The hottest Mathematics Substack posts right now

And their main takeaways
Category
Top Science Topics

Not so fast, Mr. Fourier!

lcamtuf’s thing β€’ 2332 implied HN points β€’ 12 Mar 24
πŸ”¬ Science Computing Signal Processing Mathematics Algorithms Engineering
  1. The discrete Fourier transform (DFT) is a crucial algorithm in modern computing, used for tasks like communication, image and audio processing, and data compression.
  2. DFT transforms time-domain waveforms into frequency domain readings, allowing for analysis and manipulation of signals like isolating instruments or applying effects like Auto-Tune in music.
  3. Fast Fourier Transform (FFT) optimizes DFT by reducing the number of necessary calculations, making it more efficient for large-scale applications in computing.

β€œMath is hard” β€” if you are an LLM – and why that matters

Marcus on AI β€’ 4772 implied HN points β€’ 19 Oct 23
πŸ”¬ Science Mathematics Artificial Intelligence Machine Learning
  1. Even with massive data training, AI models struggle to truly understand multiplication.
  2. LLMs perform better in arithmetic tasks than smaller models like GPT but still fall short compared to a simple pocket calculator.
  3. LLM-based systems generalize based on similarity and do not develop a complete, abstract, reliable understanding of multiplication.

Inside FunSearch: Google DeepMind’s LLM that Discovered New Math and Computer Science Algorithms

TheSequence β€’ 1106 implied HN points β€’ 18 Jan 24
πŸ•Ή Technology AI Algorithms Computer Science Mathematics Machine Learning
  1. Discovering new science is a significant challenge for AI models.
  2. Google DeepMind's FunSearch model can generate new mathematics and computer science algorithms.
  3. FunSearch uses a Language Model to create computer programs and iteratively search for solutions in the function space.

Malcolm in the middle (well, at the beginning)

Logging the World β€’ 299 implied HN points β€’ 07 Mar 24
πŸ”¬ Science Communication Mathematics Education
  1. Using interesting anecdotes or 'Malcolms' at the beginning can engage a wider audience and make complex topics more appealing.
  2. Balancing academic style writing with engaging storytelling can make science communication more effective and impactful.
  3. Integrating rhetorical tricks and interesting facts can drive curiosity and encourage broader audiences to explore complex subjects.
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Fast Biology

Asimov Press β€’ 270 implied HN points β€’ 20 Feb 24
πŸ”¬ Science Biology Physics Mathematics Experimentation
  1. The concept of viewing time differently through the lens of the Minute Man and the Millennium Man prompts questions about our understanding of speed and time in the world.
  2. Biological processes at the cellular level can occur at astonishing speeds, with enzymes performing millions of chemical reactions per second and protein 'motors' spinning thousands of times a minute.
  3. Scientists use innovative experiments to directly observe rapid biological processes, such as watching ATP synthase spin or tracking ribosomes moving along messenger RNA strands, to gain a deeper understanding of the intricate workings of life.

Philosophy is the mathematics of language

Wood From Eden β€’ 816 implied HN points β€’ 23 Dec 23
πŸ“– Philosophy Language Mathematics Concepts Logic Epistemology
  1. Philosophy is the art of clarifying concepts and finding links between them.
  2. Philosophy is similar to mathematics in that it explores relationships between concepts, just as mathematics explores relationships between numbers.
  3. Concepts in philosophy change over time, making it a field that evolves constantly unlike mathematics which is built on stable concepts.

Gain of Function.

News Items β€’ 471 implied HN points β€’ 18 Jan 24
πŸ”¬ Science AI Mathematics Reasoning Training
  1. AlphaGeometry AI system solves complex geometry problems as well as a human Olympiad gold-medalist.
  2. AlphaGeometry combines neural language model with a rule-bound deduction engine for reasoning.
  3. Development of AlphaGeometry highlights AI's logic reasoning progress and ability to discover and verify new knowledge.

mathiness instills confidence

bad cattitude β€’ 165 implied HN points β€’ 22 Feb 24
πŸ”¬ Science Mathematics Psychology Cognitive Science Research Climate Science
  1. Mathiness can make people feel more confident, especially if they aren't familiar with math.
  2. Adding complex math or 'mathiness' to information can influence how people perceive its quality, especially if they lack knowledge in math and models.
  3. It's important to be cautious of trusting information just because it includes numbers or complex equations; don't assume accuracy or rigor without verifying.

