DSpace community: 數學研究所

一維度可壓縮流歐拉-泊桑方程中的解一次微分的推導;Derivation of Riccati Equations for the Global Existence of Classical Solutions to Compressible Euler–Poisson Equations

title: 一維度可壓縮流歐拉-泊桑方程中的解一次微分的推導;Derivation of Riccati Equations for the Global Existence of Classical Solutions to Compressible Euler–Poisson Equations abstract: 在本篇論文中，我們推導出修正版可壓縮流歐拉-泊桑方程的全域古典解所滿足的 Riccati 方程。在歐拉座標下的歐拉-泊桑方程所推導出的 Riemann 不變量方程，構成一個 2×2 的雙曲平衡律系統。利用廣義 Lax 變換，並對這個 2×2 的 Riemann 不變量系統進行微分，我們可以得到一組與 Riemann 不變量的空間導數相關的 2×2 Riccati 方程。;In this thesis， we derive the Riccati equations for the global classical solutions to the modified version of compressible Euler-Poisson equations. The equations for the Riemann invariants which will be derived from the Euler-Poisson equations in Eulerian coordinates， form a two-by-two hyperbolic system of balance laws. Using the generalized Lax transform， and differentiating the two-by-two system of Riemann invariants， we are able to obtain the two-by-two Riccati equations related to the spatial derivatives of the Riemann invariants.

QS 世界大學排名改革之影響分析與師生比指標之數學建模及策略研究;Impact Analysis of the QS World University Rankings Reform and Mathematical Modeling of the Faculty-Student Ratio Indicator

title: QS 世界大學排名改革之影響分析與師生比指標之數學建模及策略研究;Impact Analysis of the QS World University Rankings Reform and Mathematical Modeling of the Faculty-Student Ratio Indicator abstract: 近年來，QS 世界大學排名已成為衡量高等教育機構國際競爭力的重要指標。2024 年 QS 正式由六指標制度擴展為九指標制度，新增國際研究網絡、就業成果及永續發展三項指標，並調整既有指標權重。然而，制度改革對大學排名之影響，以及師生比指標之評分機制仍缺乏系統性研究。此外，QS 師生比指標之實際評分函數尚未公開，使大學難以評估改善師生比所能帶來之實際效益。因此，本研究旨在分析 QS 排名制度改革對全球大學排名之影響，並建立師生比指標之數學模型，以提供大學資源配置與策略規劃之參考。本研究蒐集 2022 年至 2026 年 QS 世界大學排名、教師與學生人數及各項指標資料。第一部分重新建構六指標與九指標排名，比較制度改革前後之排名變化，並分析在制度改革下的受益學校，採用分群標準化、Welch’s t 檢定、邏輯思迴歸（Logistic Regression）等方法，分析影響受益之關鍵指標。第二部分則以 4，227 筆有效樣本，利用基於遺傳編程（Genetic Programming）之符號回歸（Symbolic Regression）建立師生比與 QS 師生比指標分數之函數模型，並透過邊際效益分析評估教師資源投入效益，並以台灣四所大學作為實際案例分析。研究結果顯示，約 58.43% 的學校於九指標制度下排名提升，其中永續發展、就業成果及國際研究網絡為受益之主要正向因素，而師生比則呈顯著負向關聯。此外，本研究成功建立具良好預測能力之師生比非線性函數模型，有效描述師生比與指標分數之關係，並發現教師資源投入存在最大邊際效益區間，其後分數增益將逐漸遞減，案例研究進一步表明，所提出的模型能夠有效量化教師資源投入的效益，提供大學政策規劃之參考。整體而言，本研究不僅揭示 QS 制度改革對全球大學排名之影響與主要驅動因素，亦建立可量化評估師資投入效益之數學模型，可作為高等教育機構進行排名策略、資源配置及校務發展決策之參考。;The QS World University Rankings have become an important indicator for evaluating the international competitiveness of higher education institutions. In 2024， QS expanded its evaluation framework from a six-indicator system to a nine-indicator system by in- troducing International Research Network， Employment Outcomes， and Sustainability. However， the impacts of this reform and the scoring mechanism of the Faculty–Student indicator remain insuﬀiciently understood. Therefore， this study investigates the effects of the QS ranking reform and develops a mathematical model for the Faculty–Student indicator to support university resource allocation and strategic planning. This study col- lected QS World University Rankings data， faculty and student data， and indicator scores from 2022 to 2026. The first part reconstructed the six-indicator and nine-indicator rank- ings to analyze ranking changes and identify universities that benefited from the reform using statistical analysis and logistic regression. The second part employed Symbolic Regression based on Genetic Programming to establish a mathematical model relating the Faculty–Student Ratio to the QS Faculty–Student indicator score using 4，227 valid samples. Marginal effect analysis and case studies of four Taiwanese universities were further conducted. The results show that approximately 58.43% of universities improved their rankings under the nine-indicator system. Sustainability， Employment Outcomes， and International Research Network were identified as the major positive factors associ- ated with beneficiary status， whereas the Faculty–Student indicator showed a significant negative association. The proposed symbolic regression model successfully captured the relationship between the Faculty–Student Ratio and indicator score， and marginal ef- fect analysis revealed a maximum-benefit region followed by diminishing returns. Case studies further demonstrated that the model can effectively quantify the benefits of fac- ulty resource investment， providing practical support for university policy and resource planning.

