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We proved in this paper that 14 triangles are necessary to triangulate a square with every angle no more than 72◦, answering an unsolved problem proposed by Professor D.Eppstein. At first, computer is used to search for topologically feasible solutions of triangulations of a rectangle. Then the geometric feasibility is analyzed case by case by plane geometry. We also proved that any rectangle can be 72◦ triangulated if the number of triangles is not limited.

Starting from the roots of the equation 3^x + 4^x = 5^x discussed in the Tenth Grade, we utilized the method, the algebraic extension of the rational number eld, to produce the ways to judge whether the roots of the basic exponential equation with a form as a^x+b^x = c^x are rational or not. For equations with more than two terms on the left side, as is the equation a_1^x+a_2^x+
···

+a_n^x = d^x, the determination of whether the root was irrational was comparatively dicult.Therefore, we provided a prevalent method for the examination of the root of a three-term equation as well as a conclusion that if the equation doesn't have integer roots, the roots won't be a rational number with a denominator of two. Finally, based on the method, the algebraic extension of the rational number eld, we concluded that under special occasions, the root of the equation can't be some rational numbers with certain denominators.

Pell equation is an important research object in elementary number theory of indefinite equation. its form is simple, but it is rich in nature. Many number theory problems can be transformed into the problem of Pell equation’s solvability. However, the previous methods in determining the Pell equation’s solvability are sophisticated for calculation, which leads to the lack of efficiency. This paper gives new and more widely used methods to determine the solvability of Pell equation, including several necessary conditions, sufficient conditions and necessary and sufficient conditions.

I heard the concept of the ”equal‐sum point” in a math summer camps and found it very interesting. After searching on the Internet,I found that the ”equal‐sum point” was associated with Soddy point. Soddy point is a magic point in plane geometry, which was found by British physicist, chemist Frederick.Soddy. Some research about the Soddy point has been done in foreign countries, but the properties are not comprehensive. In China, Teacher HuasongHuang raised the concept of the “equal‐sum point” and the “equal‐difference point” and drew some properties, I found that these two points were very similar to the Soddy point, but they did not link the two points with Soddy point. In this article ,I will connect the Soddy point with the “equal‐sum point” and the “equal‐difference point” and study their properties more deeply and find some new discoveries.

Free kicks play an important role in a modern soccer match. A skillful soccer player usually uses the opportunity of a free kick to score a goal. There are mainly two ways to give a free kick, falling ball and banana ball. The last thesis analyzes the path of falling soccer in two-dimensional plane. This thesis mainly focuses on using geometrical ways to generate a graph in three-dimensional coordinate system to analyze the path of banana ball and find out the most suitable distance to give a free kick.

Based on Leslie Matrix, the paper tries to predict the population fluctuation of Nanjing by considering the regional distinction and latest policy impacts, and hereby comes up with four conclusions: a) The population of Nanjing will see a relatively rapid growth between 2013 and 2020; b) Given the situation, the elderly population of the city will experience a much faster increase in the next 40 years, causing an elderly bulge that not only poses heavy burden to social welfare but also gives rise to a disproportionate population structure; c) The proposed retirement age extension in theory, will reduce the pressure on pension system by retaining active labors, and thus alleviating the problems of aging population to some extent; d) The newly-issued Two-Child Policy will not change the trends in or transform the population structure, therefore its effects on population aging is rather limited.

This paper proposes an Integrated Obesity Index (IOI) based on height, weight, waist, and hip, physical indexes that can be accessed through regular physical examinations, on top of common obesity standards, including Body Mass Index (BMI), Waist‐Hip Ratio (WHR), and Body Fat Percentage (BFP). Integrated Obesity Index is aimed to be feasible and convenient under application, as well as comprehensive, compensating for the limitations of existing classifications of obesity. Thus, this paper proposes and examines the application formula of the IOI by establishing geometric models. Adopting the data from epidemiological survey, this papers brings physical indicators, gender, BFP and other variables into the model for data analysis, applies linear regression on the resulting images, and obtains the IOI cut‐off point. By conducting comprehensive data analyses to various obesity classifications, the paper demonstrates that the IOI has significant advantages over BMI, WHP, or BFP and will compensate for the deficiencies of the existing obesity classifications to a certain extent in application.

