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In his article “Defend the Roman Empire!” (1999), Ian Stewart discussed a strategy of Emperor Constantine for defending the Roman Empire. Motivated by this article, Cockayne et al. (2004) introduced the notion of Roman domination in graphs. Let 𝐺 = (𝑉, 𝐸) be a graph. A Roman dominating function of 𝐺 is a function 𝑓 ∶ 𝑉 → {0, 1, 2} such that every vertex 𝑣 for which 𝑓(𝑣) = 0 has a neighbor 𝑢 with 𝑓(𝑢) = 2. The weight of a Roman dominating function 𝑓 is 𝑤(𝑓) =∑_𝑣∈𝑉 𝑓(𝑣). The Roman domination number of a graph 𝐺, denoted by 𝛾_𝑅(𝐺), is the minimum weight of all possible Roman dominating functions. This paper introduces a quantity 𝑅(𝑥𝑦) for each pair of non-adjacent vertices {𝑥, 𝑦 }in 𝐺, called the Roman dominating index of{ 𝑥, 𝑦} , which is defined by &𝑅(𝑥𝑦) = 𝛾_𝑅(𝐺) − 𝛾_𝑅(𝐺 + 𝑥𝑦)&. We prove that 0 ≤ 𝑅(𝑥𝑦) ≤ 1 and give a necessary and sufficient condition on {𝑥, 𝑦} for which 𝑅(𝑥𝑦)= 1. This paper also introduces the Roman-critical graph. We call 𝐺 = (𝑉, 𝐸) Roman-critical if 𝛾_𝑅(𝐺 − 𝑒) > 𝛾_𝑅(𝐺) , ∀𝑒 ∈ 𝐸 . It is proved that a Roman-critical graph can only be a star graph whose order is not equal to 2, or the union of such graphs. In addition, this paper shows that for each connected graph 𝐺 of order& 𝑛 ≥ 3,2 ≤ 𝛾_𝑅(𝐺) ≤frac{4𝑛}{5}&. A family of graphs for which the respective equality holds is also provided. Finally, this paper finds the lower bound of the Roman domination number for 3-regular graphs and the exact value of the Roman domination number for &𝐶_frac{𝑛}{2}*𝑃2& and 3-regular circulant graphs.

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.

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.

Forums have been one of the most important internet services since the 21st century. However, Forum users have to receive information in a passive way currently. The topics in forums are ordered by last update time (last reply time), thus the information that a user is interested in may be overwhelmed by a large number of other information. Users always have to scan many pages to find a minority of information they need. In this paper, based on the analysis of users' needs, I have designed a personalized forum topic ranking system. This personalized ranking system first calculates all the factors that will influence the user’s decision-making as to whether or not to view a topic by using his/her browsing history. Then the system predicts the click probability for each topic according to all the influence factors using a learned maximum entropy model. Finally, forum topics can be ranked by the predicted click probability, so as to make users find their favorite information easier. As shown by the experiments, the precision of the personalized ranking system is about 60% to 75%, which improves the traditional method (ordered by last update time) by 50% to 85%. In addition, with the normalization of the indicator functions of the maximum entropy model and the selection of the iterative endpoint, the training time can be lowered to an average of 0.01 seconds for each user. It indicates that such a model is able to meet the requirements of practical applications.

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.

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.

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 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.