Stories with Tag Mathematical Modeling

• Solving First Order Differential Equations with Julia

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  Posted: 2025-03-03 18:41:37

  Verbose tl;dr

  This blog post demonstrates how to solve first-order ordinary differential equations (ODEs) using Julia. It covers both symbolic and numerical solutions. For symbolic solutions, it utilizes the Symbolics.jl package to define symbolic variables and the DifferentialEquations.jl package's DSolve function. Numerical solutions are obtained using DifferentialEquations.jl's ODEProblem and solve functions, showcasing different solving algorithms. The post provides example code for solving a simple exponential decay equation using both approaches, including plotting the results. It emphasizes the power and ease of use of DifferentialEquations.jl for handling ODEs within the Julia ecosystem.

  This blog post provides a comprehensive introduction to solving first-order ordinary differential equations (ODEs) using the Julia programming language. It begins by establishing the fundamental concept of a first-order ODE, explaining that it involves a function and its first derivative, and highlighting its importance in modeling various dynamic processes across scientific disciplines. The post emphasizes the analytical approach initially, illustrating how separation of variables and integrating factors are employed to derive exact solutions for specific types of first-order ODEs.

  The core strength of the post lies in its demonstration of Julia's capabilities for numerical solutions when analytical methods are either cumbersome or impossible. The post introduces the DifferentialEquations.jl package, a powerful Julia library designed for solving differential equations efficiently and accurately. It meticulously explains how to set up and solve a first-order ODE using this package. The process involves defining the ODE function, specifying initial conditions, choosing a suitable numerical solver from the numerous options provided by the library, and finally generating the numerical solution. The post goes further by demonstrating how to visualize these numerical solutions using plotting functionalities, allowing for clear interpretation and analysis of the results.

  The post utilizes a concrete example of an exponential decay problem to illustrate both the analytical and numerical approaches. This example serves as a practical guide, showing how the mathematical concepts translate into Julia code. The clear and concise code examples, along with the detailed explanations, make it straightforward for readers to understand the implementation details and adapt the techniques to their own differential equation problems. The post not only provides a solution but also delves into interpreting the numerical results, emphasizing the importance of understanding the behavior of the system being modeled. In conclusion, the post serves as a valuable resource for anyone seeking to leverage the power of Julia for solving first-order differential equations, offering both theoretical understanding and practical implementation guidance.

  • Julia
  • Differential Equations
  • Ordinary Differential Equations
  • ODE
  • First Order ODE
  • Numerical Methods
  • Scientific Computing
  • Mathematical Modeling
  • programming
  • Tutorial
  • Computational Science

  Summary of Comments ( 29 )
  https://news.ycombinator.com/item?id=43245172

  Verbose tl;dr

  The Hacker News comments are generally positive about the blog post's clear explanation of solving first-order differential equations using Julia. Several commenters appreciate the author's approach of starting with the mathematical concepts before diving into the code, making it accessible even to those less familiar with differential equations. Some highlight the educational value of visualizing the solutions, praising the use of DifferentialEquations.jl. One commenter suggests exploring symbolic solutions using SymPy.jl alongside the numerical approach. Another points out the potential benefits of using Julia for scientific computing, particularly its speed and ease of use for tasks like this. There's a brief discussion of other differential equation solvers in different languages, with some favoring Julia's ecosystem. Overall, the comments agree that the post provides a good introduction to solving differential equations in Julia.

  The Hacker News post "Solving First Order Differential Equations with Julia" (https://news.ycombinator.com/item?id=43245172) has a modest number of comments, sparking a discussion around the use of Julia for solving differential equations and broader topics related to scientific computing.

  One commenter highlights the trade-off between performance and the "developer experience," suggesting that while Julia offers speed advantages, other languages like Python might be easier to work with, especially for those already familiar with the ecosystem. They specifically point out Python libraries like scipy.integrate.solve_ivp as a good alternative. This comment emphasizes the practical considerations beyond raw performance, like the learning curve and available tooling, when choosing a language for a particular task.

  Another comment chain discusses symbolic solutions for differential equations. One user mentions seeking symbolic solutions first and resorting to numerical methods only when necessary, while another introduces the Symbolics.jl package in Julia for symbolic computations. This exchange reflects a common workflow in scientific computing where exact solutions are preferred when available, and numerical methods are used as a fallback. The mention of Symbolics.jl provides a concrete resource for those interested in symbolic computing within the Julia ecosystem.

  A further comment emphasizes the educational value of the linked blog post, particularly for those unfamiliar with Julia's differential equation solving capabilities. This suggests that the post serves as a good introduction to this aspect of Julia.

  Finally, a comment thread explores alternative methods for solving differential equations, specifically mentioning finite element and finite difference methods. This broadens the discussion beyond the methods presented in the blog post and touches on other common numerical techniques for solving these types of problems.

  While the number of comments is not extensive, the discussion covers several pertinent points, including the practicality of using Julia for differential equations, the role of symbolic solutions, the educational value of the post, and alternative numerical methods. The comments offer valuable context and further avenues for exploration beyond the original blog post.

