在這篇論文中，我們希望用不同角度來重新探討一個古典的數學問題；點、線、面切割最多區域問題，雖然這個問題已經經由許多方法得到公式，例如：遞迴關係、差分方程式、歐拉公式、標準n維空間切割系統等等，並延伸出其他方面的問題，可以運用在很多地方，所以我們希望可以再找到更簡單易懂的論證方式，可以讓國中學生也能理解。 思考學生現有的數學觀念，我們發現利用不等式的數學觀念，藉由定義出一套有規則的系統以及數學歸納法，可以以更直接，簡單的理論驗證出此數學公式，最後我們更希望能將這理論推廣至n維度空間。 / In this research, we will discuss a classical mathematical question from different aspects. The question of maximizing the number of regions made up by points, lines and planes has been proved and developed many formulas, using Recurrence Relations, Difference Equations, and Euler's Formula etc., which can extend to other questions and apply to many areas. Therefore, we hope to find an easier way to prove it which may help middle school students to understand better. We find that we can use the concept of inequality from what the students learn so far. By defining a logical system and using Induction, we can prove this mathematical formula in an easier and more direct way. Finally we hope it can be generalized to n-dimensional space.

This report aimed to develop algorithms for solving the optimization problem of matchingbuy and sell orders in call auctions while minimizing the number of transactions. The developed algorithms were evaluated based on their execution time and solution accuracy.The study found that the problem was more difficult to solve than initially anticipated, and commercial solvers were inadequate for the task. The data’s characteristics werecritical to the algorithms’ performance, and the lack of specifications for instruments andexchange posed a challenge. The algorithms were tested on a broad range of datasets with different characteristics, as well as real trades of stocks from the Stockholm Stock Exchange. Evaluating the best-performing algorithm became a trade-off between time and accuracy, where the quickest algorithm did not have the highest solution accuracy. Therefore, the importance of these factors should be considered before deciding which algorithm to implement. Eight algorithms were evaluated: four greedy algorithms and four clusteralgorithms capable of identifying 2-1 and 3-1 matches. If execution time is the single most crucial factor, the Unsorted Greedy Algorithm should be considered. However, if accuracyi s a priority, the Cluster 3-1 & 1-3 Algorithm should be considered, even though it takes longer to find a solution. Ultimately, the report concluded that while no single algorithm can be definitively la-beled as the best, the Cluster 2-1 Algorithm strikes the most effective balance between execution time and solution accuracy, while also remaining relatively stable in perfor-mance for all test cases. The recommendation was based on the fact that the Cluster 2-1 Algorithm proved to be the quickest of the developed cluster algorithms, and that cluster algorithms were able to find the best solutions for all tested data sets. This study successfully addressed its purpose by developing eight algorithms that solved the given problem and suggested an appropriate algorithm that strikes a balance between execution time and solution quality.

Public health screening plays an important role in the overall healthcare system. As an example, consider newborn screening, a state-level initiative that screens newborns for life-threatening genetic disorders for which early treatment can substantially improve health outcomes. Another topical example is in the realm of infectious disease screening, e.g., screening for COVID-19. The common features of both public health screening problems include large testing populations and resource limitations that inhibit screening efforts. Cost is a major barrier to the inclusion of genetic disorders in newborn screening, and thus screening must be both highly accurate and efficient; and for COVID-19, limited testing kits, and other shortages, have been major barriers to screening efforts. Further, for both newborn screening and infectious disease screening, equity (reducing health disparities among different sub-populations) is an important consideration. We study the testing process design for newborn screening for genetic diseases, considering cystic fibrosis as a model disorder. Our optimization-based models take into account disease-related parameters, subject risk factors, test characteristics, parameter uncertainty, and limited testing resources so as to design equitable, accurate, and robust screening processes that classify newborns as positive or negative for cystic fibrosis. Our models explicitly consider the trade-off between false-negatives, which lead to missed diagnoses, and the required testing resources; and the trade-off between the accuracy and equity of screening. We also study the testing process design for infectious disease screening, considering COVID-19 as a model disease. Our optimization-based models account for key subject risk factors that are important to consider, including the likelihood of being disease-positive, and the potential harm that could be averted through testing and the subsequent interventions. Our objectives include the minimization of harm (through detection and mitigation) or maximization of testing coverage. These are complex problems. We develop novel mathematical models and characterize key structural properties of optimal solutions. This, in turn, allows the development of effective and efficient algorithms that exploit these structural properties. These algorithms are either polynomial- or pseudo-polynomial-time algorithms, and are able to solve realistic-sized problems efficiently. Our case studies on cystic fibrosis screening and COVID-19 screening, based on realistic data, underscore the value of the proposed optimization-based approaches for public health screening, compared to current practices. Our findings have important implications for public policy. / Doctor of Philosophy / Public health screening plays an important role in the overall healthcare system. As an example, consider newborn screening, a state-level initiative that screens newborns for life-threatening genetic disorders for which early treatment can substantially improve health outcomes. Another topical example is in the realm of infectious disease screening, e.g., screening for COVID-19. The common features of both public health screening problems include large testing populations and resource limitations that inhibit screening efforts. Cost is a major barrier to the inclusion of genetic disorders in newborn screening, and thus screening must be both highly accurate and efficient; and for COVID-19, limited testing kits, and other shortages, have been major barriers to screening efforts. Further, for both newborn screening and infectious disease screening, equity (reducing health disparities among different sub-populations) is an important consideration. We study the testing process design for newborn screening for genetic diseases, considering cystic fibrosis as a model disorder. Our optimization-based models take into account disease-related parameters, subject risk factors, test characteristics, parameter uncertainty, and limited testing resources so as to design screening processes that classify newborns as positive or negative for cystic fibrosis. Our models explicitly consider the trade-off between false-negatives, which lead to missed diagnoses, and the required testing resources; and the trade-off between the accuracy and equity of screening. We also study the testing process design for infectious disease screening, considering COVID-19 as a model disease. Our optimization-based models account for key subject risk factors that are important to consider, including the likelihood of being disease-positive, and the potential harm that could be averted through testing and the subsequent interventions. Our objectives include the minimization of harm (through detection and mitigation) or maximization of testing coverage. These are complex problems. We develop novel mathematical models and characterize key structural properties of optimal solutions. This, in turn, allows the development of effective and efficient algorithms that exploit these structural properties. Our case studies on cystic fibrosis screening and COVID-19 screening, based on realistic data, underscore the value of the proposed optimization-based approaches for public health screening, compared to current practices. Our findings have important implications for public policy.