, the response is termed sinusoidal response, which plays a role in the field of oscillations.


**2. Theory**

The SO process leading to the ejection of two electrons in the continuum can be exhibited graphically by an infinite sequence of diagrams presented in Figure 1a. Here, we use the following graphical symbols. A straight line with an arrow to the right represents the electron continuum states *k*, *f* whereas an arrow pointing to the left exhibits the holes in atomic shells *i*, *l*, *m*. Before photoabsorption, all the atomic shells are presumed to be fully occupied and thus closed. The atomic ground state (the "vacuum" state) contains neither holes nor excited electrons. The wavy line denotes the Coulomb interaction between the electrons. The dashed line represents an absorbed photon. A circle denotes the diagonal matrix element of the self-energy part of the Green's function Σ*i*. The SHGF self-energy is expanded graphically in more detail in Figure 1b.

**Figure 1.** (**a**) Schematic representation of the SO process leading to a two-electron continuum. The circle represents the self-energy part of the SHGF expanded in more detail in (**b**).

The corresponding DPI amplitude can be found by summing a geometric progression of terms in Figure 1a that leads to the following expression:

$$
\langle f \| M\_{\rm SO} \| i \rangle = \langle f \| r \| i \rangle \left( 1 - \frac{\Sigma\_i(\varepsilon)}{\varepsilon - \varepsilon\_i - i\delta} \right)^{-1} \frac{\langle ik \| \| L\_L \| \| lm \rangle}{\varepsilon - \varepsilon\_i - i\delta} \,. \tag{1}
$$

Here, *f ri*- is a dipole matrix element of the single-photon absorption and *ikULlm*- is a Coulomb matrix element, which includes a direct and exchange *l* ↔ *m* terms and involves a transfer of the angular momentum *L*. Both the dipole and Coulomb matrix elements are reduced to strip their angular momentum projections dependence. The symbol *ε* =  *f* − *ω* < 0 denotes an effective hole energy. In the absence of correlation, *ε* =  *i* is the orbital energy. Many-electron correlation adds a discrete spectrum of shake-up satellites and a shake-off continuum, which starts at the DPI threshold *E*∞ =  *l* +  *m*. The infinitesimally small *δ* → 0 in the energy denominator defines the pole bypass.

The imaginary part of Equation (1) gives rise to an additional phase of the DPI amplitude due to the final state correlation:

$$\arg M\_{\rm SO} = \arctan \frac{\mathrm{Im}\Sigma\_i(\varepsilon)}{\mathrm{Re}[\varepsilon - \varepsilon\_i - \Sigma\_i(\varepsilon)]} \approx \arctan \frac{\mathrm{Im}\Sigma\_i(\varepsilon)}{\mathrm{Re}[\varepsilon - \varepsilon\_i]} \,. \tag{2}$$

Here ImΣ*i* = *π*(2*L* + 1)−1|*ikULlm*-|2 . The approximate equality is satisfied under the condition that |*ε* −  *i*||Σ*i*(*ε*)| . This condition defines the part of the double ionized continuum sufficiently far from the main photoelectron line.

The energy resolved single-differential cross-section (SDCS) is given by Equation (9) of [25]:

$$\frac{d\sigma\_i^{2+}}{d\epsilon\_f} = \sigma\_i^+ \, \frac{1}{\pi} \frac{\text{Im}\Sigma\_i(\varepsilon)}{|\varepsilon - \varepsilon\_i - \Sigma\_i(\varepsilon)|^2} \approx \sigma\_i^+ \, \frac{1}{\pi} \frac{\text{Im}\Sigma\_i(\varepsilon)}{|\varepsilon - \varepsilon\_i|^2} \,. \tag{3}$$

By solving this equation relative to ImΣ*i*, we can express the additional phase of the DPI amplitude due to the final state correlation in the following form:

$$\arg M\_{\rm SO} = \arctan \frac{\pi}{\sigma\_i^+} \frac{d \sigma\_i^{2+}}{d \varepsilon\_f} \left| \varepsilon - \varepsilon\_i \right| \,. \tag{4}$$

We note that all the quantities entering this expression are known from the experiment, which are the single photoionization cross-section of the primary photoelectron *σ*+*i* and the energy differential DPI cross-section *dσ*2+*i*/*d f* .

