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In the last post, we embraced “good enough.” This sequel goes deeper into the math of how heuristics move across rugged fitness landscapes—the invisible mountains where every coordinate encodes a solution and its height is the objective value. If you want a warm-up, skim the intro first:
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Imagine you are standing in front of a vending machine. You are hungry, and you have exactly one minute before your bus arrives. Inside, there are 40 different snacks—chips, chocolates, nuts, granola bars.
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As renewable energy sources become increasingly integrated into power systems, a fundamental challenge emerges: how can communities design fair and efficient energy trading mechanisms that maximize renewable energy usage while ensuring all participants have incentive to participate, maintain grid stability, and operate profitably? This question becomes particularly critical as residential solar installations proliferate, creating distributed networks of energy producers and consumers who must make real-time decisions about energy trading. Unlike traditional centralized power grids, smart microgrids require sophisticated mathematical frameworks that integrate physical energy dynamics, strategic agent behavior, pricing mechanisms, and system stability—all operating under uncertainty from weather patterns, equipment failures, and market fluctuations.
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As solar photovoltaic (PV) systems become increasingly prevalent worldwide, a fundamental question emerges: are solar panel trackers—electromechanical devices that rotate panels to follow the sun—cost-effective investments, or are they merely expensive “toys” that fail to justify their additional cost? This question becomes particularly critical as installation costs, electricity prices, and system configurations vary dramatically across geographic locations, installation types (residential, commercial, industrial), and market conditions. A comprehensive mathematical framework is needed to evaluate whether the energy gains from tracking justify the additional capital costs, operational expenses, and maintenance requirements.
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As humanity has observed the night sky for millennia, different cultures have developed their own systems for dividing the celestial sphere into constellations—patterns of stars that facilitate navigation, storytelling, and astronomical identification. The modern International Astronomical Union (IAU) recognizes 88 constellations, but these traditional divisions emerged from historical and cultural contexts rather than systematic mathematical optimization. A fundamental question arises: can we redesign constellations using rigorous mathematical principles to create groupings that are simultaneously identifiable, well-separated, convenient for naked-eye observation, and characterized by recognizable patterns that facilitate memorization?
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As humanity’s presence in space expands with thousands of satellites now orbiting Earth, a critical challenge emerges at the intersection of orbital mechanics, game theory, and autonomous decision-making: how do we design collision avoidance systems that enable satellites to autonomously prevent catastrophic collisions while operating under fundamental constraints of orbital dynamics, limited fuel resources, and strategic interactions between multiple decision-makers?
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In the rapidly advancing field of computational ecology and agricultural systems modeling, a critical challenge emerges at the intersection of ecosystem dynamics, pollinator conservation, and agricultural productivity: how do we effectively model and understand the complex trade-offs between pollinator health and agricultural yield when management decisions have cascading effects across spatial and temporal scales?
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In the rapidly advancing field of scientific machine learning and computational physics, a critical challenge emerges at the intersection of deep learning, numerical methods, and inverse problem theory: how do we efficiently solve parametric partial differential equation (PDE) inverse problems—recovering unknown parameters from observed solutions—when forward solves are computationally expensive?
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This post explores a comprehensive physics-based framework I developed for modeling urban microclimates and optimizing thermal comfort through strategic tree placement. While addressing critical challenges in urban heat island mitigation and sustainable city design, this project provided an opportunity to apply rigorous mathematical modeling to real-world urban climate management problems.
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This post explores a comprehensive mathematical framework I developed for optimizing shared micro-mobility systems through advanced operations research techniques and sophisticated multi-objective optimization. While addressing complex challenges in urban transportation planning and sustainable mobility, this project provided an opportunity to apply rigorous mathematical modeling to real-world urban transportation management problems.
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This post explores a computational framework I developed for modeling and analyzing multi-species ecosystem dynamics through advanced mathematical techniques and sophisticated perturbation analysis. While addressing complex challenges in ecological stability and environmental response, this project provided an opportunity to apply rigorous mathematical modeling to real-world ecosystem management problems.
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This post explores a comprehensive computational physics framework I developed for modeling and controlling multi-degree-of-freedom robotic manipulators through advanced Lagrangian dynamics and sophisticated control strategies. While addressing complex challenges in robotic arm trajectory tracking and energy conservation, this project provided an opportunity to apply rigorous mathematical modeling to real-world robotic control problems.
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This post explores a comprehensive mathematical modeling framework I developed for optimizing urban traffic flow systems through advanced multi-objective optimization techniques. While addressing complex transportation challenges in dense urban environments, this project provided an opportunity to apply sophisticated quantitative methods to real-world traffic engineering problems.
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This post explores a comprehensive mathematical modeling framework I developed for optimizing sustainable urban mobility systems. While addressing complex transportation challenges in dense urban environments, this project provided an opportunity to apply advanced quantitative techniques to real-world urban planning problems.
