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School-based prevention frameworks for mental health in disadvantaged communities are essential for determining evidence-based action to address poorer adolescent mental health. Frameworks addressing vulnerability and sustained mental health problems in adolescence can be examined at several levels, ranging from structural and place-based factors to the role of community factors and proximal factors such as family and the school setting. The overall aim of this thesis was to develop recommendations for prevention-based practice by combining local and national predictors of depressive symptoms and then apply these recommendations to an evaluation of a school-based intervention. Study 1 examined a national dataset (N = 19,987) from the Communities That Care (CTC) study of 18 years of nationally representative data on adolescents to examine risk and protective factors predicting adolescent depressive symptoms. Results revealed several critical risk (self-blame, family conflict and friend’s use of drugs) and protective factors (good coping, school rewards for prosocial involvement and belief in the moral order) for depressive symptoms in adolescents at a national level. However, when examined in the context of several disadvantaged communities, profiles of risk and protection for depressive symptoms were found to differ. A measurement tool was developed for identifying young people with high levels of depressive symptoms based on these predictors. A prevention-focused framework was then applied to evaluating a school-based intervention, the Resilience, Emotional Awareness, Careers, and Health (REACH) program across three studies spanning 2015 to 2018. REACH aimed at targeting school- and individual-level risk and protective factors associated with mental health. Included empirical studies collected data from four yearly cohorts of students (N = 1098) who participated in the REACH program. Evaluation outcomes suggest that the REACH program was not effective for targeting

School-based prevention frameworks for mental health in disadvantaged communities are essential for determining evidence-based action to address poorer adolescent mental health. Frameworks addressing vulnerability and sustained mental health problems in adolescence can be examined at several levels, ranging from structural and place-based factors to the role of community factors and proximal factors such as family and the school setting. The overall aim of this thesis was to develop recommendations for prevention-based practice by combining local and national predictors of depressive symptoms and then apply these recommendations to an evaluation of a school-based intervention. Study 1 examined a national dataset (N = 19,987) from the Communities That Care (CTC) study of 18 years of nationally representative data on adolescents to examine risk and protective factors predicting adolescent depressive symptoms. Results revealed several critical risk (self-blame, family conflict and friend’s use of drugs) and protective factors (good coping, school rewards for prosocial involvement and belief in the moral order) for depressive symptoms in adolescents at a national level. However, when examined in the context of several disadvantaged communities, profiles of risk and protection for depressive symptoms were found to differ. A measurement tool was developed for identifying young people with high levels of depressive symptoms based on these predictors. A prevention-focused framework was then applied to evaluating a school-based intervention, the Resilience, Emotional Awareness, Careers, and Health (REACH) program across three studies spanning 2015 to 2018. REACH aimed at targeting school- and individual-level risk and protective factors associated with mental health. Included empirical studies collected data from four yearly cohorts of students (N = 1098) who participated in the REACH program. Evaluation outcomes suggest that the REACH program was not effective for targeting

In this paper we introduce a new method called the Dirac Assisted Tree (DAT) method, which can handle 1D heterogeneous Helmholtz equations with arbitrarily large variable wave numbers. DAT breaks an original global problem into many parallel tree-structured small local problems, which are linked together to form a global solution by solving small linking problems. To solve the local problems in DAT, we propose a compact finite difference method (FDM) with arbitrarily high accuracy order and low numerical dispersion for piecewise smooth coefficients and variable wave numbers. This compact FDM is particularly appealing for DAT, because the local problems and their fluxes in DAT can be computed with high accuracy. DAT with such compact FDMs can solve heterogeneous Helmholtz equations with arbitrarily large variable wave numbers accurately by solving small linear systems - $4 \times 4$ matrices in the extreme case - with tridiagonal coefficient matrices in a parallel fashion. Several numerical examples are provided to illustrate the effectiveness of DAT using the $M$th order compact FDMs with $M=6,8$ for numerically solving heterogeneous Helmholtz equations with variable wave numbers. We shall also discuss how to solve some special 2D Helmholtz equations using DAT.

Wavelet theory has been well studied in recent decades. Due to their appealing features such as sparse multiscale representation and fast algorithms, wavelets have enjoyed many tremendous successes in the areas of signal/image processing and computational mathematics. This paper primarily intends to shed some light on the advantages of using wavelets in the context of numerical differential equations. We shall identify a few prominent problems in this field and recapitulate some important results along these directions. Wavelet-based methods for numerical differential equations offer the advantages of sparse matrices with uniformly bounded small condition numbers. We shall demonstrate wavelets' ability in solving some one-dimensional differential equations: the biharmonic equation and the Helmholtz equation with high wave numbers (of magnitude $O(10^4)$ or larger).