By integrating the SDCS Equation (3) over the fast photoelectron energy, we can obtain the double-to-single photoionization cross-section ratio:

$$R = \frac{\sigma^{2+}}{\sigma^{+}} = \frac{1}{\sigma^{+}} \int\_{0}^{\infty} \frac{d\sigma^{2+}}{d\varepsilon\_{f}} d\varepsilon\_{f} \,. \tag{5}$$

This ratio is known in He, and its isoelectronic sequence of ions [30]. It will serve as a convenient reference in Section 3.2.

Following [25], we can introduce the inverse SHGF

$$F(\varepsilon) = G^{-1}(\varepsilon) = \varepsilon - \varepsilon\_i - \Sigma\_i(\varepsilon) \tag{6}$$

Then the argument of the SO amplitude Equation (2) can be rewritten as

$$\arg M\_{\rm SO} = \arg G^{-1}(\varepsilon) = \arg F(\varepsilon) \tag{7}$$

This expression allows to use the integral rule presented by Equation (5) of [25], which relates the energies of the discrete shake-up satellites with the time delay:

$$\sum\_{k=0}^{\infty} (\varepsilon\_k - E\_k) = \frac{1}{\pi} \int\_{-\infty}^{E\_{\text{SO}}} \varepsilon \tau(\varepsilon) \, d\varepsilon \text{ , where } \tau(\varepsilon) = \frac{\partial}{\partial \varepsilon} \arg M\_{\text{SO}} = \frac{\partial}{\partial \varepsilon} \arg F(\varepsilon) \tag{8}$$

Here *Ek* =  *l* +  *m* −  *k* and *E*0 =  *i* are non-correlated energies of the shake-up excitations calculated from the HF orbital energies. Meanwhile, *εk* are the corresponding energies shifted by the final-state correlation and found as the poles of the SHGF. The integral time delay rule Equation (8) presents an analytical test for the numerical SO time delay values. This test will be conducted in Section 3.3.


**2. Why bullying is a public health problem**

Bullying in school is the most common type of violence among adolescents, recognized as a serious concern for students and has become a public health problem and global concern. Although the prevalence of bullying is difficult to estimate because of the different measures used throughout the studies, researchers have agreed that bullying is a pervasive and significant problem in our schools today. In 2016 nationwide survey, 20.8% of students report being bullied [7]. In 2015 survey of students ages 12–18, 21% reported being bullied at school, 13% reported being made fun of, called names, and insulted; 12% were the subject of rumors; 5% were pushed, shoved, tripped, or spit on; and 5% of students were socially excluded. In the same survey, 4% of students experienced threatened with harm, 3% were forced to do things they did not like to do, and finally, 2% reported that their belonging was damaged on purpose [7]. In the same survey, females (23%) more than males (19%) reported being bullied and females (15%) were more than males (9%) being the subject of rumors. However, male (5%) more than female (3%) students reported threatened with harm. Black students (25%) and white students (22%) more than Hispanic students (17%) reported being bullied in school [7].

The rate of bullying throughout the survey studies ranged from 9 to 98%. The bullying rates from 80 studies for students age 12–18 year was 35% for traditional bullying and 15% for cyberbullying [8]. Only 36% of children were bullied reported the bullying, and 64% did not report it [9]. Among high school students in 2015, 16% reported cyberbullying, and 20% are bullied on school propriety. For middle school students, 24% are cyberbullied, and 45% are inside the school. From 2007 to 2016, the rate of cyberbullying is doubled from 18 to 34% [3]. Students with special needs are two to three times more likely to be bullied than the normal students. Moreover, they found to report feeling unsafe in school compared to their normal peers [10].

