2024年

本研究探討了柏拉圖多面體滾漆問題，並且延伸至半正多面體進行研究。正多面體中分別為正四面體、正六面體、正八面體、正十二面體、正二十面 體。研究中證明柏拉圖多面體滾漆問題中的地圖限制以及有解充要條件。為了簡化問題難度，我們先簡化遊戲規則使「掉落」情況不被討論，再轉換回原規則的問題。在研究中我們於正四、六、八、二十面體使用原本的遊戲地圖，而其他延伸研究則自創地圖進行討論。

對於任意正整數m和大於1的正整數p，將集合{m,m+1,...,pm-1}中的每一個元素用p進位制表示。令h為介在1到p-1的正整數，將上述集合在p進位制下有i個h的元素個數記為fh,i(m,p)。本文引進一個創新的想法，讓函數 fh,i(m,p)公式解的推導變得可行且簡單。 再者，當 p=2 時，令 fi(m)= f1,i(m,2)，由公式解可以推得對怎樣的正整數n，原像集合the preimage fi-1({n})之元素個數為1。

關於三角形的分割子三角形之內切圓問題，我們設定了幾個新的研究項目並且得出豐富的成果。第一，子三角形的切圓的切點重合性質；第二，子三角形的切圓「半徑長度乘積不變量」；第三，兩點圓心連線性質以及三點圓心連線三角形的面積不變量。首先，我們給出一般化的切圓之切點重合分割存在唯一性。在此條件下，其切圓的圓心均落在一個拋物線，這是很有趣結果。我們再依序探討兩個、三個到多個子三角形，先給出內切圓與旁切圓半徑長度乘積與邊長的關係式，並且發現了不變量。最後探討三點圓心連線三角形面積不變量。值得一提的是，看似不相關的「圓心連線三角形的面積比值」與「半徑長度乘積比值」居然是等價，這是本研究亮點。最後我們完整給出分割為三個子三角形的所有面積不變量的所有組合。

本研究主要目的是研究機器人以等距行進方式從起點前進回到起點的回程路徑問題。 主要以三角函數、遞迴數列、數學歸納法等數學模型來探究、推演與論證，同時利用數學電腦軟體來計算與驗證。一開始先利用Geogebra來探索原始題目，初步得到一些性質和結果；接著，再延伸題目，依不同的起始點位置，探討步數和角度的關係、行進過程中各落點位置的遞迴關係與回程路徑。最後，再將兩條相交直線延伸成三條共點的直線，並依其間的夾角度數，探討以等距行進方式從起點前進回到起點的回程路徑數問題。 本研究根據所建構的數學模型，依不同起始點位置，構作出機器人以等距行進方式回到起始點的回程路徑圖，並得到此回程路徑上相對位置的遞迴關係。本研究在研究過程中，得到一些有趣的數學理論，期望這些成果未來能夠應用於AI機器人運動模式的相關領域。

楊氏矩陣是由有限多個相鄰的方格排列而成的表格，各橫列的左邊對齊，格子數由下而上遞增，而標準楊氏矩陣中每列與每行的元素皆嚴格遞增。我們將楊氏矩陣的「方格」變形為「三角形」，制定與原先楊氏矩陣相似的規則，並命名之為三角楊氏陣列。 本篇研究中，我們首先求得了將某些特定形狀的兩列三角楊氏陣列的遞迴關係式、生成函數、一般項，其與組合學上著名的卡特蘭數亦有相關。後來更是一般化至任意的兩列三角楊氏陣列，得到能夠求其方法數的通式。研究中使用的推算邏輯與方法，也許對未來再研究更一般(或云更多列)的三角楊氏陣列會有所幫助，另外，此研究與偏序集合(Partially Ordered Set)有關，可能可以應用於資訊領域的排序問題。

將平面上A1A2A3的三邊以逆(順)時針方向向內旋轉相同角度後，若三線交於一點，則稱該點為此三角形的正(負)布洛卡點。任意三角形必定同時存在正、負布洛卡點[5]，我們的研究主要是推廣布洛卡點的定義到所有的折線多邊形，並進一步研究布洛卡折線多邊形的充要條件與性質。我們定義調和多邊形為正多邊形的反形，並統整出其與特定條件下的圓內接多邊形互為充要的性質，進而發現了圓內接布洛卡多邊形與調和多邊形之間的充要關係，這是目前中外文獻中沒有的。最後，我們試著利用中垂面與垂面將布洛卡點推廣到三維空間。