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This post explores a mathematical modeling exercise I recently completed during the summer break. While the scenario may seem “utopian,” it provided a fascinating opportunity to apply advanced quantitative techniques to a complex problem.
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This post presents my comprehensive mathematical modeling framework for optimizing NYC’s Citi Bike system expansion. Through an integrated approach combining advanced forecasting, multi-objective optimization, and systems thinking, I transformed a fragmented planning challenge into a cohesive decision-support system that dramatically improves both operational performance and policy outcomes.
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This is a blog post that comes from my insights on the casino industry in Macau. Since I have been supervising two teams for participating the IMMC 2025, I ain’t gonna share the blog posts for the problems in the competition. However, I get inspired by some discussions from the elderlies during the lunch break at Cha Chaan Teng. The story is the elderlies were discussing the casino industry in Macau. They were quite satisfied with the current situation of the industry. However, they were also discussing the potential risks and challenges that the industry might face in the future. I was quite interested in the topic and I decided to write a blog post about it.
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This is the fifth blog post in the series of numerical analysis. The series aims to provide a comprehensive overview of numerical methods, their theoretical foundations, and practical applications. In this post, we explore the challenges and solutions of high-dimensional interpolation, focusing on modern approaches that overcome the limitations of traditional methods. We analyze the fundamental mathematical barriers of high-dimensional interpolation, discuss the theoretical insights, and examine the practical impact of these methods in scientific computing. The post concludes with a discussion of future directions and references for further study.
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This is the fourth blog post in the series of numerical analysis. In this blog post, we will discuss interpolation beyond polynomials, focusing on the theory and applications of radial basis functions (RBFs). We will explore the limitations of traditional interpolation methods, the theoretical foundations of RBFs, and their practical advantages in handling scattered data and high-dimensional problems. We will also discuss advanced topics, future directions, and best practices in RBF interpolation.
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This is the third blog post of the series on Numerical Analysis. In this post, we will discuss Iterative Methods for Large Linear Systems. The post will cover the theoretical foundations of iterative methods, including matrix splitting methods, Jacobi and Gauss-Seidel iterations, and the Conjugate Gradient method. We will also explore the convergence analysis of these methods and their practical implementation in Python. The post will conclude with a discussion of real-world applications and future directions in the field of iterative methods.
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This is another blog post of the series of numerical analysis. In this blog post, we will discuss advanced eigenvalue methods and their applications in scientific computing. We will cover the theoretical foundations of eigenvalue problems, including the power method, QR algorithm, and related theorems. We will also explore practical implementations in Python and discuss applications in engineering, quantum chemistry, and machine learning. Finally, we will examine computational challenges and future directions in the field of eigenvalue problems.
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This is the first blog post series of Numerical Analysis. In this series, we will discuss some advanced topics in scientific computing. We will cover the following topics:
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It has been a while since I last posted a blog; yet, I have finally finished all three problems of IMMC 2025 Autumn. Problem B is the blog post here, and Problem C is the blog post here. Blog post for Problem C is quite short since the problem is for junior students. In this blog post, I will discuss Problem A of IMMC 2025 Autumn.
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This is blog post for the HiMCM 2024 competition. The blog post is about the environmental footprint of High-Powered Computing (HPC) and the challenges and solutions for a sustainable digital future. The blog post covers the problem background, analysis, solution, discussion, and conclusion of the environmental impact of HPC systems. The analysis includes mathematical modeling, scenario analysis, Monte Carlo simulation, and visualization insights to address the environmental challenges of HPC. The solution provides actionable recommendations for reducing the environmental footprint of HPC while ensuring sustainable technological advancement.
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Follows the previous blog post, this is also another one about the IMMC 2025 competition. This time, I will be discussing the solution to Problem C, focusing on household plastic recycling sustainability. This fascinating problem combines environmental science with mathematical modeling to determine whether household plastic recycling truly helps combat global warming.
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Never thought I have not written any blog post for whole November. I have been busy with my work. But anyway, I am back to write a blog post for the International Mathematical Modeling Challenge (IMMC) 2025. This blog post is for the Problem B of the IMMC 2025. The problem is about the wave height measurement system developed for the Hong Kong University of Science and Technology (HKUST) landing step project in Sai Kung. The system integrates Acoustic Doppler Current Profilers (ADCP) with advanced mathematical modeling to provide accurate nearshore wave height predictions under extreme weather conditions. The analysis covers the mathematical model, system architecture, equipment requirements, and visualization insights, highlighting the system’s robust performance and real-time processing capabilities. The blog concludes with a discussion of the system’s impact, limitations, and future perspectives, emphasizing its potential for marine structure design and safety management.