Ethnic minorities are more likely to be bullied than others, African-American students (24.7%) more likely to report being bullied than Hispanic students (17.2%), followed by Asian student (9%). About 74 and 36.2% of students were verbally and physically bullied because of their sexual orientation, and 55.2 and 22.7% of students were verbally and physically bullied because of their gender expression [10]. The analysis of students from 19 countries (lowand-middle income countries) that participated in the Global School-based Students Health Survey (GSHS), 34.2% of students reported being bullied in the last month, 55.6% of them had been victimized 1 or 2 days, and 7.9% all 30 days in the last month. The prevalence of bullying for each country ranged from 20 to 61% [11].

#### **2.1. Health consequences of bullying**

Most definitions of bullying have similarities regarding the features of bullying such as being repeated over time, intent to inflict harm, an imbalance in power, and all highlight the chronicity of bullying. The Center for Disease Control (CDC) defined bullying in school as "any unwanted aggressive behaviors by another youth or group of youths who are not siblings or current dating partners that involves observed or perceived power imbalance and is repeated

A new emerging type of bullying called electronic aggression or cyberbullying. Bullying through electronic media becoming a critical concern through the increasing virtual interactions among adolescents and children. This type of bullying is known as an aggressive online behavior in the digital space and defined as repetitive, aggressive behavior using technology through cell phones and social media. Students who experienced cyberbullying bullied through email, chat rooms, instant messages, websites, or texting [4, 5]. This type of bullying is different from the traditional bullying in that it is difficult for school staff and administrators to control because it happens outside schools and mostly in the forms of anonymous messages posted online through social media. Students who experience cyberbullying have

Bullying in school is the most common type of violence among adolescents, recognized as a serious concern for students and has become a public health problem and global concern. Although the prevalence of bullying is difficult to estimate because of the different measures used throughout the studies, researchers have agreed that bullying is a pervasive and significant problem in our schools today. In 2016 nationwide survey, 20.8% of students report being bullied [7]. In 2015 survey of students ages 12–18, 21% reported being bullied at school, 13% reported being made fun of, called names, and insulted; 12% were the subject of rumors; 5% were pushed, shoved, tripped, or spit on; and 5% of students were socially excluded. In the same survey, 4% of students experienced threatened with harm, 3% were forced to do things they did not like to do, and finally, 2% reported that their belonging was damaged on purpose [7]. In the same survey, females (23%) more than males (19%) reported being bullied and females (15%) were more than males (9%) being the subject of rumors. However, male (5%) more than female (3%) students reported threatened with harm. Black students (25%) and white students (22%) more than Hispanic students (17%) reported being bullied in school [7]. The rate of bullying throughout the survey studies ranged from 9 to 98%. The bullying rates from 80 studies for students age 12–18 year was 35% for traditional bullying and 15% for cyberbullying [8]. Only 36% of children were bullied reported the bullying, and 64% did not report it [9]. Among high school students in 2015, 16% reported cyberbullying, and 20% are bullied on school propriety. For middle school students, 24% are cyberbullied, and 45% are inside the school. From 2007 to 2016, the rate of cyberbullying is doubled from 18 to 34% [3]. Students with special needs are two to three times more likely to be bullied than the normal students. Moreover, they found to report feeling unsafe in school compared to their normal peers [10].

multiple times or is highly likely to be repeated" [3].

48 Health and Academic Achievement

similar consequences as traditional bullying [5, 6].

**2. Why bullying is a public health problem**

Scientific research indicated that experiencing bullying has short and long-term psychological and emotional impact on both victims and perpetrator [5, 12, 13]. Victims of bullying reported poor mental and physical health, more symptoms of anxiety, depression; feeling sad, being loneliness; vomiting; sleep disturbance; nightmares; body ache; a headache; abdominal pain, and frequent illnesses. This, in turn, increases students' absenteeism either from direct physical or indirect psychological impact [12, 14].

Students, whether they are bullies, a victim, or bystanders, report a suicidal behavior. Among students age15–29, suicide is the second leading cause of death [15]. Students who experienced bullying are two times at risk for have suicidal ideation, and 2.6 times more likely to attempt suicide than other students who did not experience bullying [16]. According to the Youth Risk Behavior Survey (YRBS), 17.7% of school-aged students had attempting suicide within the past year [17]. Depression, violent behavior, and substance abuse are among the most mediated factors between bullying and suicide [12, 18].