As the global demand for sustainable agriculture practices and renewable energy sources continues to rise, the integration of solar power technology with irrigation systems has gained significant attention. This abstract presents an overview of an innovative solution known as the "Automatic Solar Panel Sprinkler Irrigation System," which combines solar panels and smart irrigation technology to efficiently manage water resources in agricultural settings. The proposed system leverages solar panels to generate electricity and simultaneously operate an automated sprinkler irrigation system. Solar panels are strategically positioned in proximity to crop fields, utilizing photovoltaic cells to convert solar energy into electrical power. This energy is then harnessed to power the irrigation system, providing a sustainable and eco-friendly method for crop hydration. In Solar Power Generation the system consists of photovoltaic panels designed to capture solar energy during daylight. This renewable energy source is converted into electrical power, which is stored in batteries for subsequent use. Automated Sprinkler Irrigation is an advanced control system manages the irrigation process, ensuring efficient water distribution based on crop requirements. Soil moisture sensors and weather data are integrated to optimize irrigation scheduling. In Remote Monitoring and Control, farmers can remotely monitor and control the irrigation system through a user friendly interface, accessible via smartphones or computers. This feature enables real-time adjustments and ensures that water resources are utilized optimally. In Water Conservation the system is designed to minimize water wastage by delivering precise and targeted irrigation, reducing over-watering and the associated environmental impact. In Cost Savings the harnessing solar power, this system reduces electricity expenses, making it an economically viable solution for farmers, particularly in regions with ample sunlight. Using a tracker with an automatic solar panel sprinkler irrigation system can be a smart and efficient way to optimize the system's performance. And the most important thing is that in my prototype it is under the panel and will track the location of the Sun and with that it will lead to the Automatic movement of the panel from east to west and when the evening starts it will go back to its position.

This paper describes a power-profile-based approach to CanSat picosatellite energy usage estimation. We measured the power profile of individual CanSat components on a test bench. The collected data was used to create a power consumption model for a simple flight control algorithm. The power consumption model was implemented in Python programming language and was used to simulate the CanSat flight. The simulation results were compared with the measured power profile of a fully integrated CanSat running the same flight control algorithm. The model showed an error of 2.2% and proved to be suitable for use in battery capacity optimization.

Substance losses, burns and injuries arising from various causes represent a constant problem encountered by homo sapiens throughout its existence. Rudimentary treatments, relatively effective and less effective, have left their mark on the way we perceive the presence and treatment of wounds caused by various factors. Searching through medical archives, we can trace the specific protocols for these medical conditions back to 2200 BC, when they were structured in three steps: 1) cleaning the skin lesion, 2) applying a dressing (from glue to various preparations) and 3 ) bandaging the skin lesion. Currently, the appearance of wounds on the skin is caused both by accidents in the performance of various activities and by certain diseases that manifest themselves through skin rashes or skin lesions. Their frequency is in the thousands, according to the latest statistics, affecting the majority of the population non-selectively. It is vital that skin lesions receive the necessary care and attention, commensurate with their severity. Being open wounds on the surface of the skin, it is essential that the treatment be meticulous and appropriate to their type, as skin lesions represent a threat to the patient's life. From infections to hydroelectrolytic imbalances specific to burns, the multitude of factors that influence healing highlight the need for a dressing that can be easily customized according to the specificity of the wound, the needs of the patient and that is affordable both from the point of view of production cost as well as its use, making death from skin lesions easily avoidable through an intelligent approach. One of the most complex biological processes and indispensable to humans is the healing of skin lesions. Healing involves a carefully regulated series of biochemical and cellular activities in tandem. Traditional therapies and substances of natural origin have been used to facilitate the regeneration process and accelerate the wound healing process, being applied with encouraging results. Despite the fact that these generally present a low cost, they can be more expensive than contemporary treatments and can be influenced by regional, seasonal factors, showing fluctuations from batch to batch, which could lead to unpredictable allergic reactions, side effects and inconsistent clinical findings. Currently, the standard of care for skin lesions is to clean the wound with antiseptic solutions to prevent infection, apply a dressing followed by bandaging to keep the dressing in place, and if necessary excision of the tissue that has become non-viable. In the case of diabetic ulcers, it is necessary to excise the tissue that has become non-viable and to maintain control over the level of glucose in the body.

Infections by the Dengue virus (DENV) cause a disease amonghumansreferred to as Dengue fever, which causes thousands of fatalities globally. There is no existing treatment as of yet that successfully targets DENV. Among the factors thatdeterminetheentry of the virus and severity of the disease is the envelope(E) protein of DENV. This study aimed to examine forty antiviral phytochemicals enumeratedinpaststudiesaspossibleinhibitorstotheEprotein of DENV to provide candidates to aid in drug discovery against DENV. The phytochemicals were screened for their likelihood of inhibition of the E protein using AutoDock Suite and LigPlot+. Seven phytochemicals produced favorable binding affinities to the E protein, which are based on the interactions between the phytochemicals and amino acidsintheactivesiteoftheEprotein.Lipinski’s rule of 5 was then used to screen the seven phytochemicals for oral bioavailability. Glabridin has a binding affinity of -7.6 kcal/mol and was predicted to be orally bioavailable. This phytochemical interacts with amino acids in the E protein active site through hydrogen bonds to Asn355, andPhe337, as well as ten hydrophobic interactions. These interactions ensure that glabridin is able to specifically target and fit intotheactivesiteoftheEprotein, preventing its binding to the host cell and activating its viral proliferation. Glabridin is known to be found in the roots of licorice plants, providing anatural source for a possible cure for Dengue fever.