Emerging Stars

Compounding Quality β€’ 216 implied HN points β€’ 08 Feb 24
πŸ”¬ Science Physics Mathematics
  1. Isaac Newton is famous for his Laws of Motion in physics and mathematics.
  2. Newton's Laws of Motion are fundamental to classical mechanics and still widely used today.
  3. This post about Newton's Laws of Motion is for paid subscribers.

Next phase, next stage, next craze, next wave

Logging the World β€’ 538 implied HN points β€’ 02 Dec 23
πŸ”¬ Science Mathematics COVID-19 Exponential growth Epidemiology
  1. Understanding exponential growth in infection rates can help predict future COVID trends.
  2. Individual growth rates of different strains impact the overall daily growth rate, following a weighted average principle.
  3. Market share of strains, not just reaching a specific percentage threshold, influences overall infection growth.

105 trillion digits of pi

A Piece of the Pi: mathematics explained β€’ 60 implied HN points β€’ 15 Mar 24
πŸ”¬ Science Mathematics Algorithms
  1. The number pi has now been calculated to 105 trillion decimal places using the Chudnovsky algorithm over 75 days.
  2. Ramanujan's formula for pi has been expanded and improved upon over the years, with the Chudnovsky brothers developing a formula that computes pi to 13 decimal places.
  3. Bellard's formula and the BBP formula provide ways to compute specific digits of pi without having to calculate all earlier digits, making validations faster and more efficient.

Fun with Serial and Parallel Sum Reductions

CPU fun β€’ 121 implied HN points β€’ 22 Feb 24
πŸ”¬ Science Mathematics Computer Science Algorithms
  1. Floating point arithmetic can be more complex than expected, especially due to limited mantissa bits, affecting the accuracy of calculations.
  2. Complaining about OpenMP reductions giving 'the wrong answer' is misguided; the issue likely existed in the serial code and is now being exposed.
  3. Changing the type of the accumulator to 'double' can help resolve issues with floating point arithmetic and accuracy during sum reductions.

The devil is old

Wednesday Wisdom β€’ 113 implied HN points β€’ 21 Feb 24
πŸ•Ή Technology Programming Mathematics Physics Concurrency Software Development
  1. Experience and age often bring wisdom, knowledge, and a unique perspective.
  2. In technology, while tools and capabilities have evolved, fundamental principles like people dynamics, math, and physics remain constant.
  3. Despite advancements, people still struggle with basic math, concurrent programming, and effective communication in group settings.

So... what is the lottery ticket hypothesis [Math Mondays]

Technology Made Simple β€’ 159 implied HN points β€’ 05 Feb 24
πŸ•Ή Technology Deep Learning Neural Networks Mathematics AI Machine Learning
  1. The Lottery Ticket Hypothesis proposes that within deep neural networks, there are subnetworks capable of achieving high performance with fewer parameters, leading to smaller and faster models.
  2. Successful application of the Lottery Ticket Hypothesis relies on iterative magnitude pruning strategies, with potential benefits like faster learning and higher accuracy.
  3. The hypothesis works due to factors like favorable gradients, implicit regularization, and data alignment, but challenges like scalability and interpretability remain towards practical implementation.

cool story, dad

moontower: a stoner dad explains options trading to his kids β€’ 117 implied HN points β€’ 11 Feb 24
🚌 Education Parenting Mathematics Investing Personal Growth Learning
  1. Kids often learn from their parents through modeling and osmosis rather than direct instruction.
  2. Teaching someone else's child may sometimes be more effective in imparting knowledge and lessons.
  3. Finding learning moments in everyday activities or tailored to a child's interests can help them grasp concepts effectively and make the lessons their own.

Fast Computation of Fibonacci Numbers: Part II

Confessions of a Code Addict β€’ 288 implied HN points β€’ 12 Nov 23
πŸ•Ή Technology Mathematics Programming Algorithms Education
  1. A new method to compute Fibonacci numbers using a closed-form expression without having to resort to floating point arithmetic.
  2. Representation of irrational numbers using two parts can be done in code allowing for precise computation of Fibonacci numbers.
  3. Understanding rings and implementing arithmetic operations within it can help in computing Fibonacci numbers without any loss of precision.

