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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.

The optimal mass transportation problem, proposed by Monge, is dominated by the Monge-Ampere equation. In general, as the Monge-Ampere equation is highly non-linear, this type of partial dierential equations is beyond the solving ability of a conventional nite element method. This diculty led Gu et al. alternatively to the discrete optimal mass transportation problem. They developed variational principles for this problem and reported the calculation for the optimal mapping, based on theorem that among all possible cell decompositions, with constrained measures, the trans- portation cost of the discrete mapping from cells to the corresponding discrete points is minimized by the decomposition induced by a power Voronoi diagram. Their research inspired us to consider a similar discrete optimal mass transportation problem. Here we replace the L2 Euclidean distance by the length of the shortest path connecting two points on a line network. To solve the optimal trans- portation problem on a line network, we study a type of Voronoi diagram on undirected and connected networks and propose an elegant construction algorithm. We further consider weighted distances in a network and develop a method to compute the centroidal Voronoi tessellation (CVT) for a network. By using real geographic data, the method proposed is proved ecient and eective in several practical applications, including charging station distribution in a trac network, and trash can distribution in a park.

In this paper, we firstly re-present the results of Bondar et al. (2013). Then, following the approach taken in it, we use the method of Operational Dynamic Modeling (ODM) to deal with the situation where thebackground mechanics is varying between quantum and classical mechanics.

A physical method is developed to analyze the Riemann zeta function and the L-function. First, a physical model of an elastic bar fixed at both ends and subjected to symmetric axial loads with respect to the mid-span is constructed. The axial force of the bar at the left end is calculated with two methods: the equilibrium of forces and the principle of minimum potential energy. With the help of the orthogonality of {sinix} (i 1, 2, 3, ) , the Parseval identity and the Bessel inequality of the physical model are obtained. Further, it is proven that, in all the possible displacements which satisfy the boundary conditions, the real one minimizes the total potential energy of the bar. With the equivalence of the two methods, an identity of a type of infinite series is derived. Based on the identity, L(1) , L(3) , a recurrence formula of L(2k 1) (k  3) ,  (2) ,  (4) , and a relationship between  (2k) (k  2) and L(2i 1) (i  1, 2, ..., k) are then deduced with power axial load functions. The upper and lower bounds for the Riemann zeta function  (2k -1) and the L-function L(2k) are given. The numerical results show that the upper and lower bounds are in excellent agreement with the accurate value. Finally, two improvement methods are proposed for estimating the circumference ratio. The numerical results show that the two methods can estimate the circumference ratio with high accuracy and that the L -function is more efficient than the Riemann zeta function in estimating the circumference ratio.

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Our paper begins with a simple but beautiful subject given by Martin Gardner (Dividing ten gold coins and ten silver coins into two containers which look the same outside, then randomly choose one of the containers and take out one coin from the chosen container, Is the probability of getting golden coin becomes higher when the distributive method changes). And we change and consider the problem from some other points of view. 1. Researching the original subject’s probability. 2. Discussing the minimal unit of distributing golden coins. 3. Researching the number of containers. 4. Researching the minimal and maximal probability of getting golden coin which is the function expression of containers’ number. As a result, we will understand the subject more deeply. The essence of the subject tells us: the reason of probability’s changing can not only be interpreted by “Sample Space difference and Sample Point difference”, but also “Classical Probability’s infection to Geometry Probability”. According to this theory, we can model this subject: Gold→A and Silver→not A. Then we use the model in our example, successfully interpreting what J·BERTRAND had found. And we also get a conjecture and prove it. Further, we did some transformation of the original formulation according to the actual situation. This is our paper’s main content.

The Durfee conjecture, proposed in 1978, relates two important invariants of isolated hypersurface singularities by a famous inequality; however, the inequality in this conjecture is not sharp. In 1995, Yau announced his conjecture which proposed a sharp inequality. The Yau conjecture characterizes the conditions under which an affine hypersurface with an isoalted singularity at the origin is a cone over a nonsingular projective hypersurface; in other words, the conjecture gives a coordinate-free characterization of when a convergent power series is a homogeneous polynomial after a biholomorphic change of variables. In this project, we prove that the Yau conjecture holds for n = 5. As a consequence, we have proved that &5!pg≤μ-p(v)&, where &p(v) = (v-1)^5-v(v-1) ……(v-4)& and pg,μ,and v are, respectively, the geometric genus, the Milnor number, and the multiplicity of the isolated singularity at the origin of a weighted homogeneous polynomial. In the process, we have also defined yet another sharp upper bound for the number of positive integral points within a 5-dimensional simplex.