結合徑向邊界修正之 U-Net 分割方法應用於囊胚孵化程度量化研究;A U-Net Segmentation Method with Radial Boundary Correction for Quantifying Blastocyst Hatching Degree

title: 結合徑向邊界修正之 U-Net 分割方法應用於囊胚孵化程度量化研究;A U-Net Segmentation Method with Radial Boundary Correction for Quantifying Blastocyst Hatching Degree abstract: 囊胚孵化狀態為體外受精療程中評估胚胎發育程度的重要形態特徵之一。臨床上，early hatching 與 late hatching 多由胚胎師根據顯微影像進行人工判讀，雖具有臨床可解釋性，但容易受到影像品質、觀察角度、胚胎師經驗與分類標準差異影響。因此，如何將囊胚孵化狀態由主觀類別判讀轉換為客觀、可重現且可計算之量化指標，為本研究之主要問題。本研究提出一套基於影像分割之囊胚孵化程度量化流程。首先，使用 U-Net 雙通道分割模型對囊胚顯微影像進行分割，分別取得透明帶內緣遮罩與胚胎外緣遮罩。接著，針對透明帶邊界可能出現之局部尖刺、凹陷或不連續問題，本研究提出中心點徑向局部中位數邊界修正方法。此方法以透明帶遮罩之中心點為基準，將邊界表示為不同角度下之半徑序列，並利用鄰近角度之局部中位數修正異常半徑，以提升透明帶邊界穩定性，同時避免橢圓擬合造成過度平滑。完成邊界修正後，本研究將胚胎外緣遮罩中位於修正後透明帶外側之區域定義為孵化區域，並進一步計算 area-based hatching ratio 與 diameter-based hatching ratio，作為囊胚孵化程度之幾何量化指標。實驗結果顯示，中心點徑向局部中位數邊界修正方法在分割與邊界距離指標上優於 raw U-Net 與橢圓輕度修正方法。該方法可有效降低透明帶局部邊界誤差，並提升後續孵化比例計算之穩定性。在胚胎師一致性分析中，以 area-based hatching ratio 並採用 predefined cutoff = 1.0 進行 early/late hatching 分組時，模型分組與胚胎師人工判讀之整體一致率為 86.67%，顯示本研究所提出之量化指標能在多數案例中反映人工判讀邏輯。此外，本研究進一步分析 hatching group 與臨床結果之關聯。結果顯示，area-based 與 diameter-based hatching group 皆與 PGT-A euploid status 呈現顯著統計關聯，其中 area-based hatching group 之 odds ratio 為 2.08，95% confidence interval 為 1.48–2.94，p-value 為 0.00003。相較之下，hatching group 與 implantation outcome 之關聯未達統計顯著，顯示囊胚孵化形態可能與胚胎染色體狀態具有一定關聯，但其與著床結果之關係仍可能受到母體條件、子宮內膜狀態、植入流程與其他臨床因子影響。綜合而言，本研究建立了一套由 U-Net 分割、徑向邊界修正、孵化區域擷取至 hatching ratio 計算之完整影像量化流程。此方法可將傳統 early/late hatching 人工判讀轉換為可重現之幾何指標，並提供後續胚胎形態量化分析與臨床關聯研究之基礎。關鍵字：囊胚孵化、透明帶、U-Net、影像分割、徑向邊界修正、孵化比例、胚胎形態量化;Blastocyst hatching status is an important morphological feature for evaluating embryonic development in in vitro fertilization. In clinical practice， early hatching and late hatching are commonly assessed by embryologists through visual inspection of microscopic images. Although this approach is clinically interpretable， it may be affected by image quality， viewing angle， embryologist experience， and differences in classification criteria. Therefore， converting blastocyst hatching status from subjective categorical assessment into an objective， reproducible， and computable quantitative index is the main problem addressed in this study. This study proposes a segmentation-based framework for blastocyst hatching quantification. First， a dual-channel U-Net segmentation model is used to segment blastocyst microscopic images and generate two masks: the zona pellucida inner mask and the embryo outer mask. Since the predicted zona pellucida boundary may contain local spikes， concavities， or discontinuities， this study further proposes a centroid-based radial local median boundary refinement method. The proposed method represents the zona pellucida boundary as a sequence of radial distances from the centroid at different angles and corrects abnormal radii using the local median of neighboring angles. This design improves the stability of the zona pellucida boundary while avoiding the excessive smoothing that may occur with ellipse-based fitting. After boundary refinement， the hatched portion is defined as the region of the embryo outer mask located outside the corrected zona pellucida mask. The area-based hatching ratio and diameter-based hatching ratio are then calculated as geometric quantitative indicators of blastocyst hatching degree. Experimental results show that the proposed radial local median boundary refinement method outperforms raw U-Net and ellipse-based correction in both segmentation overlap metrics and boundary distance metrics. The method effectively reduces local zona pellucida boundary errors and improves the stability of subsequent hatching ratio calculation. In the embryologist agreement analysis， using the area-based hatching ratio with a predefined cutoff of 1.0 to classify embryos into early and late hatching groups achieved an overall agreement of 86.67% with embryologist assessment. This result indicates that the proposed quantitative indicator can reflect embryologist interpretation in most cases. This study further analyzed the association between hatching group and clinical outcomes. Both area-based and diameter-based hatching groups were significantly associated with PGT-A euploid status. For the area-based hatching group， the odds ratio was 2.08， with a 95% confidence interval of 1.48–2.94 and a p-value of 0.00003. In contrast， the association between hatching group and implantation outcome was not statistically significant. These findings suggest that blastocyst hatching morphology may be associated with chromosomal status， whereas its relationship with implantation outcome may be influenced by maternal conditions， endometrial status， embryo transfer procedures， and other clinical factors. In summary， this study establishes a complete image-based quantification pipeline consisting of U-Net segmentation， radial boundary refinement， hatched portion extraction， and hatching ratio calculation. The proposed framework converts conventional embryologist-based early/late hatching assessment into a reproducible geometric indicator and provides a quantitative basis for subsequent embryo morphology analysis and clinical association studies. Keywords: blastocyst hatching， zona pellucida， U-Net， image segmentation， radial boundary refinement， hatching ratio， embryo morphology quantification

