費瑪也瘋狂-平面上存在障礙時連接三定點的最佳網絡問題
在一個有障礙的平面上，給三個定點，我們探討連接此三點的最佳網絡。我們討論了諸如直線、射線、線段、圓、網格狀、三角形……等類的障礙，當網絡每穿越障礙一次，就必須付出代價，例如「拖延5 分鐘」。所以，設網絡穿越障礙的次數為y ，則網絡除了原本的總長度之外，還額外加入y 倍某固定數值的損耗。我們以費瑪點的各種性質及三角形不等式等方法為工具，就不同的穿越障礙次數綜合比較，而找出最佳網絡。在某些情況下，最佳網絡不是以費瑪點來連接三點，而是在障礙(如：直線)上找出符合某種與餘弦值相關特殊性質的點，以該點來連接三點，而此網絡可用GSP 軟體相當精確地作出。另外，我們也探討在考慮障礙造成損耗的情況下，兩點間的「實際距離」為何。 最後，我們考慮「混合障礙」問題。在此類問題中，除了前面所討論的障礙，還另加了如同「河流」的兩平行直線間區域之障礙，在這種障礙區域中，網絡的長度要乘以數倍來計算。我們發現，此類問題的最佳網絡也可用特定的正弦條件配合GSP 而相當精確地作出來。;Considering various kinds of obstacles in a plane, such as a line, a segment, a ray, a circle, a triangle or chessboard grids, which function like a red light, we research into the problem of finding the optimal network connecting three given points A, B, C in the plane amidst obstacles described above. Each time when the network crosses an obstacle, it will cause losses, such as five minute’s delay or a loss of one hundred dollars. Taking advantage of Fermat points, some basic inequalities concerning triangles and some special qualities about sine or cosine functions, we obtain the optimal networks in different situations. Besides, we consider what the “real distance” between two points is when there are obstacles in a plane. We also put another obstacle, including a line and a weighted region between two parallel lines, into consideration. In the region, like a river or a muddy ground in real life, the length of the network should be multiplied by a fixed time. Furthermore, we can use GSP to make the networks very accurately.
電解與磁場的秘密.
金屬離子在磁場中的流動速率會略有改變，尤其是在強磁場中時，其影響更是顯著，即 【MHD 磁流動效應】，造成整體電解液中離子的流動，此流動比擴散速率佔優勢。再利用「磁 矩」具有向量性質，探討不同金屬離子（Na+、K+、Fe3+、Al3+、Mg2+）及MnO4 -在磁場角度 相同但強度不同的情況下；及磁場角度不同但強度相同的情況下，對電解速率的影響。 經實驗發現有以下結論： 一、由法拉第電解第一定律出發，加以實驗數據分析，可推導出一關係式： 電解速率Rρ= k ×∫〈∣H 向量∣×∣cosθ∣〉×∣E 向量∣ （k 單位：g / C˙weber˙s） 二、電解效率隨價數增高而增快。 三、較強的電解質，其對磁場的感應也越大，如果就同一族而言，往 下其活性越強，對磁場的感應也越強。 The flowing rate of Metal ion changes slightly in magnetic field. This influence is especially remarkable while the magnetic force is very strong, that’s【MHD (magneto Hydrodynamic Effect ), which gives rise to ionic flowing all over the electrolyte. This flowing rate is superior to expanding rate. Further, basing on the magnetic torque ’s vector trait, this research studies how electrolysis velocity affects different metal ions (Na+、K+、Fe3+、Al3+、Mg2+) and MnO4 － under following situations: Some results are found through the experiment. 1. Begin with Farad Electrolysis First Law, and take the experimental analysis into account, then a relative formula comes out as bellow. Electrolysis Rate Rρ= k ×∫〈∣H∣×∣cosθ∣〉×∣E∣ （k：g / C˙weber˙s） 2. Electrolysis efficiency accelerates by the increasing price amount. 3. Active electrolytes get strong response to the magnetic field. For the same group, the more active the electrolytes are, the stronger it responds to magnetic filed.
Do SAT Problems Have Boiling Points？
The Boolean Satisfiability problem, called SAT for short, is the problem of determining if a set of constraints involving Boolean (True/False) variables can be simultaneously satisfied. SAT solvers have become an integral part in many computations that involve making choices subject to constraints, such as scheduling software, chip design, decision making for robots (and even Sudoku!). Given their practical applications, one question is when SAT problems become hard to solve. The problem difficulty depends on the constrainedness of the SAT instance, which is defined as the ratio of the number of constraints to the number of variables. Research in the early 90’s showed that SAT problems are easy to solve both when the constrainedness is low and when it is high, abruptly transitioning (“boiling over” ) from easy to hard in a very narrow region in the middle. My project is aimed at verifying this surprising finding. I wrote a basic SAT solver in Python and used it to solve a large number of randomly generated 3SAT problems with given level of constrainedness. My experimental results showed that the percentage of problems with satisfying assignment transitions sharply from 100% to 0% as constrainedness varies between 4 and 5. Right at this point, the time taken to solve the problems peaks sharply. Similar behavior also holds for 2SAT and 4SAT. Thus, SAT problems do seem to exhibit phase transition behavior; my experimental data supported my hypothesis.