About "artificial intuition"

Silicon Reckoner β€’ 98 implied HN points β€’ 11 Jan 24
πŸ”¬ Science Mathematics Artificial Intelligence Research Intuition Collaboration
  1. Mathematicians have two sides to their work: creating new ideas and proving statements.
  2. Artificial General Intelligence (AGI) could potentially encompass all human competences, including mathematical creativity.
  3. Artificial Intuition is being explored to assist mathematicians in generating new ideas and collaborating with AI.

BA.2.86 growth rates

Logging the World β€’ 418 implied HN points β€’ 23 Aug 23
πŸ”¬ Science Statistics Genetics Epidemiology Research Mathematics
  1. New COVID variant BA.2.86 has mutations that suggest fast growth, but estimating its growth rate is tricky.
  2. Statisticians use models and likelihood functions to estimate parameters like growth rates, but uncertainty exists in the estimates.
  3. The work of statistician C.R. Rao, like the Fisher information, shows fundamental limits to parameter estimation and the role of geometry in statistics.

A Bridging of Dimensions

α΄‹ΚŸα΄€α΅Ύs β€’ 628 implied HN points β€’ 15 Jun 23
πŸ”¬ Science Physics Mathematics Biology Neuroscience Consciousness
  1. Former government officials have revealed details about UFO crash retrieval programs involving non-human intelligence and advanced materials.
  2. The use of topological materials in UFO technology could explain exotic properties, like strange isotopes and materials able to deform into higher dimensions.
  3. Connections between the human brain's multi-dimensional functions and UFO phenomena could suggest a link between consciousness and unexplained aerial phenomena.

Introducing QF's Advanced Mathematics GPT

Quantum Formalism β€’ 59 implied HN points β€’ 02 Feb 24
πŸ•Ή Technology Mathematics
  1. QF has introduced an Advanced Mathematics GPT as a learning assistant for graduate-level mathematics.
  2. The new GPT covers advanced branches of pure mathematics like Abstract Algebra, Algebraic Geometry, and Differential Geometry, allowing users to ask questions on various topics.
  3. Access to the Advanced Mathematics GPT is currently restricted to users with ChatGPT Plus or higher subscription plans.

Maths until 18

Logging the World β€’ 418 implied HN points β€’ 15 Aug 23
🚌 Education Mathematics Teaching Curriculum
  1. The proposal for compulsory math education until age 18 in the UK received mixed reactions, highlighting the importance of making math appealing and accessible to a wide audience.
  2. Implementing math education until 18 requires consideration of factors like shortage of math teachers and effective delivery methods such as leveraging online resources.
  3. Math education should cover areas such as practical number skills, understanding uncertainty and randomness, and exploring connections between math and other subjects like art and music.

The Dragon Every Math Teacher Has to Slay Eventually

Mathworlds β€’ 569 implied HN points β€’ 22 Jun 23
🚌 Education Teaching Mathematics
  1. Students often feel worse about math class compared to other subjects because of the pressure to only have one correct answer for each question.
  2. Math should be taught as a creative discipline that embraces human subjectivity, not just a set of memorized steps.
  3. Teachers can help students deconstruct the idea of one right way to do math by introducing activities that show multiple paths lead to the same solution.

Some results on income distributions and total utility

Splitting Infinity β€’ 59 implied HN points β€’ 28 Jan 24
πŸ”¬ Science Mathematics Statistics Economics
  1. The type of income distribution models used like Pareto or lognormal can impact total utility calculations in economics
  2. There is an interesting relationship observed where the degree of inequality doesn't directly correlate with total utility in certain scenarios
  3. Introducing more risk-averse utility functions can bring the focus back on the importance of inequality in calculations

Picturing symmetry

A Piece of the Pi: mathematics explained β€’ 24 implied HN points β€’ 04 Mar 24
πŸ”¬ Science Symmetry Mathematics Group Theory
  1. The order in which symmetries are applied can significantly affect the final result, as shown through reflections and rotations of a square.
  2. Using Cayley graphs can help visualize and calculate products of symmetries.
  3. In symmetry operations, combining reflections and rotations follows specific rules, similar to adding odd and even numbers. Grouping rotations and reflections can simplify understanding complex symmetries.

