The blog post details the surprisingly difficult journey to find the exact value of the integral of the secant cubed function, a problem that stumped mathematicians for a century. While the integral's relevance to the Mercator projection made an approximate solution necessary for navigation since the 1600s, the precise antiderivative remained elusive. The post highlights the unexpected complexity arising from seemingly simple trigonometric functions and celebrates how a collaborative effort using symbolic computation software finally cracked the problem in 2020, yielding a closed-form solution involving complex logarithms. This solution, though exact, is less practical for computation than previously known series approximations. The post ultimately underscores the persistent challenge of symbolic integration and the power of modern computational tools.
This blog post provides a gentle introduction to automatic differentiation (AD), explaining how it computes derivatives of functions efficiently. It focuses on the forward mode of AD, building the concept from basic calculus and dual numbers. The post illustrates the process with clear, step-by-step examples, calculating derivatives of simple functions like f(x) = x² + 2x + 1 and more complex composite functions. It demonstrates how to implement forward mode AD in Python, emphasizing the recursive nature of the computation and how dual numbers facilitate tracking both function values and derivatives. The post concludes by hinting at the reverse mode of AD, a more efficient approach for functions with many inputs.
HN users generally praised the article for its clear explanation of automatic differentiation (AD), particularly its focus on building intuition and avoiding unnecessary jargon. Several commenters appreciated the author's approach of starting with simple examples and progressively building up to more complex concepts. Some highlighted the article's effectiveness in explaining the difference between forward and reverse mode AD. A few users with experience in machine learning frameworks like TensorFlow and PyTorch pointed out that understanding AD's underlying principles is crucial for effective use of these tools. One commenter noted the article's relevance to fields beyond machine learning, such as scientific computing and optimization. A minor point of discussion revolved around the nuances of terminology, specifically the distinction between "dual numbers" and other approaches to representing derivatives.
This post provides a gentle introduction to stochastic calculus, focusing on the Ito Calculus. It begins by explaining Brownian motion and its unusual properties, such as non-differentiability. The post then introduces Ito's Lemma, a crucial tool for manipulating functions of stochastic processes, highlighting its difference from the standard chain rule due to the non-zero quadratic variation of Brownian motion. Finally, it demonstrates the application of Ito's Lemma through examples like geometric Brownian motion, used in option pricing, and illustrates its role in deriving the Black-Scholes equation.
HN users largely praised the clarity and accessibility of the introduction to stochastic calculus, especially for those without a deep mathematical background. Several commenters appreciated the author's approach of explaining complex concepts in a simple and intuitive way, with one noting it was the best explanation they'd seen. Some discussion revolved around practical applications, including finance and physics, and different approaches to teaching the subject. A few users suggested additional resources or pointed out minor typos or areas for improvement. Overall, the post was well-received and considered a valuable resource for learning about stochastic calculus.
"Matrix Calculus (For Machine Learning and Beyond)" offers a comprehensive guide to matrix calculus, specifically tailored for its applications in machine learning. It covers foundational concepts like derivatives, gradients, Jacobians, Hessians, and their properties, emphasizing practical computation and usage over rigorous proofs. The resource presents various techniques for matrix differentiation, including the numerator-layout and denominator-layout conventions, and connects these theoretical underpinnings to real-world machine learning scenarios like backpropagation and optimization algorithms. It also delves into more advanced topics such as vectorization, chain rule applications, and handling higher-order derivatives, providing numerous examples and clear explanations throughout to facilitate understanding and application.
Hacker News users discussed the accessibility and practicality of the linked matrix calculus resource. Several commenters appreciated its clear explanations and examples, particularly for those without a strong math background. Some found the focus on differentials beneficial for understanding backpropagation and optimization algorithms. However, others argued that automatic differentiation makes manual matrix calculus less crucial in modern machine learning, questioning the resource's overall relevance. A few users also pointed out the existence of other similar resources, suggesting alternative learning paths. The overall sentiment leaned towards cautious praise, acknowledging the resource's quality while debating its necessity in the current machine learning landscape.
The blog post demonstrates how to implement symbolic differentiation using definite clause grammars (DCGs) in Prolog. It leverages the elegant, declarative nature of DCGs to parse mathematical expressions represented as strings and simultaneously construct their derivative. By defining grammar rules for basic arithmetic operations (addition, subtraction, multiplication, division, and exponentiation), including the chain rule and handling constants and variables, the Prolog program can effectively differentiate a wide range of expressions. The post highlights the concise and readable nature of this approach, showcasing the power of DCGs for tackling symbolic computation tasks.
Hacker News users discussed the elegance and power of using definite clause grammars (DCGs) for symbolic differentiation, praising the conciseness and declarative nature of the approach. Some commenters pointed out the historical connection between Prolog and DCGs, highlighting their suitability for symbolic computation. A few users expressed interest in exploring further applications of DCGs beyond differentiation, such as parsing and code generation. The discussion also touched upon the performance implications of using DCGs and compared them to other parsing techniques. Some commenters raised concerns about the readability and maintainability of complex DCG-based systems.
This post provides a gentle introduction to stochastic calculus, focusing on the Ito integral. It explains the motivation behind needing a new type of calculus for random processes like Brownian motion, highlighting its non-differentiable nature. The post defines the Ito integral, emphasizing its difference from the Riemann integral due to the non-zero quadratic variation of Brownian motion. It then introduces Ito's Lemma, a crucial tool for manipulating functions of stochastic processes, and illustrates its application with examples like geometric Brownian motion, a common model in finance. Finally, the post briefly touches on stochastic differential equations (SDEs) and their connection to partial differential equations (PDEs) through the Feynman-Kac formula.
