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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.

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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Nowadays, pictures have more and more applications in various fields. Because different situations may have different commands of the ratios of pictures, we present a program which can resize the pictures according to their contents. Compared with the traditional method which only cares about the geometric constraints, our program can reserve the important contents of the pictures, which means that the visual effect of the pictures will not be damaged after the resizing. In order to change the ratio of pictures while avoiding distorting their main contents, we use a method called seam carving. Based on the energy function used for finding useless pixels and dynamic programming used for finding some continuous seams which will be deleted later, our program can finally change the image to a new ratio perfectly. Due to some deficiencies of the basic program, we also make some improvements. The first one is the improvement of the algorithm: we add face detection to the basic program. Not only does this improvement can solve many problems of the basic program automatically, but also it is our own innovation of the study of content-aware resizing. The second one is the addition of customization: we allow users to select the specific contents that they want to reserve, which can meet the different needs of users. It’s also our own innovation which will be very useful in practical situations. Our program can have many practical applications. We have some ideas about how to put this algorithm into real use, which is one of our practical innovations of this topic. In terms of different ratios of photo printing, graphic designing and screen displaying, this program will have various uses, which will make a groundbreaking progress in practice.

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?

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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.

A topic for study derives from an interesting problem on the internet.Some significative conclusions about the problem of number ring transformation are made in the paper. Firstly, an assumption that a given starting number ring could be transformed to the end ring in finite degree is proved. Secondly, transformation degree is proved to be unconcerned with the order of the transformation. Furthermore, a function of computing transformation degree is worked out. Finally, the same end ring is gained with a given starting number ring transformed in different sequence.

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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.

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.

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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.

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.