MathSciDoc: An Archive for Mathematician ∫

In this paper, we will discuss some upper bound formulas for Ramsey numbers.At first, we will derive a more unified form from two parameter formulas which was obtained by Yiru Huang etc. Then, similar to their methods, we introduce two new bound formulas for Ramsey numbers in this paper. At last, as the applications of our new bound formulas, some upper bounds for small Ramsey numbers will be obtained.

In a class of mathematic competition in our school, Mr. Yang explained for us a problem in 2006 Iranian Maths Olympics (Example 1) and the problem of the 176 in Medium Mathematics 2006.4.At the end of which he raised a question that whether we can make up an inequality chain, which interested us very much. So with the instruction of Mr. Yang, we obtained Theorems 1-6 by means of Bottema 2009 which is mdae by Professor Yang Lu ,a researcher of Chinese Academy of Social Sciences ), computers and calculators. And in this line, we considered that whether they could be extended to situations of high power. After making great endeavors, we endeavor Theorem 7. Finally, we thought about whether it could be extended to high dimensional situation and obtained the creative Theorem 8. The objective of this paper is to further study a type of inequalities of the sum of equal powers of right triangle’s 3 edges to the and has successfully obtained a series of inequalities to the power of 3 to 6 and some of higher order, among which some inequalities of lower order have been proved true by means of the software of Bottema 2009 that developed by Doctor Yang Lu, who is a researcher of Chinese Academy of Science. And we have obtained the inequalities about ndimensional simplex. Meanwhile we studied the application of these inequalities in mathematical competitions and teaching. In the last section, we also proposed two conjectures of more generality,among which conjecture 2 is about n dimensional simplex, for those who are interested in it. The study of this type of problems is beneficial to mathematical competitions and researches of primary and advanced mathematics. As for the application of these theorems, we can discover their value in geometries like right-angled triangle, rectangle,round,ellipse, hyperbola,cube, hypocycloid,right-angled triangular pyramid, globe, ellipsoid, hyperboloid of two sheet, elliptic cone and so on. We can also study about their value in teaching and competitions. In the end, we proposed more general conjectures while the second discusses about situations of n dimensional simplex and both of which are for those readers who are interested in it. This type of problems is beneficial for competitions and study of primary and advanced mathematics. Highlight：We systematically study the inequality of the sum of 3 right-angled triangular sides（Theorems1-6）and extend them to the right-angled tetrahedron and propose some difficult conjectures. Innovation：We get some inequalities of the sum of 3 right-angled triangular sides of high order (4 and above(Theorem 7))，and extend them to the n dimensional space (theorem 8)，and proposed conjectures in n dimensional space (conjecture 1,2) with creative methods .

Lights Out is a video game, which was released by Tiger Toys in 1995. Generally, there are 3×3, 4×4, 5×5, 10×10 grids of games. Under the normal circumstances, it’s easy to deal with some special configurations manually. However, the possibility of solving the randomly-generated configuration by hand is small, for it’s a bit difficult to find the law. Out of interest and enthusiasm about the game, our team reads some reference and gathered four mathematic methods to solve the game. What’s more, we extend the methods to m×n games, the cycling game, the lit-only game and N-color game. We independently propose a method to classify all configurations in the game for 5×5, which can also be extended to general cases. We also put forward the issues about N-order hexagon games, and provide some feasible solutions.

A circle family is a group of separate or tangent circles in the plane. In this paper, we study how many parts at most a plane can be divided by several circle families if the circles in a same family must be separate (resp. if the circles can be tangent). We also study the necessary conditions for the intersection of two circle families. Then we primarily discuss the similar problems in higher dimensional space and in the end,raise some conjectures.

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The traditional kaleidoscope uses the prism as a basic unit. Through the reflections of the mirrors on the inner surfaces of a prism, we can see many symmetric axles and patterns. However, traditional kaleidoscope only presents two-dimensional images. The three-dimensional kaleidoscope we designed uses the pyramid as a basic unit. We can see three-dimensional images via mirror faces on the inner surfaces of the pyramid. Our examination shows that not all pyramids are able to show three-dimensional images but only pyramids with specific angles do. The kaleidoscope we designed differs from the traditional kaleidoscope in the following ways: ①Our three-dimensional kaleidoscope not only shows three-dimensional space, but also shows it through two-dimensional planes. ②Our three-dimensional kaleidoscope is based on three kinds of pyramid: three-sided pyramid (tetrahedron), four-sided pyramid (square pyramid) and five-sided pyramid (pentagonal pyramid). Each kind of pyramid has its own charm in showing its reflected images. ③Each kind of pyramid is separated from its entire geometry. However, it can present the whole view of the geometry after mirrored. Our exploration shows that two dimensional and three dimensional worlds can change into one another under specific conditions. Perhaps before long we can discover more dimensions and explain the relationships between each kind of dimension. This treatise mainly involves the principles of Euclidean Geometry and Projective Geometry.