#### **2.2. School bullying and academic achievement**

Bullying in school not only affects students' physical, emotional and psychosocial lives, it is also found to affect their academic achievements and their classwork directly. Theoretical and empirical literature has been supported the direct association between bullying and student' academic achievements [5, 19–23]. A study among secondary level students shows that traditional as well as cyberbullying has a significant negative impact on students' academic performance when controlling the socioeconomic status of the student [24]. Other findings among students of 13–18 years old revealed that academic achievement negatively correlated with bullying [25]. The research examined the achievement scores of 46 schools and found that peer harassment was negatively correlated with achievement and that students who are teasing they miss school and missing educational opportunities [26]. A recent study among primary school children revealed that verbal bullying among female students was associated with poor academic performance on writing skills. Physical bullying was negatively associated with performance on numeracy, and writing for both males and females. Students who report being verbally and physically bullied had poor achievements in reading, writing, spelling, grammar, and punctuations [23].

Research to date indicates that bullying can seriously affect adolescents' psychosocial functioning, school adjustment particularly academic performance. Peer victimization and low academic performance often correlated because children who are chronically victimized experience negative emotional and psychological outcomes that can inhibit their engagement in the classroom and thus affected their academic achievement. Although small percentages of students are chronic victims of bullying in school, but also temporary victimization can seriously impaired students' academic performance and achievement [5, 21, 27–29].

bully, and bully-victim), found that student who bullies others has a negative attitude toward other, trouble with academic and social cognition, and come from low-income family. Victims, on the other hand, were more likely to have the characteristics of bad self- recognition, lack of social skills, isolated and rejected by peers. Victim of bullying viewed as being passive or submissive victims, often quite, careful, sensitive and may easily cry, are unsure of themselves,

Bullying in School

51

http://dx.doi.org/10.5772/intechopen.75729

Previous research found that gender of students is one reason that students get bullied, and females students more likely of being harassed by their peer [26, 29, 39, 40]. Ethnic and racial background were also identified as a reason for being bullied, minorities and Asian Americans students more likely to be a target of bullying behaviors than others. Students

The research took into consideration the mental health issue as the characteristics of those involved in school bullying. One of these critical issues is self-esteem. Although the idea that low self-esteem is a prominent characteristic of the victims, some results contradicted this idea. Some who says it is both for the victim and bully and other supported that bullies often showed lower self-worth, but higher social acceptance [12, 40]. Some research assessed selfesteem as a protective factor from involvement in bullying behaviors. Family factors such as employment status of parents show significant association with being at risk for bullying, a result of cross-sectional study among Greek students of 16–18 years old show that students of low school performance and unemployed father were more likely to be a perpetrator. Students, who have unemployed mother have the chance of being either as a victim or perpetrator of bullying [40]. An analysis of data on 5th and 9th-grade students in 1000 schools in Colombia, investigates variables associated with bullying such as socioeconomic status, family characteristics, and community characteristics. Results revealed that male students, low level of family empathy, family violence, community violence, hostile aggressiveness, and supporting beliefs

Although bullying in schools has gained the attention of research over the last three decades, there are some inconsistencies of how students perceive bullying [37]. The research found that students perceive the bully as one who has characteristics such as feel superior, seeking the attention of another, giving the way the victim behaved or look (e.g., wearing glasses, types of hairstyles or clothes), or act in a manner that makes others irritable. Moreover, students perceived the way the victim look, talk or dress as an essential leading factor for bullying. Being odd, such as wearing different clothes, eating different food, listening to different music, and having strange behaviors perceived as an exclusive characteristic associated with the victim

A recent study found that students described a bully as "one who is a coward underneath, lacks respect for other people, wants to show power, wants to impress others, and wants to feel superior". Students perceived victims of bullying as "having low self-esteem, talking or sounding different than others, shy, and having no friends" [37]. Both victims and bullies,

from lower socio-economic status are at a higher risk of being bullied [17, 41].

having low self-confidence and a negative self-image [33].

in the community are associated with bullying [19].

