We present a framework for generating street networks and parcel layouts. Our goal is the generation of high-quality layouts that can be used for urban planning and virtual environments. We propose a solution based on hierarchical domain splitting using two splitting types: streamline-based splitting, which splits a region along one or multiple streamlines of a cross field, and template-based splitting, which warps pre-designed templates to a region and uses the interior geometry of the template as the splitting lines. We combine these two splitting approaches into a hierarchical framework, providing automatic and interactive tools to explore the design space.

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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.

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.

For the purpose of isogeometric analysis, one of the most common ways is to construct structured hexahedral meshes, which have regular tensor product structure, and fit them by volumetric T-Splines. This theoretic work proposes a novel surface quadrilateral meshing method, colorable quad-mesh, which leads to the structured hexahedral mesh of the enclosed volume for high genus surfaces. The work proves the equivalence relations among colorable quad-meshes, finite measured foliations and Strebel differentials on surfaces. This trinity theorem lays down the theoretic foundation for quadrilateral/hexahedral mesh generation, and leads to practical, automatic algorithms. The work proposes the following algorithm: the user inputs a set of disjoint, simple loops on a high genus surface, and specifies a height parameter for each loop; a unique Strebel differential is computed with the combinatorial type and the heights prescribed by the user’s input; the Strebel differential assigns a flat metric on the surface and decomposes the surface into cylinders; a colorable quad-mesh is generated by splitting each cylinder into two quadrilaterals, followed by subdivision; the surface cylindrical decomposition is extended inward to produce a solid cylindrical decomposition of the volume; the hexadhedral meshing is generated for each volumetric cylinder and then glued together to form a globally consistent hex-mesh. The method is rigorous, geometric, automatic and conformal to the geometry. This work focuses on the theoretic aspects of the framework, the algorithmic details and practical evaluations will be given in the future expositions.

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.

Ricci flow deformsthe Riemannian metric proportionallyto the curvature, such that the curvatureevolves accordingto a heat diffusion process and eventually becomes constant everywhere. Ricci flow has demonstrated its great potential by solving various problems in many fields, which can be hardly handled by alternative methods so far. This work introduces the unified theoretic framework for discrete Surface Ricci Flow, including all the common schemes: Tangential Circle Packing, Thurston’s Circle Packing, Inversive Distance Circle Packing and Discrete Yamabe Flow. Furthermore, this work also introduces a novel schemes, Virtual Radius Circle Packing and the Mixed Type schemes, under the unified framework. This work gives explicit geometric interpretation to the discrete Ricci energies for all the schemes with all back ground geometries, and the corresponding Hessian matrices. The unified frame work deepens our understanding to the the discrete surface Ricci flow theory, and has inspired us to discover the new schemes, improved the flexibility and robustness of the algorithms, greatly simplified the implementation and improved the efficiency. Experimental results show the unified surface Ricci flow algorithms can handle general surfaces with different topologies, and is robust to meshes with different qualities, and is effective for solving real problems.

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Lecture notes : Introduction to Cyclic Covers in Algebraic Geometry