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This blog presents a mathematical model for fair resource allocation in a Mars research base scenario, implementing multiple distribution strategies including maximum value allocation, Nash social welfare optimization, and social interaction-adjusted distributions. I develop algorithms in Python to solve these complex allocation problems and analyze their fairness metrics.
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It has been a while since I last posted. I have been busy with work and other personal projects. I am excited to share with you a new project that I have been working on. In this blog post, I will discuss the art and science of searching for lost objects. I will explore how Archimedean spiral transforms this everyday problem into a solvable equation. Using Python and mathematical modeling, I will uncover patterns that could revolutionize how I search for lost objects.
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This is another blog post continuing the series on Mathematical Modeling and Python, the problem comes from the MCM (Mathematical Contest in Modeling) competition. The problem is about building and optimizing a trading strategy about gold and bitcoun using Python. The problem is as follows:
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This is a blog post for the HiMCM 2021 Problem B. The blog post will cover the Lake Mead water crisis, including the factors affecting water volume, modeling water levels using SARIMAX and structural time series models, and developing innovative solutions to address the drought problem. The analysis will involve importing data, exploratory data analysis, time series analysis, and forecasting future water levels using mathematical models.
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This is a blog post about the 2021 MCM Problem A on modeling fungal decomposition of woody fibers.
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This is a blog post about HiMCM 2021 Problem A. The problem statement can be found here. Briefly, the problem is about designing a solar energy storage system for a small town.
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This is a blog post about the 2019 High School Mathematical Modeling Contest(HiMCM) Problem B. The problem is about modeling the impact of banning the sale of single-serving plastic water bottles in Concord and San Francisco. To be honest, I have participated in the HiMCM contest in 2019; however, I have not solved this problem well. Therefore, it would be a good opportunity for me to revisit this problem and provide a detailed analysis and solution.
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This is another blog post for the 2023 High School Mathematical Contest in Modeling (HiMCM). By taking a glance at the title, one may understand it is about e-buses. The problem statement in some sense is “easy” to understand, but it is not easy to solve appropriately. The problem statement is as follows:
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It is a mathematical modeling problem from MCM 2023 Problem A. The problem is as follows:
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This is another blog about Mathematical Modeling, Curve Fitting, and Python. In this blog, I will discuss the Problem B of the HiMCM 2022. The problem is about CO2 and Global Warming.
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This problem unlike the previous posts, is more “recent” in some sense. However, I also encountered this problem in a MCM competition (Even though I have not enrolled). Like the problems in MCM, this problem is also quite interesting and challenging.
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This problem is quite an “ancient” problem in the HiMCM competition. The problem is about the hydrographic data problem. Even though the problem is quite old, the problem is still relevant to the current situation. Moreover, I consider this problem can bring an insight into the problem-solving process in mathematical modeling. Therefore, I will discuss the problem in this blog post.
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School Busing is quite an interesting problem in real life. For me, as an adult with no vehicle, I have to rely on public transportation. Luckily, I live near my workplace. However, for children, they might have to travel a long distance to go to school. In some cases, they have to take a bus to go to school. This is where the school busing problem comes in. The school busing problem is a classic optimization problem that involves determining the most efficient way to transport students to and from school using a limited number of buses. The goal is to minimize the total cost of transportation while ensuring that all students are picked up and dropped off on time. In this blog post, we will discuss the school busing problem and how it can be solved using heuristic algorithms.
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This post is established based on the project I have read on the book “A First Course in Mathematical Modeling” by Frank R. Giordano et al. The project is about the spread of a contagious disease. The spread of a contagious disease can be modeled by a system of differential equations.
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This post is established based on the project I have read on the book “A First Course in Mathematical Modeling” by Frank R. Giordano et al. The project is about simulating the game of darts.
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Continuing with the series of posts on numerical analysis, in this post, I am going to discuss Curve Fitting. One may wonder what the difference between interpolation and curve fitting is. To illustrate, one may consider interpolation is a method of constructing a curve to go through all the data points, while curve fitting is a method of constructing a curve that seems to best represent the data points which may not necessarily go through all the data points.
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Continuing with the series of posts on numerical analysis, in this post, I will discuss the Gibbs phenomenon. Last time I discussed the Runge phenomenon, which is a problem that occurs when interpolating a function using high-degree polynomials. The Gibbs phenomenon is a similar problem that occurs when approximating a function with discontinuities.
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To begin with, one may wonder why we need to interpolate a function. The answer is simple: interpolation is a fundamental technique in numerical analysis that allows us to approximate a function using a set of discrete data points. By constructing an interpolating polynomial that passes through these points, we can estimate the function’s values at other points within the interval. This is essential in various fields, such as engineering, physics, and computer science, where accurate function approximation is required for modeling and simulation purposes.