**4. Students' perception of bullying**

of bullying [19, 26, 33, 38, 42].

Results of longitudinal studies highlight strong correlations among peer victimization and lower grade-point averages and lower teacher-rated academic engagement across middleschool years [29]. Students who experienced bullying has 2–3 times negative school perception than those who did not involve in bullying activities [30]. Data from three African countries including participants aged 12–16 year, who participated in Trend study in mathematics and sciences. The results show that bullying is a significant problem in the three countries and is among the most common factors associated with low academic performance [31]. From Arab country, students who report being bullied have symptoms of depression and anxiety and poor academic performance [32].

Research also assessed the mediating and moderating factors that influence peer victimization, and suggested that bullying contributes to poor academic achievement through mediating influences of internalizing behaviors, such as depression, anxiety, and low self-esteem [33]. Victims of bullying are often blaming themselves for being bullied which in turn develop a negative self-perception that affects their concentrating on school work, and become potential to have lower grades and perform poorly on standardized tests [13]. The most commonly investigated factors as having a moderating role in the relationship between bullying victimization and academic performance were friendship quality, peer social support, school maladjustment, and school disconnect [21, 22].

Findings from a nationally representative sample of 7th and 8th-grade students suggest that classmate support was negatively related to victimization and positively related to academic performance for both boys and girls [34]. On the other hands, having friends with high prosocial behavior, and low social anxiety reduces the risk of victimization and enhances academic performance [35]. Bullying and a violent school culture are also interfering with students' academic performance and negatively affect their social and emotional well-being [35, 36].


**4. Conclusions**

The fate of pesticides during various household and industrial processes depends on the type of processing methods, the physicochemical properties of the pesticides, and the nature of the product. Pesticides with high octanol–water coefficients were absorbed by the wax of the orange peel, therefore they remained on the peel and could not easily be removed by the washing step. Moreover, pesticides with lower water solubility did not diffuse easily through the fruit juices from the pulp section of the fruit. The processing factor was greater than 1 for the separation of the orange peel and less than 1 for the washing step and jam and fruit juice production. The compliance of residues in processed commodities with MRLs set for RAC should be assessed by taking into account the processing factor values. Furthermore, processing factors are essential to estimate residues in processed commodities for conducting dietary risk assessment. Therefore, it is essential to reveal the processing factors with scientific studies in order to make legal arrangements. It should be indicated that the dipping of the crops into commercial formulations of the pesticides under laboratory conditions does not reflect real processing effects. Therefore, to obtain more accurate processing factors, in future studies field treatment of the crops should be applied. In field-treated crops, pesticides may penetrate into different sections of the plants based on the pre-harvest interval. Absorption and translocation of the pesticide through the crops may affect the fate of the pesticide throughout the processing. Consequently, this study can be taken as a case study and more extensive studies should be carried out for different type of foods and different type of processing methods with field-treated crops.

**Author Contributions:** Conceptualization, B.A.C. and E.K.A.; methodology, B.A.C. and E.K.A.; software, B.A.C. and E.K.A.; validation, B.A.C. and E.K.A.; formal analysis, B.A.C. and E.K.A.; investigation, B.A.C. and E.K.A.; resources, P.Y.O.; data curation, P.Y.O.; writing—original draft preparation, B.A.C. and E.K.A.; writing—review and editing, P.Y.O.; visualization, P.Y.O.; supervision, P.Y.O.; project administration, P.Y.O.; funding acquisition, P.Y.O. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the Scientific Research Project Office of Bursa Uludag University, Bursa, Turkey (Project No: HDP(Z)-2018/3).

**Data Availability Statement:** Data is contained within the article.

**Acknowledgments:** The authors thank LOTUS Food Analysis Laboratory for their technical assistance.

**Conflicts of Interest:** The authors declare no conflict of interest.