Pell equation is an important research object in elementary number theory of indefinite equation. its form is simple, but it is rich in nature. Many number theory problems can be transformed into the problem of Pell equation’s solvability. However, the previous methods in determining the Pell equation’s solvability are sophisticated for calculation, which leads to the lack of efficiency. This paper gives new and more widely used methods to determine the solvability of Pell equation, including several necessary conditions, sufficient conditions and necessary and sufficient conditions.

An order 3 magic hexagon resembles the shape of a 19-cell honeycomb, arranged in a 3 4 5 4 3 manner. The requirement is to fill the numbers 1-19 in the grids so that each row (15 in total) adds up to 38. Previously invented methods aimed at solving this problem and proving its uniqueness were either not rigorous enough or too intricate. So by analyzing its properties, I wanted to find a combinatorial solution to its construction, prove its uniqueness, and investigate whether its mathematical principles can be used in real-world applications. The difficulty depends on the viewpoint, so the first step was to label each grid in a convenient way. I chose to look at the magic hexagon as a network composed of a center and rings. Then the connections and restrictions of each number set could be found by formula derivation. In a similar fashion, symmetrical properties were also found. The next step was to analyze possible distributions of odd and even numbers. Out of the 9 configurations, only 1 proved to be usable. The final step was construction. With all the properties known, the few impossibilities were easily eliminated, and only one solution remained, thus proving its uniqueness. The procedures used on the order 3 magic hexagon may be extended to those of higher orders, providing more ease in their construction. The unique properties of magic hexagons may be used in some fields of application, such as in password systems, large-scale roof structure, composite material, national security systems and many other fields.

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Information Technology has enabled new levels of convenience to our daily life. While providing people with astonishingly good experiences, computer science has still a long way to meet the needs from the users for various scenarios. As a good example, path selection problem is an important and common problem which finds applications in location based services, internet routing and navigation systems. Choosing the most cost effective and time saving routines plays an essential role in both theory and engineering. Although a plethora of classical solutions have been made to the static shortest path problem, dynamic shortest path problem has not been thoroughly studied. To address this critical issue, this paper investigates the dynamic shortest path problem by optimizing existing work and conducting extensive experiments. The studies fall into three main variants of the problem: unweighted shortest paths, weighted shortest paths, and shortest paths on the map. We present a comprehensive comparison of our optimizations and existing solutions which demonstrates the effectiveness and efficiency of our algorithms.

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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.

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Polyhedral combinatorics has been a topic of interest in modern day’s computational geometry. The founding of Steinitz’s Theorem in 1922 revealed consequential relations between graph theory and polyhedral combinatorics. It allows us to better investigate on the topology of convex polyhedrons. In this paper, we proposed an algorithm that generates a unique sequence of points, using the vertices of a triangulated polyhedron, pre-determined by the selection of the starting 3 vertices in the sequence. Following that, we discover an interesting relation between the sequence and the volume of the polyhedron itself, in which we presented in the form of a sufficient condition. To further investigate which polyhedrons generate sequences that satisfy the sufficient condition, we study the problem in the context of graph theory, that is, the explorer walk (corresponding to the sequence of vertices) in maximal planar graphs (skeletons of triangulated convex polyhedrons). With that, we uncovered a family of maximal planar graphs, called the explorer graphs, which exhibits volumetric properties in the polyhedrons constructed from them, in regard to the explorer walk. In this paper, we also introduce generalized methods of constructing explorer graphs of higher order from explorer graphs of lower order, demonstrating the prevalence of explorer graphs. As the edges of a maximal planar graph is of great importance in tracing an explorer walk, we investigate on the line graph of maximal planar graphs, and re-establish a better definition of explorer graphs. Lastly, our paper covers the edge contraction of explorer graphs, which allows us to solve the volume of polyhedrons constructed from non-explorer graphs. For this, we presented a possible bound for the minimum number of edge contractions a non-explorer graph requires from an explorer graph. This will generalize the proposed method of finding volumes to any triangulated convex polyhedron.

We proved in this paper that 14 triangles are necessary to triangulate a square with every angle no more than 72◦, answering an unsolved problem proposed by Professor D.Eppstein. At first, computer is used to search for topologically feasible solutions of triangulations of a rectangle. Then the geometric feasibility is analyzed case by case by plane geometry. We also proved that any rectangle can be 72◦ triangulated if the number of triangles is not limited.

