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Quotient sets have attracted the attention of mathe- maticians in the past three decades. The set of quotients of primes is dense in the positive real numbers and the set of all quotients of Gauss- ian primes is also dense in the complex plane. Sittinger has proved that the set of quotients of primes in an imaginary quadratic ring is dense in the complex plane and the set of quotients of primes in a real quadratic number ring is dense in R: An interesting open question is introduced by Sittinger: Is the set of quotients of Hurwitz primes dense in the quaternions? In this paper, we answer the question and prove that the set of all quotients of Hurwitz primes is dense in the quaternions.

In this project we study the energy functional on the set of Lagrangian tori in CP2. The energy functional has been introduced in [2] as integral of the potential of 2D periodic Schrödinger operator associated to Lagrangian torus. It has been conjectured in [2] that the Clifford torus is the unique global minimum of energy functional (the statement is later referred to as the energy conjecture). Due to geometric interpretation of energy functional as linear combination of the volume andWillmore functionals, this conjecture can be seen as the CP2 analogue of the well-known Willmore conjecture for tori in R3, recently proved in [18]. The energy conjecture has been verified for two families of Hamiltonian-minimal Lagrangian tori in [2]. Results of [5] and [23] imply the conjecture for minimal Lagrangian tori of sufficiently high spectral genus and non-embedded minimal Lagrangian tori, respectively. In the present work we prove the energy conjecture for a family of Hamiltonian-minimal Lagrangian tori in CP2 constructed in [4]. In sharp distinction with cases considered in [2], the value of the energy functional for these tori can not be calculated exactly. The proof relies on analytic bounds for certain elliptic integrals arising from the induced metric of tori. Possible directions of further work are: 1. Consider local behaviour of the energy functional. Are the critical points of the energy functional governed by an integrable PDE, akin to Tzizeica equation describing minimal Lagrangian tori? The same questions for critical points under Hamiltonian variations. 2. Is there an analogue of the energy conjecture for other Kähler-Einstein surfaces? The case of K3 surface is of special interest as minimal Lagrangian tori in K3 can be related to elliptic fibrations (for instance [20]) making the conjecture amenable to algebrogeometric analysis. 3. Examples of monotone Lagrangian tori with trivial Floer cohomology were constructed in [21]. Do there exist critical points of the energy functional with trivial Floer cohomology?

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 .

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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.

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Checkpoint/restart is an important mechanism for system fault tolerance. It saves the state of an executing program periodically and recovers it after a failure. As many applications involve file operations, supporting file rollbacks is essential for checkpoint/restart. Complete backup can restore files to the correct state, but its cost is too high. In this paper we propose a behavior based file checkpointing strategy (BBFC), which provides a correct recovery of file data and ensures consistency between file state and other states of a process when a rollback is done by restarting the program from the last checkpoint. BBFC classifies details of the file operation behaviors and provides a guidance on what to be saved during file checkpointing according to those behaviors. It dramatically minimizes the overhead of file checkpointing due to the reduction of file data which need to be saved. And it is transparent and easy to use.

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The paper mainly discusses, based on the properties and related knowledge of continuous functions, the conditions that the graphs of separated binary polynomials are closed curves by the method of separation of variables. It also studies the properties of closed curves in the ways of analytic geometry, and draws a sucient and necessary condition to determine whether the graph of a separated binary polynomial is closed.

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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.

Clothing is one of the symbol of modern fashion trends, research concerning clothing detail’s recognition still remains a hot topic in the field of computer. Existing research is mainly focused on relating clothing detail features and aesthetic styles, realizing the possibilities of auto clothing aesthetic appreciation. In this paper, based on existing data sets, we will introduce the ideology of Deep Learning into the association model between clothing visual features to improve the model’s effectiveness and accuracy. By calculating the distance between aesthetic words and seed words, we are able to map the most often used aesthetic words in a two-dimensional space, building a fashion semantic space. Later, we introduce the ideology of Deep Learning into the association model between clothing detail features, and designed experiments to prove the effectiveness of the model. In the research, we discussed how different parameters and structures in Deep Learning effect the outcome of the relating model, and by comparison we choose the best parameters of the model. Based on the work mentioned above, we analyze different styles of different categories’ clothing in shopping websites, which further confirms the effectiveness of our model.

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.

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According to Van der Waerden, the sum of two natural numbers is defined as &x*1=x& &x*y^+= x*y+x& (for every x and every y) The product of two natural numbers is recursively defined as &x*1=x& &x*y^+= x*y+x& (for every x and every y) x^+ ( y^+ ) here means the successor(consequent) of x( y) in the set of natural numbers.