This article applies functional knowledge and computer software to describe the relation between people distribution and evacuation efficency and safety in high-rise buildings.The people distribution contains two parts:the allocation of people in different storeys and the distribution of people in a certain storey. Through this mathematical model,we attempt to find the optimum plan of people distribution and use it to guide the distribution of different functional areas in high-rise buildings.

It is well known that group is a set with an algebra operation, which is often denoted by G . A simple group is the group which has only the 2 trivial normal subgroups. The importance of the simple groups may analogue to the prime numbers in the number theory. The classification of the finite simple groups was the central problem of 20th century’s’ algebra. In this research report, we use Sylow theorem, Burnside theorem, group action and some other elementally group theory methods to obtain some criteria of simple group . By using these criteria we determine the possible simple groups whose order less than 700. Generalizes the results in the reference [1]，[2]， [3]，[4].We conclude that Theorem A group G of order less or equal than 700 could not be simple except for |G|∈{60,168,360,504,660 and all primes}.

Computational techniques derived from digital signal processing are playing a significant role in the security and digital copyrights of audio, video, and visual arts. In light of the quantum computing, the corresponding algorithms are becoming a new research direction in today’s high-technology world. The nature of quantum computer guarantees the security of quantum data, so a safe and effective quantum watermarking algorithm is in demand. Quantum watermarking is the technique that embeds the invisible quantum signal into quantum multimedia data for copyright protection. This research presents a quantum watermarking algorithm in discrete M-band wavelet domain. Different from most traditional algorithms, we propose a new algorithm which applies quantum watermarking in M-band wavelet domain. Assured by the Heisenberg uncertainty principle and quantum no-cloning theorem, the security of quantum watermark can reach a very high-level standard. In other words, this watermarking algorithm can defeat nearly all attackers, no matter using classical computer or quantum computer.

The aim of this study is to figure out the optimization of survival probability of the generals in the card game "Legends of the Three Kingdoms". First, based on combinatorics and probability theory, the PASCAL program was performed to analyze the probabilities of success when several popular generals perform their special skills in the game. The results are as follows: Second, seven typical generals, including Zhou Tai, Zhang Jiao, Zhen Ji, Lu Xun, Ma Chao, Xiahou Dun, and Guo Jia, were chosen, and their abilities including attacking and defense were arbitrarily quantified according to a model constructed in the present study. The scores are as follows: Card-drawing Changing Attacking Defense Blood-selling Zhen Ji 5.62 Zhou Tai 1.04 Ma Chao 1.68 Xiahou Dun 1.19 Guo Jia 3.36 Lu Xun 3.55 Zhang Jiao 2.55 The abilities of different generals were sorted, and by comparison with the statistical record online, the reliability of the model was proved. Thus, the irrationality of the game is identified in this study, and we recommend the game designer to modify the abilities of some generals and to obtain an optimized value according to the output of the model, by which the game would be more popular.

Cartographical projection is mathematical cartography. Map projection is the mathematical model of geoid. Gauss-Kruger projection is unable to realize seamless splicing between adjoining sheet maps. While Mercator projection, especially web Mercator projection, is able to realize seamless splicing but has larger measurement error. How to combine with the advantage of the two projections, apply the method of map projection transformation and realize accurate measurement under the framework of a global map, is the purpose of thisarticle. In order to demonstrate that the problem of measurement could be solved by map projection transformation, we studied the mathematical principle of Gauss - Kruger Projection and the Mercator Projection and designed the algorithm model based on characteristics of both projections. And established simulated data in ArcMap, calculated the error of the three indicators-the area, the length and the angle- under areas in different latitude(0°-4°、30°-34°、60°-64°)and longitude (120°-126°)range which validated the algorithm model. The conclusion suggests that it is feasible that using web Mercator Projection is capable to achieve a global map expression. Meanwhile, calculations of area, distance and angle using map projection transformation principle are feasible as well in Gauss - Kruger Projection. This algorithm combines the advantage of the two different kinds of projection, is able to satisfy uses' high demand and covers the shortage of measurement function of online map services in China. It has certain practical value.