Introducing QF GPT

Quantum Formalism β€’ 59 implied HN points β€’ 26 Jan 24
🚌 Education Quantum Mechanics Mathematics
  1. QF GPT is a learning assistant tool for those studying quantum mechanics and quantum information science, especially those struggling with mathematical definitions.
  2. QF GPT utilizes ChatGPT's capabilities to provide accurate responses most of the time and encourages users to correct errors to solidify their understanding.
  3. Access to QF GPT is currently limited to users with ChatGPT Plus or higher subscription plans, but the hope is to make it accessible to the entire QF community in the future for advanced mathematical learning in QM and QIS.

The infinite monkey theorem

A Piece of the Pi: mathematics explained β€’ 18 implied HN points β€’ 11 Mar 24
πŸ”¬ Science Mathematics Literature Research Experiment Evolution
  1. The infinite monkey theorem states that given enough time and randomness, a monkey could type out the complete works of Shakespeare on a keyboard.
  2. Generating longer phrases by random means, as shown in simulations, becomes exponentially more difficult as the phrase length increases.
  3. The famous infinite monkey paradox has been explored through history, including Cicero's speculation in 45 BC and modern computer simulations using actual monkeys with disappointing results.

How To Fix Math(s) Education

Insight Axis β€’ 276 implied HN points β€’ 11 Sep 23
🚌 Education Mathematics Teaching Problem Solving Curriculum
  1. Math education should focus on real-world problems to make it interesting and meaningful for students.
  2. Students should be taught a structured process of defining, abstracting, computing, and interpreting problems in math.
  3. School math should prioritize applied mathematics to show the practical utility of math, cater to the majority, and prepare students for the future.

Genius in turbulent times

Logging the World β€’ 418 implied HN points β€’ 05 Jul 23
πŸ”¬ Science Mathematics History Biology Diversity
  1. Genius can be found in lesser-known figures like Kolmogorov, who made significant contributions to mathematics and other fields.
  2. Kolmogorov's work on probability theory and the Kolmogorov-Arnold theorem had a lasting impact on mathematics and even underpins modern AI algorithms.
  3. Kolmogorov's life was not only marked by academic achievements but also by navigating personal challenges, such as opposing Lysenkoism and living as an openly gay man in Stalinist Russia.

A Linear Algebra Trick for Computing Fibonacci Numbers Fast

Confessions of a Code Addict β€’ 158 HN points β€’ 05 Nov 23
πŸ”¬ Science Mathematics Computing Algorithms Linear Algebra Complexity
  1. A linear algebra technique can be applied to compute Fibonacci numbers quickly with a logarithmic time complexity.
  2. Efficient algorithms like repeated squaring can compute powers of matrices in logarithmic time, improving performance for Fibonacci number calculations.
  3. A closed form expression using the golden ratio offers a direct method to compute Fibonacci numbers, showing different approaches with varied performance.

Four ways of thinking

Logging the World β€’ 199 implied HN points β€’ 28 Sep 23
πŸ”¬ Science Mathematics Complexity Statistics Chaos Evolution
  1. The book 'Four Ways of Thinking' by David Sumpter discusses four philosophies that map onto the four types of cellular automata identified by Stephen Wolfram, with historical anecdotes and life lessons.
  2. The book explores statistical, interactive, chaotic, and complex ways of thinking, connecting topics like cellular automata, chaos theory, and modern statistics with practical applications.
  3. David Sumpter's book introduces the complexity of modern mathematical research, showcasing the emergence of complicated behavior from simple rules and the fascinating concept of quantifying complexity in patterns.

Edge colourings and Archimedean lattices

A Piece of the Pi: mathematics explained β€’ 24 implied HN points β€’ 18 Feb 24
πŸ”¬ Science Mathematics Graphs Symmetry
  1. Edge colorings of graphs are not just recreational, but have practical applications in quantum technology.
  2. Graphs can be colored either by edges or by vertices, with different requirements for each coloring approach.
  3. Vizing's Theorem states that a graph can be edge colored with either the maximum degree or the maximum degree plus one colors.

How We Set Math Class up to Fail

Mathworlds β€’ 373 implied HN points β€’ 26 May 23
🚌 Education Mathematics Teaching Learning Abstract
  1. Math class often focuses on moving students towards abstract concepts, neglecting the value of concrete understanding.
  2. Teachers who can help students transition between concrete and abstract knowledge effectively engage students in math.
  3. Including both concrete and abstract elements in math problems can make learning more engaging and effective.