Sign language bridges the communication between the deaf-mute and the rest of the world. With the development of computer technology, the research on sign language recognition is prospering. Currently, there are two kinds of research methods. One is based on machine vision, the other is based on wearable input devices. Comparing with machine vision, wearable devices have the advantage of being able to get real-time information of the bend of the fingers and the movement of the hand. The essay applies recent technology in the field of voice recognition, and aims at finding out the best algorithm for medium vocabulary continuous sign language recognition. By building a hybrid Deep Neural Network-Hidden Markov Model, and designing and making a low-cost digital glove, the essay successfully compares the features between Dynamic Time Warping (DTW), Gaussian Mixture Model-Hidden Markov Model (GMM-HMM) and Deep Neural Network-Hidden Markov Model (DNN-HMM) as of solving the problem of sign language recognition. Tests show that in terms of calculating the observation probability, DNN has a much better performance than HMM, especially when syntax is not provided, meaning that DNN better suits the developing trend of sign language recognition. Real-time recognition is achieved on the intelligent terminal with an accuracy of over 97%, using the trained model and a decoding program.

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}.

The development of modern technology is closely related to the use of metals, whose demand has been shifting from common industrial metals to a variety of minor metals, such as cobalt or indium. The industrial importance and limited geological availability of some minor metals have led to them being considered more “critical.” In this research, we develop a novel approach to investigate metal criticality. Instead of quantifying the supply risk of critical metals with static indicators, as used in conventional criticality measures, we propose a dynamic method by looking at the stock trend of the major critical metal producers, who would, in turn, influence the future supply. This allows us to utilize high-frequency time series data in creating machine learning models that predict the stock prices using metal-related features. Specifically, we use regularized linear regression, logistic regression, support vector machine and gradient boosted trees as base models. An ensemble of them is then used for the final prediction, with methods including majority vote, logistic regression stacking, and gradient boosted trees stacking. Based on a misclassification error of 34% in the validation set, we further develop a stock trading strategy, which leads to a back-tested return of 313%, or an excess return of 147%. More importantly, a set of significant features selected by our models leads to insightful suggestions that certain unseeming features, such as markets of other metal cycles and currency exchange rates, may play important roles in influencing the supply of critical metals.

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Structural variations in genome could be more critical than SNP (Single-Nucleotide Polymorphism) in causing tumor cells, immune system diseases and fatness. It is necessary and important to detect structural variations with next generation sequencing data by means of efficient and accurate algorithms. This paper introduced a new algorithm using k-mers (short fragments of genome), string algorithms and graph algorithms, which combines similarity check, variation detection with sequence assembly. Our results showed that our method has high accuracy and it is more reliable than methods using reference and more efficient than popular algorithms.

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Characterization of homogeneous polynomials with isolated critical point at the origin follows from a study of complex geometry. Yau previously proposed a Numerical Characterization Conjecture. A step forward in solving this Conjecture, the Granville-Lin-Yau Conjecture was formulated, with a sharp estimate that counts the number of positive integral points in ndimensional (n≥3) real right-angled simplices with vertices whose distance to the origin are at least n-1. The estimate was proven for n≤6 but has a counterexample for n = 7. In this project we come up with an idea of forming a new sharp estimate conjecture where we need the distances of the vertices to be n. We have proved this new sharp estimate conjecture for n≤7 and are in the process of proving the general n case.

This research, initiated by a problem excerpted from Baidu, one of the most famous internet search engines in China, explores two types of movements, “converging curve” and “tracing curve” named by this research group, which really act in a regular pattern. In the research, by applying scores of mathematical methods, we found that the key to converging curves is to spot the invariant elements from the changeful motions. For tracing curves, we adopt estimating equations to get the solution. Due to the limit of time and the relevant knowledge of the research group, many unexpected problems constantly appeared which are quite far beyond our ability. Yet we still tried to apply some promising mathematical thoughts, and have achieved some exciting results. In the process of discovering and rediscovering, we did not succeed in finding a perfect solution to the problem. After all, as a group of middle school students fascinated by mathematics, creating and discovering more are the greatest delight of us all.

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This essay mainly involves a method to count the number of permutations of size n with length of the longest increasing subsequence equal to 2. The question is raised as the research project of Algebraic Combinatorics in 2016 Tsinghua Math Camp. This essay will show the relationshi between a permutation's longest increasing subsequence andits correspondingRobinson-Schensted map. Moreover, theessay will develop a general formula to count thenumberwith the help of the hook length formula.

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