The problem of the existence of infinitely many prime values of a number-theoretic function f (x) has been one of the most important topics in Number Theory. Note that if f (x) represents infinitely many primes, then we can get this necessary condition: for any positive integer h , there exists a positive integer k such that ( f (k), h) =1and f (k) >1. Naturally, we are interested in the number-theoretic functions f (x) that satisfy the aforementioned necessary condition. Thus, there must exist the least positive integer n such that (f(n),h)-1andf(k)>1. Denote this least positive integer n byF_f(h). In this paper, we mainly focus on three famous number-theoretic functions:&f(x)=2^{2^x}+1,m(x)=2^x-1 and l(x)=x^2+1& proving they satisfy the aforementioned necessary condition respectively. Furthermore, we approximately estimate the upper bound of F_f(x)(h),Fm(x)(h)，Fl(x)(h) respectively, and obtain some interesting results.

In this paper we extend and deepen a shortlist for the 37th International Mathematical Olympiad (IMO) and propose the Frog Leap Commute Theorem and the Queue Polynomial. We explore the problem from the following aspects: (1) Make use of semi-invariants and propose the Frog Leap Commute Theorem. (2) Make extensions regarding frogs leaping to opposite directions on a straight line. (3) Research frogs leaping to the same direction on a straight line and solve the minimum number of frogs satisfying an infinite leap. (4) Extend the problem to leaps on a plane or in space. (5) Research and extend problems regarding frogs leaping on a circle. (6) Estimate the function c(n) and calculate the order of the function.

The linear congruence equations are the ancient and significant research contents. Most discussions of the linear congruence equations focus on the special cases, for example, there is a linear congruent theorem for solving the congruent linear equation in one unknown and the Chinese remainder theorem for solving the simultaneous congruent linear equations in one unknown, which are involved to find a special solution using the properties of the integer number, and some papers discuss the equation in unknowns. But all the results are not convenient and efficient to solve the equations and not adapt to solving the general system of congruent linear equations in n unknowns. There is no uniform, convenient and efficient technique and theory for the general system of congruent linear equations, like the theory of linear equations over real numbers. The inspiration arose from the elimination of variables when solving the linear equations over real numbers, Generalized the elementary transformations of matrix over real numbers to the integer numbers modulo m, the paper discussed the properties of the modular matrix under the elementary transformations and a similar equivalent transforming theorem for matrix modulom theorem was obtained that any matrix modulo m can be transformed into a canonical diagonal form by means of a finite number of elementary row and column operations. furthermore, by means of the equivalent transforming theorem, the solution criterion theorem and structure theorem were proposed for the general congruent linear equations based on the modular matrix transformations, which extended the theories of the congruent linear equations, and finally, the detailed steps of the uniform method were given for solving the general system of congruent linear equations based on the elementary transformations of matrix modulo m, it can be easily written out the solutions of the system immediately as determining solutions of the system conveniently by elementary matrix transformations. Analysis and discussions indicate that the results are most valuable in science and the proposed technique for solving the system is convenient, efficient and adaptable.

Options is an important part of global financial market，with great influence on national economies. While most classic option pricing models are based on the assumption of a constant interest rate, economic data show that interest rates in reality frequently fluctuated under the influence of varying economic performances and monetary policies. As interest rate fluctuation is closely related to the value and expected return of options, it is worth discussing option pricing under stochastic interest rate models. Since 1990s, scholars home and abroad have been conducting researches on this topic and have formulated price formulas for some types of options. However, because the pricing process involves two stochastic variables, the majority of previous studies employed sophisticated methods. As a result, their price formulas were too complicated to provide straightforward explanations of the parameters’ influence on option prices, unable to offer investors direct assistance. This paper selects Vasicek interest rate model to describe interest rate’s stochastic movement, and discusses the pricing of European equity options whose underlying asset’s price follows Geometric Brownian Motions in a complete market. The paper’s value and innovation lie in the following aspects: ① It improves and simplifies the pricing methods for options under stochastic interest rate models, applies comparatively primary mathematical methods, and attains concise price formulas; ② it 52 conducts in-depth analysis of major parameters’ financial significance, which helps investors to make better investment decisions by estimating the variations in option prices corresponding to different parameters.