HN users generally praised the clarity and accessibility of the introduction to stochastic calculus. Several appreciated the focus on intuition and the gentle progression of concepts, making it easier to grasp than other resources. Some pointed out its relevance to fields like finance and machine learning, while others suggested supplementary resources for deeper dives into specific areas like Ito's Lemma. One commenter highlighted the importance of understanding the underlying measure theory, while another offered a perspective on how stochastic calculus can be viewed as a generalization of ordinary calculus. A few mentioned the author's background, suggesting it contributed to the clear explanations. The discussion remained focused on the quality of the introductory post, with no significant dissenting opinions.
The Graphics Codex is a comprehensive, free online resource for learning about computer graphics. It covers a broad range of topics, from fundamental concepts like color and light to advanced rendering techniques like ray tracing and path tracing. Emphasizing a practical, math-heavy approach, the Codex provides detailed explanations, interactive diagrams, and code examples to facilitate a deep understanding of the underlying principles. It's designed to be accessible to students and professionals alike, offering a structured learning path from beginner to expert levels. The resource continues to evolve and expand, aiming to become a definitive and up-to-date guide to the field of computer graphics.
Hacker News users largely praised the Graphics Codex, calling it a "fantastic resource" and a "great intro to graphics". Many appreciated its practical, hands-on approach and clear explanations of fundamental concepts, contrasting it favorably with overly theoretical or outdated textbooks. Several commenters highlighted the value of its accompanying code examples and the author's focus on modern graphics techniques. Some discussion revolved around the choice of GLSL over other shading languages, with some preferring a more platform-agnostic approach, but acknowledging the educational benefits of GLSL's explicit nature. The overall sentiment was highly positive, with many expressing excitement about using the resource themselves or recommending it to others.
Karl Weierstrass’s function revolutionized mathematics by demonstrating a curve that is continuous everywhere but differentiable nowhere. This “monster” function, built from an infinite sum of cosine waves with increasingly higher frequencies and smaller amplitudes, visually appears jagged and chaotic at every scale. Its existence challenged the prevailing assumption that continuous functions were mostly smooth, with only isolated points of non-differentiability. Weierstrass's discovery exposed a deep rift between intuition and mathematical rigor, ushering in a new era of analysis focused on precise definitions and rigorous proofs, impacting fields from calculus to fractal geometry.
HN users generally express fascination with the Weierstrass function and its implications for calculus. Several comments dive into the history and significance of the function, appreciating its role in challenging intuitive notions of continuity and differentiability. Some discuss its relation to fractals and Brownian motion, while others highlight the beauty of mathematical discoveries that defy expectations. A few commenters provide additional resources, including links to visualizations and related mathematical concepts like space-filling curves. Some debate the accessibility of the original Quanta article, suggesting ways it could be more easily understood by a broader audience. A recurring theme is the wonder inspired by such counterintuitive mathematical objects.
Summary of Comments ( 42 )
https://news.ycombinator.com/item?id=43741273
HN users discuss the surprising difficulty of integrating the secant function, historically important for mapmaking. Several commenters highlight the multiple approaches to the solution, appreciating the blog post's clear explanation of the "circular reasoning" some methods employ. Some debate the true "difficulty" of the integral, pointing out that with knowledge of complex logarithms, it's straightforward. Others note the historical context, where these tools were not yet available, emphasizing the ingenuity of earlier mathematicians. A few commenters dive into specific techniques, like differentiating under the integral sign and the usefulness of Weierstrass substitutions. The overall sentiment is one of appreciation for the blog post shedding light on a seemingly simple yet historically significant integral.
The Hacker News post "100 Years to Solve an Integral (2020)" discussing the article about the secant integral has several comments exploring various aspects of the topic.
Some commenters delve into the historical context of the problem, highlighting the contributions of Isaac Newton and James Gregory in developing methods to calculate the integral, and emphasizing the long-standing interest in this mathematical puzzle. They discuss the importance of the integral in navigation and map projection, linking it to the Mercator projection used in nautical charts.
A few comments focus on the technical details of the solution, discussing different approaches to solving the integral, including the use of complex analysis and power series expansions. They also touch upon the computational aspects of evaluating the integral numerically, and the challenges involved in obtaining accurate results.
Several commenters express appreciation for the clear and accessible explanation provided in the linked article, praising the author's ability to break down a complex mathematical concept into understandable terms. They find the connection between seemingly abstract mathematics and practical applications like mapmaking particularly fascinating.
There's a discussion about the significance of the integral's solution, with some commenters emphasizing its historical importance and its contribution to our understanding of calculus. Others point out the practical implications of the integral in modern applications, including navigation and surveying.
A few commenters share their own experiences with the integral, recounting their attempts to solve it and the challenges they encountered. They also discuss the educational value of exploring such problems, highlighting the importance of mathematical curiosity and problem-solving skills.
Some commenters draw parallels between this integral and other mathematical puzzles, discussing the broader theme of unsolved problems in mathematics and the ongoing quest for solutions. They also touch upon the role of collaboration and communication in advancing mathematical knowledge.
Finally, some comments offer additional resources and references related to the integral and its applications, providing further avenues for exploration and learning. They also suggest alternative approaches to understanding the integral and its implications.