Finding Eigenvalues of Unitary Matrices

The study introduces methods of finding eigenvalues for unitary matrices and pencils. Bunse-Gerstner and Elsner ([2]) proposed an algorithm of using the Schur parameter pencil to solve eigenproblems for unitary matrices and pencils. This thesis reviews the Schur parameter pencil algorithm. The method is divided into two phases: Reducing a unitary pencil to a Schur parameter form and QR-type shifted iteration. The algorithm is proved to be backward stable and more efficient than the standard QR/QZ algorithm. However, during the process of reduction, norms of vectors are frequently compared for numerical stability, which causes a lot of extra work for computations. Based on the idea in [8], we introduce a modified Schur parameter algorithm to avoid such frequent comparison. The modified algorithm is still divided into two phases similar to the one in [2]. A detailed reduction process and shifted iteration are described in this thesis

Promoting Students' Motivation and Use of SRL Strategies in Online Mathematics Learning

Computer and information technologies have brought revolutions to many aspects of our lives. Online learning is one of the new learning methods that have emerged with the development of new technologies. Even though the advantages of online learning are very attractive to students and educators, when they are applied in real educational settings, not all individuals that participate in online learning are successful; not all programs that are offered online attain their academic goals. This study was based on an experiment involving high school students from two high schools in China over about two weeks’ time. It evaluated the effectiveness of motivational and SRL design implemented in the online trigonometry function instructional system. Two hundred and thirty-six students participated in the study to test the effectiveness of the instructional design and 183 students completed the tasks through to the end. Participants were randomly divided into four groups: the motivational design group (MD); the SRL intervention group (SI); the SRL intervention and motivational design group (MDSI); and the control group (CT). Three sets of two-way ANOVA (Analysis of Variance) were used to test the effectiveness of the motivational design and SRL intervention with the independent variables of group membership and gender. Research results showed that the motivational design using the ARCS (Attention, Relevance, Confidence and Satisfaction) model for students to learn trigonometry online is effective in enhancing students’ learning motivation online. The IMPROVE method was effective as an SRL intervention in the instructional design to promote students’ use of SRL strategies during their online learning activity. Both motivational design and SRL intervention were positively correlated to students’ academic achievement. But for students learning math online, motivation played a more important role in improving students’ academic achievement than SRL intervention. However, the group with both motivational design and SRL intervention showed the highest academic improvement compared with other two treatment groups

PROPHETIC: Prospective Identification of Pneumonia in Hospitalized Patients in the Intensive Care Unit

BACKGROUND: Pneumonia is the leading infection-related cause of death. Using simple clinical criteria and contemporary epidemiology to identify patients at high risk of nosocomial pneumonia should enhance prevention efforts and facilitate development of new treatments in clinical trials. RESEARCH QUESTION: What are the clinical criteria and contemporary epidemiology trends helpful in identifying patients at high risk of nosocomial pneumonia? STUDY DESIGN AND METHODS: Within the intensive care units of 28 United States hospitals, we conducted a prospective cohort study among adults hospitalized more than 48 hours and considered high risk for pneumonia (defined as treatment with invasive or noninvasive ventilatory support or high levels of supplemental oxygen). We estimated the proportion of high-risk patients developing nosocomial pneumonia. Using multivariable logistic regression, we identified patient characteristics and treatment exposures associated with increased risk of pneumonia development during the intensive care unit admission. RESULTS: Between February 6, 2016 and October 7, 2016, 4613 high-risk patients were enrolled. Among 1464/4613 (32%) high-risk patients treated for possible nosocomial pneumonia, 537/1464 (37%) met the study pneumonia definition. Among high-risk patients, a multivariable logistic model was developed to identify key patient characteristics and treatment exposures associated with increased risk of nosocomial pneumonia development (c-statistic 0.709, 95% confidence interval 0.686 to 0.731). Key factors associated with increased odds of nosocomial pneumonia included an admission diagnosis of trauma or cerebrovascular accident, receipt of enteral nutrition, documented aspiration risk, and receipt of systemic antibacterials within the preceding 90 days. INTERPRETATION: Treatment for nosocomial pneumonia is common among intensive care unit patients receiving high levels of respiratory support, yet more than half of patients treated do not fulfill standard diagnostic criteria for pneumonia. Application of simple clinical criteria may improve the feasibility of clinical trials of pneumonia prevention and treatment by facilitating prospective identification of patients at highest risk