An estimated 30% of individuals with autism spectrum disorders (ASD) remain minimally verbal into late childhood, but research on cognition and brain function in ASD focuses almost exclusively on those with good or only moderately impaired language. Here we present a case study investigating auditory processing of GM, a nonverbal child with ASD and cerebral palsy. At the age of 8 years, GM was tested using magnetoencephalography (MEG) whilst passively listening to speech sounds and complex tones. Where typically developing children and verbal autistic children all demonstrated similar brain responses to speech and nonspeech sounds, GM produced much stronger responses to nonspeech than speech, particularly in the 65–165 ms (M50/M100) time window post-stimulus onset. GM was retested aged 10 years using electroencephalography (EEG) whilst passively listening to pure tone stimuli. Consistent with her MEG response to complex tones, GM showed an unusually early and strong response to pure tones in her EEG responses. The consistency of the MEG and EEG data in this single case study demonstrate both the potential and the feasibility of these methods in the study of minimally verbal children with ASD. Further research is required to determine whether GM's atypical auditory responses are characteristic of other minimally verbal children with ASD or of other individuals with cerebral palsy.

An estimated 30% of individuals with autism spectrum disorders (ASD) remain minimally verbal into late childhood, but research on cognition and brain function in ASD focuses almost exclusively on those with good or only moderately impaired language. Here we present a case study investigating auditory processing of GM, a nonverbal child with ASD and cerebral palsy. At the age of 8 years, GM was tested using magnetoencephalography (MEG) whilst passively listening to speech sounds and complex tones. Where typically developing children and verbal autistic children all demonstrated similar brain responses to speech and nonspeech sounds, GM produced much stronger responses to nonspeech than speech, particularly in the 65–165 ms (M50/M100) time window post-stimulus onset. GM was retested aged 10 years using electroencephalography (EEG) whilst passively listening to pure tone stimuli. Consistent with her MEG response to complex tones, GM showed an unusually early and strong response to pure tones in her EEG responses. The consistency of the MEG and EEG data in this single case study demonstrate both the potential and the feasibility of these methods in the study of minimally verbal children with ASD. Further research is required to determine whether GM's atypical auditory responses are characteristic of other minimally verbal children with ASD or of other individuals with cerebral palsy.

An estimated 30% of individuals with autism spectrum disorders (ASD) remain minimally verbal into late childhood, but research on cognition and brain function in ASD focuses almost exclusively on those with good or only moderately impaired language. Here we present a case study investigating auditory processing of GM, a nonverbal child with ASD and cerebral palsy. At the age of 8 years, GM was tested using magnetoencephalography (MEG) whilst passively listening to speech sounds and complex tones. Where typically developing children and verbal autistic children all demonstrated similar brain responses to speech and nonspeech sounds, GM produced much stronger responses to nonspeech than speech, particularly in the 65–165 ms (M50/M100) time window post-stimulus onset. GM was retested aged 10 years using electroencephalography (EEG) whilst passively listening to pure tone stimuli. Consistent with her MEG response to complex tones, GM showed an unusually early and strong response to pure tones in her EEG responses. The consistency of the MEG and EEG data in this single case study demonstrate both the potential and the feasibility of these methods in the study of minimally verbal children with ASD. Further research is required to determine whether GM's atypical auditory responses are characteristic of other minimally verbal children with ASD or of other individuals with cerebral palsy.

An estimated 30% of individuals with autism spectrum disorders (ASD) remain minimally verbal into late childhood, but research on cognition and brain function in ASD focuses almost exclusively on those with good or only moderately impaired language. Here we present a case study investigating auditory processing of GM, a nonverbal child with ASD and cerebral palsy. At the age of 8 years, GM was tested using magnetoencephalography (MEG) whilst passively listening to speech sounds and complex tones. Where typically developing children and verbal autistic children all demonstrated similar brain responses to speech and nonspeech sounds, GM produced much stronger responses to nonspeech than speech, particularly in the 65–165 ms (M50/M100) time window post-stimulus onset. GM was retested aged 10 years using electroencephalography (EEG) whilst passively listening to pure tone stimuli. Consistent with her MEG response to complex tones, GM showed an unusually early and strong response to pure tones in her EEG responses. The consistency of the MEG and EEG data in this single case study demonstrate both the potential and the feasibility of these methods in the study of minimally verbal children with ASD. Further research is required to determine whether GM's atypical auditory responses are characteristic of other minimally verbal children with ASD or of other individuals with cerebral palsy.

In the framework of the Jacobi-weighted Sobolev space, we design the a-posterior error estimators and error indicators associated with residuals and jumps of normal derivatives on internal edges with appropriate Jacobi weights for the p-version of the finite element method. With the help of quasi Jacobi projection operators, the upper bounds and the lower bounds of indicators and estimators are analyzed, which shows that such a-posteriori error estimation is quasi optimal. The indicators and estimators are computed for some model problems and programmed in C++. The numerical results show the reliability of our indicators and estimators.

In the framework of the Jacobi-weighted Sobolev space, we design the a-posterior error estimators and error indicators associated with residuals and jumps of normal derivatives on internal edges with appropriate Jacobi weights for the p-version of the finite element method. With the help of quasi Jacobi projection operators, the upper bounds and the lower bounds of indicators and estimators are analyzed, which shows that such a-posteriori error estimation is quasi optimal. The indicators and estimators are computed for some model problems and programmed in C++. The numerical results show the reliability of our indicators and estimators.