Dynamics and H�lder Continuity of Green Functions with Applications to Arithmetic Distance;Dynamics and H�lder Continuity of Green Functions with Applications to Arithmetic Distance

title: Dynamics and H�lder Continuity of Green Functions with Applications to Arithmetic Distance;Dynamics and H�lder Continuity of Green Functions with Applications to Arithmetic Distance abstract: 本論文探討算術動力學中格林函數的解析性質與算術距離的極值行為。首先，我們研究與定義在具備絕對值 $v$ 之體上的有理映射相關的修正格林函數 $\hat{\mathcal G}_{\Phi， v}$。透過分析相關態射相對於其次數 $d$ 的李普希茲常數 $L_\phi$，我們建立了該函數關於弦度量 $\rho_v$ 的連續性。具體而言，當 $L_\phi > d$ 時，我們證明了 $\hat{\mathcal G}_{\Phi， v}$ 滿足赫爾德連續（H\"older continuous），並給出明確的指數 $\frac{\log d}{\log L_\phi}$ 以及對應的赫爾德常數。此外，我們證明了當 $L_\phi < d$ 時，該修正格林函數為李普希茲連續（Lipschitz continuous），並為此李普希茲常數提供了一個明確的上界。在第二部分，本論文探討有理映射間的算術距離 $\hat{\delta}(\phi， \psi)$，其定義為兩者所關聯的動力學規範高度 $\hat{h}_\phi$ 與 $\hat{h}_\psi$ 之差的上確界（supremum）。對於具備良好約化（good reduction）的有理映射，我們證明了可以使用修正格林函數將此距離重新表述為有利於計算的形式。具體而言，此問題可化簡為計算以下實射影直線上的上確界： \begin{align*} \hat{\Delta}(\Phi， \Psi) = \sup_{[x:y] \in \mathbb P^1(\mathbb R)} \left| \hat{\mathcal G}_{\Phi， \infty}(x， y) - \hat{\mathcal G}_{\Psi， \infty}(x， y) \right|. \end{align*} 為了有效計算此全域上確界，我們開發了一種新穎的演算法：「具動態座標卡切換之雙趟網格搜尋演算法」（Two-Pass Grid Search Algorithm with Dynamic Chart Switching）。此混合方法利用精確的理論指數來安全地過濾非最佳區域，並採用動態仿射座標卡來防止因無界仿射座標所造成的數值溢位。將此演算法應用於具備固定提升 \begin{align*} \Phi_c(x， y) = (x^d + cy^d， y^d)\quad \text{與}\quad \Psi(x， y) = (x^d， y^d) \end{align*} 的特定有理函數族時，我們根據次數 $d$ 的奇偶性與參數 $c$ 的正負號，識別並嚴格證明了截然不同的極值行為。對於偶數次數 $d$，我們證明了極值行為中存在著根本的分岔：當 $c \in \mathbb N$ 時，算術距離 $\hat{\Delta}(\Phi_c， \Psi)$ 隨 $c$ 嚴格遞增，並在常數座標 $(\pm 1， 1)$ 處達到全域上確界；然而，當 $c \in -\mathbb N$ 時，上確界轉移至對稱座標 $(\pm p， 1)$，其中 $p$ 對應於仿射映射 $\Phi_c(x， 1)$ 的唯一正實不動點。反之，對於奇數次數 $d$ 與參數 $c \in \mathbb Z \setminus \{0\}$，我們證明上確界必定在某點 $(x^*， 1) \in \mathbb A^2(\mathbb R)$ 處達到，其中 $x^*$ 恰好與該系統的一個實不動點重合。總結來說，這些發現建立了算術距離的全域最大值與底層動力系統的不變代數結構（特別是實週期點與前週期點）之間深刻的關聯。;This thesis investigates the analytic properties of Green functions and the extremal behavior of arithmetic distance in arithmetic dynamics. First， we study the modified Green function $\hat{\mathcal G}_{\Phi， v}$ associated with a rational map over a field equipped with an absolute value $v$. We establish its continuity properties with respect to the chordal metric $\rho_v$ by analyzing the Lipschitz constant $L_\phi$ of the associated morphism relative to its degree $d$. Specifically， when $L_\phi > d$， we prove that $\hat{\mathcal G}_{\Phi， v}$ is H\"older continuous with an explicit exponent $\frac{\log d}{\log L_\phi}$ and