Variable Shape Parameter Strategies in Radial Basis Function Methods

The Radial Basis Function (RBF) method is an important tool in the interpolation of multidimensional scattered data. The method has several important properties. One is the ability to handle sparse and scattered data points. Another property is its ability to interpolate in more than one dimension. Furthermore, the Radial Basis Function method provides phenomenal accuracy which has made it very popular in many fields. Some examples of applications using the RBF method are numerical solutions to partial differential equations, image processing, and cartography. This thesis involves researching Radial Basis Functions using different shape parameter strategies. First, we introduce the Radial Basis Function method by stating its history and development in Chapter 1. Second, we explain how Radial Basis Functions work in Chapter 2. Chapter 3 compares RBF interpolation to polynomial interpolation. Chapters 4 and 5 introduce the idea of variable shape parameters. In these chapters we compare and analyze the variable shape parameters in one and two dimensions. In Chapter 6, we introduce the challenges in interpolations due to errors in boundary regions. Here, we try to reduce the error using different shape parameter strategies. Chapter 7 lists the conclusions resulting from the research

Mathematical Modeling: Issues and Challenges in Mathematics Education and Teaching

Mathematics education researchers and policy documents in the United States have expressed the need to improve the teaching and learning of mathematical modeling at the K&amp;ndash;12 levels so that students can apply their knowledge of mathematics to solve real-world situations. Unfortunately, most practicing teachers (PTs) and preservice teachers (PSTs) acquire didactical and pedagogical styles that do not support effective modeling practices. To investigate these dilemmas, this study examined PTs&amp;rsquo; pedagogical experiences in and PSTs&amp;rsquo; perspectives on mathematical modeling practices. Participants included 62 PTs and 18 PSTs from a Midwestern region of the United States. Data originated from questionnaire items and open-ended questions, which were analyzed quantitatively and qualitatively. Varied participants&amp;rsquo; ideas on mathematical modeling practices were identified, recorded, and summarized. Results indicated that most of these PTs and PSTs have little to no experiences with mathematical modeling practices and associated pedagogies. Such results along with a supplemental discussion have implications for teacher education programs and professional development centered on mathematical modeling education.</jats:p

Assessment of Symptom, Disability, and Financial Trajectories in Patients Hospitalized for COVID-19 at 6 Months

ImportanceIndividuals who survived COVID-19 often report persistent symptoms, disabilities, and financial consequences. However, national longitudinal estimates of symptom burden remain limited.ObjectiveTo measure the incidence and changes over time in symptoms, disability, and financial status after COVID-19–related hospitalization.Design, Setting, and ParticipantsA national US multicenter prospective cohort study with 1-, 3-, and 6-month postdischarge visits was conducted at 44 sites participating in the National Heart, Lung, and Blood Institute Prevention and Early Treatment of Acute Lung Injury Network's Biology and Longitudinal Epidemiology: COVID-19 Observational (BLUE CORAL) study. Participants included hospitalized English- or Spanish-speaking adults without severe prehospitalization disabilities or cognitive impairment. Participants were enrolled between August 24, 2020, and July 20, 2021, with follow-up occurring through March 30, 2022.ExposureHospitalization for COVID-19 as identified with a positive SARS-CoV-2 molecular test.Main Outcomes and MeasuresNew or worsened cardiopulmonary symptoms, financial problems, functional impairments, perceived return to baseline health, and quality of life. Logistic regression was used to identify factors associated with new cardiopulmonary symptoms or financial problems at 6 months.ResultsA total of 825 adults (444 [54.0%] were male, and 379 [46.0%] were female) met eligibility criteria and completed at least 1 follow-up survey. Median age was 56 (IQR, 43-66) years; 253 (30.7%) participants were Hispanic, 145 (17.6%) were non-Hispanic Black, and 360 (43.6%) were non-Hispanic White. Symptoms, disabilities, and financial problems remained highly prevalent among hospitalization survivors at month 6. Rates increased between months 1 and 6 for cardiopulmonary symptoms (from 67.3% to 75.4%; P = .001) and fatigue (from 40.7% to 50.8%; P &amp;amp;lt; .001). Decreases were noted over the same interval for prevalent financial problems (from 66.1% to 56.4%; P &amp;amp;lt; .001) and functional limitations (from 55.3% to 47.3%; P = .004). Participants not reporting problems at month 1 often reported new symptoms (60.0%), financial problems (23.7%), disabilities (23.8%), or fatigue (41.4%) at month 6.Conclusions and RelevanceThe findings of this cohort study of people discharged after COVID-19 hospitalization suggest that recovery in symptoms, functional status, and fatigue was limited at 6 months, and some participants reported new problems 6 months after hospital discharge.</jats:sec