Stories with Tag Infinite Series

• Weierstrass's Monster

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  Posted: 2025-01-24 03:02:10

  Verbose tl;dr

  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.

  In the realm of mathematical analysis, a field dedicated to the study of change and continuity, there exists a peculiar function known as the Weierstrass function, often dramatically dubbed "Weierstrass's Monster." This function, formulated by the eminent German mathematician Karl Weierstrass in 1872, presented a groundbreaking challenge to the conventional understanding of calculus, specifically the concept of differentiability. It possesses a truly remarkable property: while being undeniably continuous, meaning its graph can be drawn without lifting the pen, it is nowhere differentiable. This means that at no point on the curve can a tangent line be drawn, as the function experiences infinitely rapid oscillations at every scale.

  Before Weierstrass's discovery, mathematicians generally believed that continuous functions were differentiable almost everywhere, meaning they possessed derivatives at all but a few exceptional points, such as sharp corners or cusps. Weierstrass's function shattered this assumption, demonstrating the existence of functions that are continuous everywhere yet differentiable nowhere. This revelation sent shockwaves through the mathematical community, prompting a re-evaluation of the foundations of calculus and a deeper exploration of the subtle relationship between continuity and differentiability.

  The Weierstrass function is typically represented as an infinite sum of cosine functions with carefully chosen frequencies and amplitudes. These individual cosine waves, while smooth and differentiable on their own, combine in a way that produces the function's characteristic jaggedness and non-differentiability. This intricate construction demonstrates the power of infinite series to generate highly complex and counterintuitive mathematical objects.

  The discovery of the Weierstrass function was a pivotal moment in the history of mathematics. It exposed limitations in the intuitive understanding of continuity and differentiability, highlighting the importance of rigorous mathematical proof. It also spurred the development of more precise definitions and a deeper understanding of the subtleties of real analysis. Furthermore, the Weierstrass function served as a catalyst for the development of new branches of mathematics, such as fractal geometry, which deals with complex shapes exhibiting self-similarity at different scales. The "monster" ultimately helped mathematicians tame the wild and intricate landscape of functions, leading to a richer and more nuanced understanding of the mathematical universe. While initially perceived as a pathological example, it ultimately enriched and expanded the field, demonstrating the beauty and sometimes surprising nature of mathematical exploration.

  • mathematics
  • calculus
  • analysis
  • Weierstrass function
  • continuous function
  • nowhere differentiable
  • fractal
  • history of mathematics
  • Karl Weierstrass
  • Infinite Series
  • mathematical analysis

  Summary of Comments ( 81 )
  https://news.ycombinator.com/item?id=42810103

  Verbose tl;dr

  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.

  The Hacker News post titled "Weierstrass's Monster," linking to a Quanta Magazine article about the Weierstrass function, has a moderate number of comments discussing various aspects of the function and its historical context.

  Several commenters focus on the function's counterintuitive properties. One commenter highlights the disconnect between intuitive notions of smoothness and the mathematical definition of continuity, noting that the Weierstrass function is continuous everywhere but differentiable nowhere. This leads to a discussion about how mathematical definitions can sometimes defy our intuition based on real-world experience. Another user echoes this sentiment by explaining how the function subverts the expectation that a continuous curve should have a tangent at every point, pointing out the "infinite jaggedness" at all scales.

  Another thread discusses the historical significance of the function. A commenter notes that the Weierstrass function challenged the prevailing assumptions of 19th-century mathematicians about the relationship between continuity and differentiability, opening up new avenues of research in analysis. This sparked a brief exchange about other mathematical "monsters" and their role in advancing mathematical understanding. Someone further elaborates on the historical impact, mentioning that the function's discovery contributed to a more rigorous and formal approach to mathematical analysis, moving away from purely intuitive arguments.

  A few comments delve into the practical implications or applications of the function. While some acknowledge that direct real-world applications might be limited, one commenter suggests its potential relevance to modeling phenomena exhibiting fractal-like behavior, where roughness persists at arbitrarily small scales.

  Some comments offer additional resources or related concepts. One user points to a 3Blue1Brown video that visually explains the concept of the Weierstrass function. Another commenter connects the function to Brownian motion, mentioning the similar nowhere-differentiable nature of Brownian paths.

  Finally, some comments provide technical clarifications or corrections. One commenter points out a subtle distinction between nowhere differentiability and infinite jaggedness, emphasizing that infinite jaggedness is not a formally defined mathematical concept. Another user clarifies the historical context, pointing out that while Weierstrass's example is the most famous, he was not the first to discover such a function.

• Capacitors Meet Geometric Series

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  Posted: 2025-01-22 05:34:15

  Verbose tl;dr

  The post explores how the seemingly simple problem of calculating the equivalent capacitance of an infinite ladder network of capacitors can be elegantly solved using the concept of geometric series. By recognizing the self-similar nature of the circuit as sections are added, the problem is reduced to a quadratic equation where the equivalent capacitance of the infinite network is expressed in terms of the individual capacitances. This demonstrates a practical application of mathematical concepts to circuit analysis, highlighting the interconnectedness between seemingly disparate fields.

