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.

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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.

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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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.

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Free kicks play an important role in a modern soccer match. A skillful soccer player usually uses the opportunity of a free kick to score a goal. There are mainly two ways to give a free kick, falling ball and banana ball. The last thesis analyzes the path of falling soccer in two-dimensional plane. This thesis mainly focuses on using geometrical ways to generate a graph in three-dimensional coordinate system to analyze the path of banana ball and find out the most suitable distance to give a free kick.

Automatic hexahedral mesh generation plays a fundamental role in CAD/CAE fields, especially for IGA (isogeometric analysis). Recently, an automatic hex-mesh generation method has been proposed, which is based on the equivalence among three key concepts: colorable quadrilateral meshes, finite measured foliations and Strebel differentials. This work focuses on the computational aspect of Strebel differentials on high genus surfaces, which is based on graph-valued harmonic mapping. The algorithmic pipeline is as follows: first, the user inputs an admissible curve system, which induces a cylindric-decomposition graph; then the user specifies the lengths of the edges of the graph; third, the algorithm finds the unique harmonic map from the surface to the metric graph by a non-linear heat flow; finally, the harmonic map induces a Strebel differential, which can be further utilized to generate the hex-mesh. The method has solid theoretic foundation, which guarantees the existence and the uniqueness of the graph-valued harmonic map. The algorithm is capable of handling surfaces with complicated topologies, and producing all possible Strebel differentials on the surface. The user has full control of the combinatorial type and the geometry of the Strebel differential. The computational pipeline is automatic. The experimental results show the efficiency and efficacy of the algorithm, and demonstrate the great potential for automating the structured hexahedral mesh generation.

Based on Leslie Matrix, the paper tries to predict the population fluctuation of Nanjing by considering the regional distinction and latest policy impacts, and hereby comes up with four conclusions: a) The population of Nanjing will see a relatively rapid growth between 2013 and 2020; b) Given the situation, the elderly population of the city will experience a much faster increase in the next 40 years, causing an elderly bulge that not only poses heavy burden to social welfare but also gives rise to a disproportionate population structure; c) The proposed retirement age extension in theory, will reduce the pressure on pension system by retaining active labors, and thus alleviating the problems of aging population to some extent; d) The newly-issued Two-Child Policy will not change the trends in or transform the population structure, therefore its effects on population aging is rather limited.

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In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the $(1,1)$- component of the curvature $2$-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Hermitian manifolds and the background Riemannian manifolds. Moreover, we study non-K\"ahler Calabi-Yau manifolds by using the first Aeppli-Chern class and the Levi-Civita Ricci-flat metrics. In particular, we construct explicit Levi-Civita Ricci-flat metrics on Hopf manifolds $\S^{2n-1}\times \S^1$. We also construct a smooth family of Gauduchon metrics on a compact Hermitian manifold such that the metrics are in the same first Aeppli-Chern class, and their first Chern-Ricci curvatures are the same and nonnegative, but their Riemannian scalar curvatures are constant and vary smoothly between negative infinity and a positive number. In particular, it shows that Hermitian manifolds with nonnegative first Chern class can admit Hermitian metrics with strictly negative Riemannian scalar curvature.