As the barcode becomes more widely used, its applications and data capacity demands grow, increasing the need for barcodes with greater data density. Utilizing the quick response (QR) code–one of the many types of barcodes–we developed two algorithms. The first algorithm creates a color QR code that stores more information than a standard QR code and embeds extra data with limited access privilege. The second algorithm denoises a noisy color QR code. These algorithms consist of three techniques: (1) enlarging the data capacity of a compact QR code image by stacking multiple classical QR codes to form a color barcode, (2) embedding information into the color QR code using pseudo quantum signals in an M-band wavelet domain and selecting the discrete 4-band wavelet transforms to compress the QR images, and (3) applying Discrete M-band Wavelet Transform (DMWT) and Patch Group Prior based Denoising (PGPD) methods to denoise noisy QR code images. The peak-signal-to-noise-ratio (PSNR) summary indicates that information in a color QR code can be efficiently stored and retrieved with these methods. Moreover, it shows that our denoising algorithm effectively removes heavy noise from the noisy color QR code. Our algorithms are implemented in a flexible framework, which allows for further modifications to improve both the data capacity of a color QR code and the effectiveness of signal extraction from noisy data to meet future demands.

Motivated by an extra credit problem from our Linear Algebra class, we study the invertibility probability of binary matrices (the number of invertible binary matrices divided by the total number of binary matrices). Binary matrices are of interest in combinatorics, information theory, cryptology, and graph theory. It is known that the invertibility probability of n × n binary matrices goes to 1 as n→∞. We conjecture that this probability monotonically increases as the size of the binary matrix increases, and we investigate this by exploring how n×n binary matrices of rank n and rank (n−1) can be enlarged to (n + 1) × (n + 1) invertible binary matrices. Calculating this explicitly for the identity matrix, we obtain a probable bound that would show that, in a sense, our conjecture is asymptotically true. With the use of a computer, we also computed how many (n + 1) × (n + 1) invertible binary matrices can be enlarged from n×n matrices of rank n and rank (n−1) for small n. In addition, we study the invertibility probability of matrices with entries in Z_q.

Cartographical projection is mathematical cartography. Map projection is the mathematical model of geoid. Gauss-Kruger projection is unable to realize seamless splicing between adjoining sheet maps. While Mercator projection, especially web Mercator projection, is able to realize seamless splicing but has larger measurement error. How to combine with the advantage of the two projections, apply the method of map projection transformation and realize accurate measurement under the framework of a global map, is the purpose of thisarticle. In order to demonstrate that the problem of measurement could be solved by map projection transformation, we studied the mathematical principle of Gauss - Kruger Projection and the Mercator Projection and designed the algorithm model based on characteristics of both projections. And established simulated data in ArcMap, calculated the error of the three indicators-the area, the length and the angle- under areas in different latitude(0°-4°、30°-34°、60°-64°)and longitude (120°-126°)range which validated the algorithm model. The conclusion suggests that it is feasible that using web Mercator Projection is capable to achieve a global map expression. Meanwhile, calculations of area, distance and angle using map projection transformation principle are feasible as well in Gauss - Kruger Projection. This algorithm combines the advantage of the two different kinds of projection, is able to satisfy uses' high demand and covers the shortage of measurement function of online map services in China. It has certain practical value.

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In this paper we first introduce a fractional form formula among a number of Euler’s formulas. We then extend the formula and with mathematical induction prove the case when the number of terms increases and the exponent is integer. Afterwards, we study the connection between Euler’s formula and Lagrange interpolating polynomial and use the latter to prove part of the extended formula. We then obtain a new formula from this connection. At last, we derive a set of new equations from the extended formula.

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.

This paper researches on the judgment theorem and proof of the equivalency condition of a class of symmetric inequalities. By controlling two elementary symmetric polynomials and using the monotonicity of functions and Jensen inequality, it finds the necessary and sufficient condition of the equivalency a class of three-variable and n-variables symmetric inequalities. And we illustrate the application of this method in proof of these inequalities. Then we obtain several judgment theorems on symmetric and cyclic inequalities.