2025年

本研究先觀察著名的密克定理（Miquel theorem）與密克點（Miquel point），我們創新給出了新的研究項目，關注密克點𝑃與密克三角形的頂點所構成直線和原三角形𝐴𝐵𝐶三邊直線的其餘六個交點，這是前人沒有觸及的研究項目，從而定義旁接三角形與衍伸三角形。 我們先針對特殊型（直角）的構圖，發現滿足兩個衍伸三角形的有向面積 [𝐴1𝐵1𝐶1]=±[𝐴2𝐵2𝐶2] 時，𝑃 點形成的軌跡為原三角形的 Kiepert hyperbola 與外接圓，這個是有趣且重要發現，我們也進一步給出其幾何必然性。進一步考慮 [𝐴1𝐵1𝐶1]=𝑟[𝐴2𝐵2𝐶2] 時，則刻劃出 𝑃 點軌跡為圓錐曲線系。在前面的基礎下，再針對一般型（任意角）的構圖，若 𝑃 點位於原三角形外接圓及Kiepert hyperbola 與 Steiner circumellipse 的線性組合曲線上，此時兩個衍伸三角形 𝐴1𝐵1𝐶1 與 𝐴2𝐵2𝐶2 的有向面積比值為定值，且兩者恆為相反數。

這是一份將近持續四年的研究，而這一年佩爾質數的出現，讓我們的討論「突飛猛進」。 佩爾方程式是形如𝑥2−𝑚𝑦2=1的方程式，其中𝑘不為完全平方數之正整數。我們定義廣義佩爾方程式是形如𝑥2−𝑚𝑦2=𝑛 的方程式。在過去的研究中，我們主要從𝑥2−𝑘𝑦2=𝑝 (𝑘,𝑝 皆為互質的奇質數) 的正整數解開始研究，接著延伸到 𝑥2−𝑘𝑦2=2𝑚𝑝1𝑛1𝑝2𝑛2⋯𝑝𝑗𝑛𝑗，進而得到了解的唯一分解性質。而本次的研究，延續之前的工作，對佩爾質數展開了討論。利用蜈蚣彘，我們成功地發現了一些佩爾質數，猜測出一些可能的結果並證明；同時我們對佩爾質數的生成結構做了相當程度的了解。作為結束，設法利用分析的方法解決的之前的問題，以及對方程式的不可約解，是否存在較低次方根解，給出了必要條件。

This paper presents a simplified but mostly accurate model for the acoustic mechanism of Seebeck sirens. We investigate the impact of key parameters, including the number and size of holes, as well as the angular speed of the disk, on the characteristics of the produced sound. The disk is fabricated using fused deposition modelling 3D printing, and we used a brushless motor, an air compressor, and a shotgun microphone to capture the generated sound. An order of magnitude analysis was conducted on the Navier-Stokes equation to formulate a simplified version. These simplifications allowed for a low computational intensity model relating volume flow rate to sound pressure level, which is used to predict the waveform of sound produced. Our findings reveal that the fundamental frequency of the sound can be precisely predicted by only the rotational frequency of the disk and the number of holes, a relationship validated experimentally. Notably, observed asymmetry in the waveform was attributed to skin drag effects, and this hypothesis was experimentally verified. Our model computes a solution in less than half a second on average: far less than the 21h 47min needed for a k−ω turbulent model to compute the same phenomenon. The research presents and verifies a simplified model of acoustic mechanics for the sound generated by rotating systems that require little computational resources, which can prove useful in situations where absolute precision is not required, in exchange for ease of computation. For more precise systems, this model serves as a foundation for quickly generating an initial design, paving the way for subsequent iterations using more comprehensive models. The developed model not only serves as a foundation for efficient preliminary designs but also contributes valuable insights into the intersection of fluid dynamics and sound production.