Dijkstra and Bellman-ford are the two basic methods to solve Single-source shortest paths problem，and SPFA is the optimized version of Bellman-ford. However, since the algorithm time complexity is 0 (kn), SPFA’s time efficiency has some instabilities. This paper will compare the time efficiency between Dijkstra optimized with heap and three kinds of SPFA in different density graphs. Meanwhile, random data is created to test optimized Dijkstra and 3 types of SPFA. Finally, according to the calculation result of two methods’ time efficiency, some suggestion is given for their application under different situation.

The problem we study in this paper originated from a simple high school math competition (AMC 12) question on a geometric probability model: “A frog makes 3 jumps, each exactly 1 meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog’s final position is no more than 1 meter from its starting position?” [1] From here, we changed the number of jumps the frog made, the length of each jump and the dimensions of the space. Using recursion, we obtained the recursive formula of the probability of a frog landing x meters within its original spot after jumping m 1-meter jumps in an N -dimension space by integrating the probability distribution of the former step. We also suggested a concept “the intensity of the probability field” to describe the relative probability of each spot and gave an expression of the intensity of each spot of a 2-dimensional space when the frog makes m jumps.

The log-optimal strategy in gambling and asset allocation problem attempts to maximize the expectation of the logarithmic rate of return but not the gross wealth itself. This strategy has been shown to have long termsuperiority overother strategies in theory. Another remarkable virtue is it allows updating the real-time market information quickly which could increase of the logarithmic rate of return. We will prove these properties ourselves in a simple way. We further consider how to do asset allocation by applying the log-optimal strategy in practice. The distribution function of the returns is usually unknown and need to be estimated from the history in a real world. The estimation error will affect the asset allocation directly, but such effect mayresult in a "butterfly effect" which could bring an investment disaster. Therefore, it is an important issue to answer whether there is logoptimal strategy resisted "butterfly effect". We develop a new log-optimal strategy based on the allocation utility which is a quadratic approximation to the expectation of the logarithmic rate of return. We show that the upper bound of the allocation utility deviation can be controlled by the l_1-norm of portfolio and l_∞-norm of the bias of variance matrix estimator. This turns out that our proposed strategy is robust. We apply our strategy for NYSE data. The results showed that our strategy has high performance in return.

The subject of counting positive lattice points in n-dimensional simplexes has interested mathematicians for decades due to its applications in singularity theory and number the- ory. Enumerating the lattice points in a right-angled simplex is equivalent to determining the geometric genus of a singularity of a weighted homogeneous complex polynomial. It is also a method to shed insight into large gaps in the sequence of prime numbers. Seeking to contribute to these applications, this research project proves the Yau Geometric Con- jecture in six dimensions, a sharp upper bound for the number of positive lattice points in a six-dimensional tetrahedron. The main method of proof is summing existing sharp upper bounds for the number of points in 5-dimensional simplexes over the cross sections of the six- dimensional simplex. This research project paves the way for the proof of a fully general sharp upper bound for the number of lattice points in a simplex. It also moves the mathematical community one step closer towards proving the Yau Geometric and Yau Number-Theoretic Conjectures in full generality.

Matrix factorization methods are widely used for extracting latent factors for low rank matrix completion and rating prediction problems arising in recommender systems of on-line retailers. Most of the exist- ing models are based on L2 delity (quadratic functions of factorization error). In this work, a coordinate descent (CD) method is developed for matrix factorization under L1 delity so that the related minimization is done one variable at a time and the factorization error is sparsely distributed. In low rank random matrix completion and rating predic- tion of MovieLens 100k datasets, the CDL1 method shows remarkable stability and accuracy under gross corruption of training (observation) data while the L2 delity based methods rapidly deteriorate. A closed form analytical solution is found for the one-dimensional L1-delity sub- problem, and is used as a building block of CDL1 algorithm whose con- vergence is analyzed. A connection with robust principal component analysis is drawn.

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