formulate the corresponding H\"older constant. Furthermore， we demonstrate that when $L_\phi < d$， the modified Green function is Lipschitz continuous， and we provide an explicit upper bound for the Lipschitz constant. In the second part， the thesis explores the arithmetic distance $\hat{\delta}(\phi， \psi)$ between rational maps， defined as the supremum of the difference between their associated dynamical canonical heights $\hat{h}_\phi$ and $\hat{h}_\psi$. For rational maps with good reduction， we demonstrate that this distance can be reformulated into a computationally favorable form using modified Green functions. Specifically， the problem reduces to evaluating the supremum \begin{align*} \hat{\Delta}(\Phi， \Psi) = \sup_{[x:y] \in \mathbb P^1(\mathbb R)} \left| \hat{\mathcal G}_{\Phi， \infty}(x， y) - \hat{\mathcal G}_{\Psi， \infty}(x， y) \right|. \end{align*} To effectively compute this global supremum over the real projective line， we develop a novel algorithm: the Two-Pass Grid Search Algorithm with Dynamic Chart Switching. This hybrid approach utilizes exact theoretical exponents to filter suboptimal regions securely， and employs dynamic affine charts to prevent numerical overflow associated with unbounded affine coordinates. Applying this algorithm to the specific family of rational functions with fixed lifts \begin{align*} \Phi_c(x， y) = (x^d + cy^d， y^d)\quad \text{and}\quad \Psi(x， y) = (x^d， y^d)， \end{align*} we identify and rigorously prove distinct extremal behaviors based on the parity of the degree $d$ and the sign of the parameter $c$. For even degrees $d$， we demonstrate a fundamental bifurcation in the extremal behavior: when $c \in \mathbb N$， the arithmetic distance $\hat{\Delta}(\Phi_c， \Psi)$ strictly increases with $c$ and globally attains its supremum at the constant coordinates $(\pm 1， 1)$; however， when $c \in -\mathbb N$， the supremum shifts to the symmetric coordinates $(\pm p， 1)$， where $p$ corresponds to the unique positive real fixed point of the affine map $\Phi_c(x， 1)$. Conversely， for odd degrees $d$ and parameters $c \in \mathbb Z \setminus \{0\}$， we demonstrate that the supremum is invariably attained at a point $(x^*， 1) \in \mathbb A^2(\mathbb R)$， where $x^*$ precisely coincides with a real fixed point of the system. Collectively， these findings establish a profound connection between the global maximum of the arithmetic distance and the invariant algebraic structures—specifically， the real periodic and pre-periodic points—of the underlying dynamical systems.