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The treatment of missing data on placement tools for predicting success in college algebra at the University of AlaskaThis project investigated the statistical significance of baccalaureate student placement tools such as tests scores and completion of a developmental course on predicting success in a college level algebra course at the University of Alaska (UA). Students included in the study had attempted Math 107 at UA for the first time between fiscal years 2007 and 2012. The student placement information had a high percentage of missing data. A simulation study was conducted to choose the best missing data method between complete case deletion, and multiple imputation for the student data. After the missing data methods were applied, a logistic regression with fitted with explanatory variables consisting of tests scores, developmental course grade, age (category) of scores and grade, and interactions. The relevant tests were SAT math, ACT math, AccuPlacer college level math, and the relevant developmental course was Devm /Math 105. The response variable was success in passing Math 107 with grade of C or above on the first attempt. The simulation study showed that under a high percentage of missing data and correlation, multiple imputation implemented by the R package Multivariate Imputation by Chained Equations (MICE) produced the least biased estimators and better confidence interval coverage compared to complete cases deletion when data are missing at random (MAR) and missing not at random (MNAR). Results from multiple imputation method on the student data showed that Devm /Math 105 grade was a significant predictor of passing Math 107. The age of Devm /Math 105, age of tests, and test scores were not significant predictors of student success in Math 107. Future studies may consider modeling with ALEKS scores, and high school math course information.

Tsunami runup in U and V shaped baysTsunami runup can be effectively modeled using the shallow water wave equations. In 1958 Carrier and Greenspan in their work "Water waves of finite amplitude on a sloping beach" used this system to model tsunami runup on a uniformly sloping plane beach. They linearized this problem using a hodograph type transformation and obtained the KleinGordon equation which could be explicitly solved by using the FourierBessel transform. In 2011, Efim Pelinovsky and Ira Didenkulova in their work "Runup of Tsunami Waves in UShaped Bays" used a similar hodograph type transformation and linearized the tsunami problem for a sloping bay with parabolic crosssection. They solved the linear system by using the D'Alembert formula. This method was generalized to sloping bays with crosssections parameterized by power functions. However, an explicit solution was obtained only for the case of a bay with a quadratic crosssection. In this paper we will show that the KleinGordon equation can be solved by a spectral method for any inclined bathymetry with power function for any positive power. The result can be used to estimate tsunami runup in such bays with minimal numerical computations. This fact is very important because in many cases our numerical model can be substituted for fullscale numerical models which are computationally expensive, and time consuming, and not feasible to investigate tsunami behavior in the Alaskan coastal zone, due to the low population density in this area

Vertex arboricity of trianglefree graphsThe vertex arboricity of a graph is the minimum number of colors needed to color the vertices so that the subgraph induced by each color class is a forest. In other words, the vertex arboricity of a graph is the fewest number of colors required in order to color a graph such that every cycle has at least two colors. Although not standard, we will refer to vertex arboricity simply as arboricity. In this paper, we discuss properties of chromatic number and kdefective chromatic number and how those properties relate to the arboricity of trianglefree graphs. In particular, we find bounds on the minimum order of a graph having arboricity three. Equivalently, we consider the largest possible vertex arboricity of trianglefree graphs of fixed order.