  The blog post "Capacitors Meet Geometric Series" delves into the fascinating interplay between electrical circuits involving capacitors and the mathematical concept of geometric series. It begins by establishing the fundamental principles of capacitance, reminding the reader that capacitors store electrical energy by accumulating charge, and that the voltage across a capacitor is directly proportional to the stored charge, with the capacitance acting as the constant of proportionality. The post then introduces the core problem: determining the equivalent capacitance of an infinite ladder network composed of identical capacitors. This network consists of an infinite repetition of a simple "L" shaped unit, formed by one capacitor in series with the parallel combination of another capacitor and the rest of the infinite network.

  The author elegantly tackles this problem by utilizing the self-similar nature of the infinite ladder. Since the network stretches infinitely, removing the first "L" section leaves a structure identical to the original, implying that the remaining network possesses the same equivalent capacitance as the entire network. This realization allows the author to formulate an equation relating the equivalent capacitance of the whole network to the capacitance of the individual capacitors forming the "L" section. This equation takes the form of a quadratic equation, as it involves the equivalent capacitance both linearly and squared, due to the series and parallel combinations within the “L” structure.

  The post proceeds to solve the quadratic equation, deriving two possible solutions for the equivalent capacitance. The author then meticulously explains why only one of these solutions is physically meaningful. The negative solution is discarded as capacitance is a positive physical quantity representing the ability to store energy, and a negative capacitance would imply a negative energy storage capability, which is not physically realizable. The remaining positive solution is presented as the equivalent capacitance of the infinite ladder network. This solution exhibits a direct relationship with the individual capacitance and also contains the golden ratio, a mathematical constant with a rich history and numerous appearances in fields ranging from art and architecture to natural phenomena.

  Finally, the post concludes by highlighting the surprising connection between the seemingly disparate fields of electrical circuit analysis and pure mathematics, specifically the concept of geometric series. The equivalent capacitance is identified as being related to a geometric series through the quadratic equation’s relationship to the golden ratio, further enriching the exploration of the intricate link between these two domains. The author implicitly suggests that this example serves as a testament to the power of mathematical tools in understanding and simplifying complex physical systems.

  • Capacitors
  • Electronics
  • Circuit Analysis
  • Geometric Series
  • mathematics
  • physics
  • electrical engineering
  • Series and Parallel Circuits
  • RC Circuits
  • Infinite Series
  • Convergence
  • Impedance
  • Reactance

  Summary of Comments ( 0 )
  https://news.ycombinator.com/item?id=42789372

  Verbose tl;dr

  HN commenters generally praised the article for its clear explanation of how capacitors work, particularly its use of the geometric series analogy to explain charging and discharging. Some appreciated the interactive diagrams, while others suggested minor improvements like adding a discussion of dielectric materials and their impact on capacitance. One commenter pointed out a potential simplification in the derivation by using the formula for the sum of a geometric series directly. Another highlighted the importance of understanding the underlying physics rather than just memorizing formulas, praising the article for facilitating this understanding. A few users also shared related resources and alternative explanations of capacitor behavior.

  The Hacker News post "Capacitors Meet Geometric Series" discussing the article on capacitors and geometric series sparked a moderately active discussion with several insightful comments.

  One commenter appreciated the clear explanation of how the infinite series formula applies to this capacitor setup, mentioning how this helped solidify their understanding of the concept. They specifically enjoyed seeing the connection between the theoretical formula and a practical application, finding it more engaging than just rote memorization.

  Another user delved into the practical limitations of this theoretical model. They pointed out that the infinite series model assumes ideal components and doesn't account for real-world factors like parasitic capacitance and inductance, which become more significant at higher frequencies. They also noted the physical impossibility of truly infinitely small capacitors, suggesting that the model breaks down as the capacitor size approaches the atomic scale.

  Furthering this practical discussion, another comment highlighted the potential dangers of high voltage experiments involving capacitors, particularly when dealing with large capacitances. They emphasized the importance of safety precautions and a deep understanding of the involved energies.

  One commenter shared a related anecdote about using capacitors to model diffusion processes, showcasing another practical application of these electrical components in understanding different physical phenomena.

  There was also a brief discussion about visualizing the current flow in such a setup, with a user suggesting a diagram depicting how the current progressively diminishes through each capacitor in the series. Another commenter raised a question about the convergence of the series in relation to the physical sizes of the capacitors, but this did not receive a direct response.

  Finally, one comment simply expressed appreciation for the elegance and simplicity of the presented analysis.

  Overall, the comment section provided a blend of appreciation for the clear explanation of the core concept, alongside insightful considerations of practical limitations and real-world applications. While not extensive, the discussion adds valuable context and depth to the original article's presentation of capacitors and geometric series.