Identities for the Multiple Polylogarithm Using the Shuffle Operation

At the beginning of my research, I understood the shuffle operation and iterated integrals to make a new proof-method (called a combinatorial method). As a first work, I proved an combinatorial identity 2 using a combinatorial method. While proving it, I got four identities and showed that one of them is equal to an analytic identity 1 which is found at the paper [2] written by David M. Bradley and Doug Bowman. Furthermore, I derived an formula involving nested harmonic sums. Using Maple (a mathematical software), I found a new combinatorial identity 3 and derived two formulas: One is related to multiple polylogarithms and the other is related to rational functions. Since letters in the identities represent differential 1-forms which converge, I can find new formulas - - - if I get a proper setting. - My research was developed by considering a combinatorial identity 4 given by David M. Bradley, thesis advisor. Though it looked very complicated, the implication for the identity was very interesting to me. Using a combinatorial proof-method, I proved it. Even though I just derived one formula involving nested harmonic sums in this thesis, the identity has potentiality because, if I 6nd a new setting for differential 1-forms, I can derive a new formula involving multiple polylogarithms. It was not very easy to prove the combinatorial identity 4 even though I used the combinatorial proof-method as I did at the proofs of the combinatorial identity 2 and 3. The reason is that the result of the identity 4 is more complicated than those of the identities 3 and 4. So, Lemma 5 is needed to complete the proof of the identity 4, which step is not needed in the proofs of the identities 3 and 4. When formulating the identity 4, I had a trouble in defining the notations because of their complexity. When I formulated the identity 4, it was a beautiful formula. As we can see in the paper [3], there are various conjectures related to multiple zeta values whose incompletion is a sign that both the mathematics and physics communities do not yet completely understand the field. At this situation, this combinatorial proof-method can play a crucial role in developing other fields such as knot theory and quantum field theory as well as combinatorics

Longitudinal Model Building Using Latent Transition Analysis: An Example Using School Bullying Data

Applications of latent transition analysis (LTA) have emerged since the early 1990s, with numerous scientific findings being published in many areas, including social and behavioral sciences, education, and public health. Although LTA is effective as a statistical analytic tool for a person-centered model using longitudinal data, model building in LTA has often been subjective and confusing for applied researchers. To fill this gap in the literature, we review the components of LTA, recommend a framework of fitting LTA, and summarize what acceptable model evaluation tools should be used in practice. The proposed framework of fitting LTA consists of six steps depicted in Figure 1 from step 0 (exploring data) to step 5 (fitting distal variables). We also illustrate the framework of fitting LTA with data on concerns about school bullying from a sample of 1,180 students ranging from 5th to 9th grade (mean age = 12.2 years, SD = 1.29 years at Time 1) over three semesters. We identified four groups of students with distinct patterns of bullying concerns, and found that their concerns about bullying decreased and narrowed to specific concerns about rumors, gossip, and social exclusion over time. The data and command (syntax) files needed for reproducing the results using SAS PROC LCA and PROC LTA (Version 1.3.2) (2015) and Mplus 7.4 (Muthén and Muthén, 1998–2015) are provided as online supplementary materials

The Efficacy of Conjoint Behavioral Consultation on Parents and Children in the Home Setting: Results of a Randomized Controlled Trial

The present study is a large-scale randomized trial testing the effects of a family-school partnership model (i.e., Conjoint Behavioral Consultation, CBC) for promoting behavioral competence and decreasing problem behaviors of children identified by their teachers as disruptive. CBC is a structured approach to problem solving that involves consultants, parents, and teachers. The effects of CBC on family variables that are commonly associated with important outcomes among school-aged children (i.e., family involvement and parent competence in problem solving), as well as child outcomes at home, were evaluated. Participants were 207 children with disruptive behaviors from 91 classrooms in 21 schools in kindergarten through grade 3 and their parents and teachers. Results indicated that there were significantly different increases in home-school communication and parent competence in problem solving for participants in the CBC relative to control group. Likewise, compared to children in the control group, children in the CBC group showed significantly greater decreases in arguing, defiance, noncompliance, and tantrums. The degree of family risk moderated parents’ competence in problem solving and children’s total problem behaviors, teasing, and tantrums

Congruence within the Parent-Teacher Relationship: Associations with Children’s Functioning

Meaningful interactions between families and schools benefit multiple facets of children’s functioning including their academic, social, and behavioral adjustment (Christenson & Sheridan, 2001). Positive relationships between parents and teachers predict children’s enhanced social-emotional functioning and academic adjustment across time (Izzo, Weissberg, Kasprow, & Fendrich, 1999). Studies of parent-teacher relationships often focus on the association of child outcomes with separate parent or teacher reports of their relationship quality. Little attention has focused on the congruence of perceptions within parent-teacher dyads. It may be the case that when parents and teachers view their relationship in a similar positive light, better connections or partnerships across the home and school environments result, thereby enhancing children’s functioning. Conversely, when parents and teachers hold discrepant views about their relationship, or both view it negatively, they may be less likely to communicate and share goals for children; this disconnect may impede children’s functioning. This study examined the degree to which congruity and incongruity in parent and teacher views of their relationship are related to children’s academic, social, and behavioral functioning. Congruity was examined using a categorical approach: o Positive congruence: parents and teachers share positive views about their relationship o Non-positive congruence: parents and teachers share non-positive views about their relationship o Incongruence: parents and teachers hold differing views about the quality of their relationship Research Question and Hypothesis Is congruence/incongruence between parents and teachers in their views of their relationship related to children’s academic, social, and behavioral functioning? It was hypothesized that congruent, positive views of the parent-teacher relationship would be associated with children’s enhanced academic, social, and behavioral functioning to a greater extent than non-positive congruent or incongruent views