MAGALH˜AES, Eduardo De Mˆonaco. PiezoPioggia: Energy Harvesting with Raindrops. 2024. 24 p. Research report – Scientific Apprentice Program, Col´egio Dante Alighieri, S˜ao Paulo, 2024. This project wishes to study and analyze the possibility of generating clean and accessible energy with the plain impact of droplets in the ground. Therefore, it was necessary to use piezoelectric devices in order to convert the kinetic energy of each droplet into electric energy throughout piezoelectric energy harvesting processes, (PEH). Piezoelectricity is a method of clean and sustainable energy generation, developed and explored by several scientists worldwide. Thus, while studying the proprieties of those devices, the project evaluates the present situation of electricity harvesting in Brazil, the benefits of piezoelectric technology and the possibilities it presents to economy and society. Throughout the development the project builds itself upon mathematical equations and experimental results, analyzing the deformation and generated tensions of piezos. Brand new data on the behavior of rain, as well as about the potential it presents for PEH are highlighted throughout the research, reinforcing the value of such process as a sustainable energy generation method alongside with its investment potential, both from governmental and private institutions. The project also deeply characterizes the piezoelectric device studied, diving deeply in its characteristics and evaluating the deformation of the device and treating the data sets with statistical analysis methods, in order to improve the precision of the data presented. All in all, the opportunities of piezoelectric energy harvesting in the rain, nella pioggia, shall be discussed profoundly throughout the project.

One of the main challenges with today’s batteries is their relatively low volumetric and specific capacities. The highest specific capacity can be achieved with lithium-air batteries, which use metallic lithium as the anode and typically some form of porous carbon as the cathode. To enhance performance, aerogels—among the world’s lightest solid materials—are ideal candidates for cathodes. Resorcinol-formaldehyde (RF)-based carbon aerogels, for example, serve this purpose well. In my work, I utilized two types of carbon aerogels as cathode materials: one derived from pyrolyzed resorcinol-formaldehyde polymer and the other a graphene-oxide-modified version of this carbon gel. I integrated the carbon aerogels I had pyrolyzed into lithium-air batteries to improve the cell’s performance, energy density, and capacity compared to cells using activated carbon. In my research, I examined the pore structure and surface properties of these materials in aqueous media using NMR (nuclear magnetic resonance) relaxometry and cryoporometry, exploring their impact on battery efficiency. I found that the graphene-oxide-containing sample's pores filled with water in a layered manner, indicating a more hydrophilic surface, which suggests a denser arrangement of oxygen-containing functional groups compared to the unmodified carbon aerogel. The pore sizes were reduced after adding graphene oxide, resulting in an increased specific surface area for the sample. Incorporating the reduced graphene-oxide-containing carbon aerogel enabled the creation of a more efficient, higher-capacity battery than with the RF carbon aerogel. This improved performance is likely due to the aerogel’s higher oxygen content and altered morphology. The increased oxygen content provides more active sites for oxygen reduction, meaning that a greater specific power output can be obtained from the battery.

According to the 2024 American Cancer Risk Survey, one in 24 individuals is at high risk of developing colon cancer. This condition is linked to gut microbiome instability. Consequently, there is a pressing need for a more effective and precise approach to maintaining gut microbiome stability, which this research aims to solve by finding the most crucial bacteria species in maintaining the stability of the gut microbiome through the application of Optimal Linear Feedback Control. Two of its variants being applied in this research are Sparsity Promoting Linear Quadratic Regulator (LQRSP) with a variety range of  (0.05, 44.58, and 49.84) and Linear Quadratic Regulator (LQR) ( = 0) along with other supporting methods; Controllability Gramian and Network Theory (graph analysis). The finding in this research shows that bacteria species Bacteroides hydrogenotrophica, Bacteroides uniformis, Bacteroides vulgaris, Bacteroides thetaiotaomicron, Escherichia lenta, and Dorea formicigenerans have an important role for preventing and medicating a variety of gut-related diseases. This conclusion is reinforced by the analysis conducted using the Controllability Gramian, displaying five of the chosen bacteria with the highest controllability index, which demonstrates that the system can be effectively controlled. This finding suggests a potential for enhancing therapeutic strategies, rendering them more precise and systematic. To gain deeper insights into the relationship between each bacteria and the rationale behind the selection of these bacteria by LQRSP, this study also employs network theory, which successfully elucidates the choice of Bacteroides uniformis despite its low controllability index. Additionally, to further validate the efficacy of these bacteria, the research develops a simulation that compares the controlled system with the uncontrolled system, utilizing two types of disturbances. The results indicate a significant difference in robustness against disturbances between the controlled and uncontrolled systems. The findings from this research can be used as a foundation for a more efficient and systematic intervention strategy findings. By researching gut microbiome composition regulation using a mathematical approach, it opens new opportunities for new method discoveries aiming to increase the health of the gut microbiome which is beneficial for the medical field and prevention of gut related diseases.

