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In Ramsey theory (a branch of combinatorics) one often proves that a function exists by giving enormous bounds on it. The function's actual values may be much smaller; however, this is not known. In this paper we explore variants and special cases of these functions where we obtain much smaller bounds. The following is known: no matter how the lattice points of a coordinate plane are colored red and blue, there exists a square whose corners are the same color (a monochromatic square). In fact, using more than just two colors will still guarantee a monochromatic square (one whose vertices are the same color). So for all c (the number of colors), there is a number G(c) where all colorings with c colors of the lattice points of a G(c)*G(c) grid will contain a monochromatic square. Unfortunately, the necessary number of points is unknown, but bounds are known. These bounds are enormous,however. So we look at a variant of this problem in order to work with more manageable bounds. Looking for rectangles or right triangles rather than squares makes these bounds polynomial. We explore which grids contain a specic number of rectangles or right triangles. It is known that if the naturals are colored with two colors, there will be two of the same color that are a square apart. Once again, this is true for any number of colors. It is also true for any polynomial whose constant term is 0 and for multiple polynomials. This is known as the Polynomial Van der Waerden Theorem, or PolyVDW. The bounds on the number of naturals that have to be colored grow very quickly. However, for quadratic polynomials and 2 or 3 colors, there are some cases where the theorem is false, and others where it is true with polynomial bounds. Since PolyVDW does not hold in its entirety for all such polynomials, we try to classify for which of these polynomials and for which numbers of colors the theorem holds and try to nd bounds for them. We explore monochromatic rectangles and right triangles rather than squares in order to get reasonable bounds on the grid sizes necessary to guarantee the existence of these monochromatic shapes. We also explore the necessary grid sizes for multiple rectangles and right triangles. In fact, we nd many exact numbers. In addition, we look at variants of PolyVDW and obtain much better upper bounds for these numbers than are known for the standard PolyVDW numbers.

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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}.

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In the early 1980s of twenty century, Professor Zhang Zhongfu, an expert in graph theory, raised a question in his research: how many points of intersection of diagonal lines are there inside a regular N polygon, hoping to find out a formula of count. It has been more than 20 years since the question was raised, which has aroused the interest of quite a lot of experts, scholars and those who love mathematics, but still remains unsolved. When N is an odd number, proposition 1 can be derived from formula of counting based on proposition 2, which can also be verified by the programme the author has made. Proposition 1: when N is an odd number, there is no concurrence of 3 or more than 3 diagonal lines of regular N polygon. Proposition 2: when N is an odd number, the number of intersection points of inner diagonal lines of regular N polygon is: &a_n=c^{4}_{n}=frac{1}{24}(n-1)(n-2)(n-3)$ But when N is an even number, it becomes quite complicated. When N are some special even numbers referred to by Chang Jianming, there are some laws: the research is made from the perspective of prime factorization of figures. The writer primarily studies the issue by making use of geometrical drawing and imitation by computer programme, so that the programme is improved and the imitating solutions of the numbers of intersection points of inner diagonal lines of regular N polygon with specific programmes can be solved. The solution is popularizable. The writer has worked out the numbers of intersection points of inner diagonal lines of regular 2N polygon when N 50, if conditions of computers in schools allow. By comparing the counting between drawing and computer imitation, the writer has also found out some interesting laws about the numbers of intersection points of inner diagonal lines of regular N polygon, and has put forward some hypotheses.

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.

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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.

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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.

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.

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.

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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.