• Complex Systems and Quantitative Mereology

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  Posted: 2025-02-05 14:06:44

  Verbose tl;dr

  The blog post explores the potential of applying "quantitative mereology," the study of parts and wholes with numerical measures, to complex systems. It argues that traditional physics, focusing on fundamental particles and forces, struggles to capture the emergent properties of complex systems. Instead, a mereological approach could offer a complementary perspective by quantifying relationships between parts and wholes across different scales, providing insights into how these systems function and evolve. This involves defining measures of "wholeness" based on concepts like integration, differentiation, and organization, potentially leading to new mathematical tools and models for understanding emergent phenomena in areas like biology, economics, and social systems. The author uses the example of entropy to illustrate how a mereological view might reinterpret existing physical concepts, suggesting entropy as a measure of the distribution of energy across a system's parts rather than purely as disorder.

  The blog post "Complex Systems and Quantitative Mereology," authored by Abel Jansma, explores the intersection of complex systems theory and mereology, the study of part-whole relationships. Jansma posits that a more mathematically rigorous and quantifiable approach to mereology is crucial for understanding and modeling complex systems, which are characterized by emergent properties arising from the interactions of their constituent parts. He argues that traditional, qualitative mereology, while useful for philosophical discussions, lacks the precision needed to analyze the intricate dynamics within complex systems. This necessitates the development of what he terms "quantitative mereology" or, more provocatively, "mereophysics."

  Jansma outlines several key concepts that contribute to this emerging field. He emphasizes the importance of precisely defining and quantifying the contributions of individual parts to the overall system's behavior, moving beyond simple membership or inclusion. This involves considering not only the presence or absence of a part but also its quantitative influence, possibly weighted by factors like connectivity, interaction strength, or functional relevance. He suggests exploring concepts like "fractional participation," where a part might contribute only partially to a particular system property, and "weighted averaging" of part properties to understand emergent whole properties.

  The author further elaborates on the connection between mereology and information theory, proposing that the organization and interaction of parts within a system can be understood through the lens of information processing and integration. He suggests that quantifying the information flow between parts and how this information contributes to the system's overall functionality is a vital aspect of quantitative mereology. This approach resonates with the concept of integrated information theory, potentially offering a framework for measuring the degree of "wholeness" or integrated complexity of a system.

  Jansma acknowledges the challenges inherent in developing such a framework, particularly the difficulty in defining appropriate metrics for quantifying part-whole relationships and the complexity of modeling emergent behavior. He suggests drawing inspiration from existing fields like network theory, statistical mechanics, and information theory to develop the necessary mathematical tools and conceptual frameworks. He also highlights the potential applications of quantitative mereology in diverse fields, including biology, neuroscience, and social sciences, where understanding the interplay of parts is crucial for deciphering complex phenomena. Ultimately, the author envisions quantitative mereology as a bridge between reductionist and holistic approaches to understanding complex systems, offering a rigorous framework for analyzing how emergent properties arise from the intricate interactions of their constituent components.

  • Complex Systems
  • Mereology
  • Quantitative Mereology
  • physics
  • philosophy of science
  • Metaphysics
  • Parts and Wholes
  • Formal Ontology
  • Systems Theory
  • Mathematical Modeling
  • Emergence

  Summary of Comments ( 0 )
  https://news.ycombinator.com/item?id=42948642

  Verbose tl;dr

  HN users discussed the practicality and philosophical implications of applying mereology (the study of parts and wholes) to complex systems. Some expressed skepticism about quantifying mereology, questioning the usefulness of assigning numerical values to part-whole relationships, especially in fields like biology. Others were more receptive, suggesting potential applications in areas like network analysis and systems engineering. The debate touched on the inherent complexity of defining "parts" and "wholes" in different contexts, and whether a purely reductionist approach using mereology could capture emergent properties. Some commenters also drew parallels to other frameworks like category theory and information theory as potentially more suitable tools for understanding complex systems. Finally, there was discussion of the challenge of reconciling discrete, measurable components with the continuous nature of many real-world phenomena.

  The Hacker News post titled "Complex Systems and Quantitative Mereology," linking to an article exploring the application of mereology (the study of parts and wholes) to complex systems, has generated a modest discussion with a few insightful comments.

  One commenter questions the practical applicability of mereology, expressing skepticism about its usefulness beyond philosophical discussions. They wonder if it can truly offer new insights or tools for understanding and managing complex systems, or if it's primarily an academic exercise.

  Another commenter approaches the topic from a mathematical perspective, drawing parallels between mereology and measure theory. They suggest that measure theory, with its established framework for quantifying sets and their properties, might offer a more robust and practical approach to analyzing complex systems. This comment implies that mereology might be a less developed or less powerful tool for the same task.

  A different commenter focuses on the concept of "emergence" in complex systems. They argue that mereology, by focusing on the relationships between parts and wholes, might provide a valuable framework for understanding how emergent properties arise from the interactions of simpler components. This perspective suggests a potential strength of mereology in addressing a key challenge in complex systems research.

  Finally, one commenter expresses interest in the potential of applying mereology to specific domains like biology and ecology. They suggest that understanding the relationships between different levels of organization in these systems, from individual organisms to entire ecosystems, could benefit from a mereological approach. This points to potential practical applications of the concepts discussed in the linked article.

  While the discussion is not extensive, these comments highlight some of the key considerations surrounding the application of mereology to complex systems: its practical relevance, its relationship to other mathematical frameworks, its potential for understanding emergence, and its possible applications in specific scientific fields. The overall tone is one of cautious interest, with some skepticism balanced by the recognition of potential value.