隨著網路技術不斷的進步，意見和情感分析逐漸成為人們日常生活中的一部分。儘管如此，目前人們缺乏一個方便且快速的情緒分析模型，供廣大大眾使用。 本研究旨在提供人們一個緩解憂鬱情緒的管道——當人們輸入一個需要被安慰的情境時，我們的系統將輸出安慰語句，以緩解該使用者之憂鬱情緒，達到安慰效果。為此，本研究訓練了BERT model以及 LLaMA model。BERT model能判斷使用者輸入的語句是否為需安慰語句。而LLaMA model則作為安慰語句之生成模型，以達到安慰之效果。

睡眠相關問題常見於現代緊張的社會，傳統睡眠分析方法需要腦電圖（EEG）、肌電圖（EMG）、眼電圖（EOG）等信號，量測複雜度高。本研究透過 Python 程式語言以深度學習和階層式投票機器學習方法，開發一套自動分析程式，僅透過心電圖（ECG）信號分析睡眠階段。並結合睡眠評估標準，製訂一可量化的睡眠品質評估表，提供臨床醫師判讀睡眠品質的指標。本研究的優點是僅透過一種信號便能準確、客觀、快速分析，且操作介面簡易。研究結果顯示，本研究清醒和睡眠狀態之辨識準確率高達約90%，與其他類似睡眠品質評估研究的論文比較，準確率高出10～17%，整體睡眠階段分析準確度高達87%。本研究方法未來可應用於臨床醫療，協助醫師做精準的患者睡眠品質診斷。

為了解決水污染問題，本研究探討共價有機框架（COF）作為光催化劑的應用。COF具備高度可調孔洞、高穩定性及選擇性吸附等優勢，有助於有效去除水中污染物，對未來具有前景。本實驗採用大氣常壓微電漿合成COF，此方法能在室溫下以水為溶劑，無需高溫或有毒化學品，並僅需一小時即可完成合成，具有綠色化學優勢。實驗結果顯示，成功合成的COF能有效降解水中常見染料污染物（結晶紫及亞甲藍），證明了COF的高效光催化性。在紫外-可見光光譜中，隨著光催化反應的進行，染劑吸收波峰顯著減弱並幾乎完全褪色，確認了COF優異的降解能力。掃描電子顯微鏡圖像顯示，COF的高度有序孔洞結構提升了其催化活性與穩定性。這項技術不僅能高效處理水中有機污染物，還具備廣泛應用潛力，有望為全球水污染治理與環保提供新思路。

Climate changes that include ocean acidification and global warming are serious problems in the ecosystem, affecting marine phytoplankton, including Nannochloropsis oceanica. In the effort to further explore the impact of rising temperature and carbon dioxide (CO₂) concentrations on oceanic ecosystems, the phytoplankton Nannochloropsis oceanica was used as a model organism. This study explored the effect of temperature change and CO₂ concentration on the growth of Nannochloropsis oceanica, achieving 243 samples that were tested with three different temperatures (24 degrees Celsius (°C), 28°C, 32°C) and CO₂ concentrations (0 milliliter (ml)/min, 0.4 ml/min, 0.6 ml/min), utilizing a hemocytometer counting method. Results indicate that the CO₂ concentration has a significant effect on the population of Nannochloropsis oceanica. But the temperature doesn't affect a lot. The Nannochloropsis oceanica in the lowest temperature and highest concentration of CO₂ in its environment had the highest population growth, and in the highest temperature and lowest concentration of CO₂, it had the lowest population growth. Results show the serious negative effect of climate change on the cosystem and the importance of environmental protection. Population blooms due to excess CO₂ taking up ocean resources causing dangerous ecological imbalances.