The hottest Algebra Substack posts right now

And their main takeaways
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Natural Ordinal Addition

Infinitely More • 7 implied HN points • 15 Mar 26
🔬 Science Algebra
  1. The natural sum and product (Hessenberg operations) make the ordinals into a commutative semiring, contrasting with standard ordinal arithmetic where addition and multiplication are not commutative.
  2. The natural ordinal operations match the operations on surreal numbers, so the ordinals under natural addition and multiplication form a subsemiring of the surreals.
  3. There are five independent, self-contained ways to define the natural sum and product—order-theoretic, computational, proof-theoretic, and others—and all five are equivalent, giving complementary perspectives and routes to generalization.

Analyze FIR filters using just high-school algebra

filterwizard • 39 implied HN points • 23 Sep 24
🕹 Technology Algebra
  1. FIR filters have a finite impulse response, meaning they only remember a limited amount of past input. This makes them predictable and stable, especially for applications needing fast settling times.
  2. You can think of FIR filter coefficients as a polynomial, which allows you to use algebra to analyze and create filters. This approach helps in understanding how changing coefficients affects the filter's behavior.
  3. By factoring the polynomial of an FIR filter, you can create smaller filters that combine to produce the same overall effect. This technique allows for a deeper exploration of filter design, giving you more control over the filter's characteristics.

Indecomposable Ordinals

Infinitely More • 38 implied HN points • 01 Feb 26
🔬 Science Algebra
  1. An ordinal λ>0 is additively indecomposable when the sum of any two smaller ordinals is still below λ; equivalently, it cannot be expressed as a sum of two smaller ordinals.
  2. Concrete examples: ω is additively indecomposable, ω·2 is not, and the next indecomposable after ω is ω^2, so such ordinals must exceed every finite multiple of ω.
  3. There are analogous notions for multiplication and exponentiation — multiplicative or exponential indecomposability raises similar questions about whether indecomposable equals irreducible and motivates a full characterization via tools like Cantor normal form.

Ordinal arithmetic

Infinitely More • 30 implied HN points • 22 Jan 26
🔬 Science Algebra
  1. The series develops the basics of ordinal arithmetic—standard addition, multiplication, and exponentiation—and then moves on to topics like indecomposable and irreducible ordinals, Cantor normal form, and binary ordinal representation.
  2. It introduces the natural (Hessenberg) ordinal operations, which are commutative and make the ordinals into a commutative semiring, and it will study the natural ring of ordinals ⟨Ord⟩ inside the surreal numbers, asking about expressions, algebraic properties, and unique factorization.
  3. This essay first lays a rigorous foundation by giving order-theoretic and recursive definitions of the standard ordinal operations, which the later, deeper investigations will rely on.
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Flexibility of plang programming language

Ingig • 3 HN points • 05 May 24
🕹 Technology Algebra
  1. The Plang programming language offers great flexibility, allowing for simplified and personalized app development without the need for extensive programming.
  2. FoodDiaryApp in Plang demonstrates how job-related tasks can be efficiently accomplished with minimal user input.
  3. Plang allows for the creation of variables within user input, showcasing a unique approach to designing personalized applications with user-defined logic.

An algebra of orders

Infinitely More • 17 implied HN points • 04 Feb 24
🔬 Science Algebra
  1. There is a rich algebra of orders involving operations like addition and multiplication.
  2. The disjoint sum operation creates a combined order without interactions between the two parts.
  3. The ordered sum operation combines two orders by placing one above the other, creating new orders with distinct properties.

Influential Mathematicians: Sophus Lie

Quantum Formalism • 19 implied HN points • 13 Aug 20
🔬 Science Algebra
  1. Sophus Lie was a Norwegian mathematician who made significant contributions to mathematics, developing the theory of continuous transformation groups that later led to Lie groups and Lie algebras.
  2. Lie Groups and Lie Algebras, named after Sophus Lie, are essential in the Hilbert space formalism of quantum mechanics, specifically in understanding symmetry and operators in quantum systems.
  3. Although Sophus Lie did not directly contribute to quantum formalism, his mathematical work has had a profound influence on areas of mathematics that are crucial to understanding quantum mechanics.

Solutions manual for "A computational introduction to number theory and algebra"

Thái | Hacker | Kỹ sư tin tặc • 0 implied HN points • 07 Jul 09
🚌 Education Algebra
  1. The book "A computational introduction to number theory and algebra" is recommended as an excellent resource for those interested in number theory, algebra, and cryptography, particularly from a computer science perspective.
  2. The book emphasizes computational aspects, presents algorithms, and discusses complexity analysis, making it a valuable resource for cryptography applications.
  3. The author has created a solutions manual for some chapters of the book, focusing on exercises related to basic properties of integers, congruences, and computing with large integers.

Từ đại số đến Bitcoin: 26, Fermat và tôi

Thái | Hacker | Kỹ sư tin tặc • 0 implied HN points • 15 Jul 14
🔬 Science Algebra
  1. 26 is a unique natural number sandwiched between a square and a cube, a discovery by Fermat, a notable French mathematician.
  2. Euler's proof on Fermat's equation $y^2 = x^3 - 2$ showcases the power of abstract algebra and group theory in solving complex mathematical problems.
  3. Understanding algebraic structures like groups, rings, and unique factorization plays a crucial role in various fields, from cryptography to machine learning.

Đường đến Galois

Thái | Hacker | Kỹ sư tin tặc • 0 implied HN points • 04 Nov 15
🔬 Science Algebra
  1. When working with polynomial equations with complex roots, Galois theory provides a powerful tool to understand and solve them.
  2. Field extensions, such as adding roots of numbers to the rational field, play a key role in finding all roots of a polynomial.
  3. Galois theory reveals the importance of group theory in understanding the symmetries and roots of polynomial equations.

A Whirlwind Tour of Galois Theory

Thái | Hacker | Kỹ sư tin tặc • 0 implied HN points • 06 Nov 15
🔬 Science Algebra
  1. Learning complex mathematical theories like Galois Theory can be accessible and engaging through practical problem-solving and exploring related topics.
  2. Understanding the concepts and applications of field extensions and automorphisms is crucial in grasping the fundamental ideas of Galois Theory.
  3. Galois Theory provides powerful tools to solve mathematical problems, such as finding the minimal polynomial with specific roots, and has practical implications in fields like cryptography.

Influential Mathematicians: Emmy Amalie Noether

Quantum Formalism • 0 implied HN points • 29 Jun 20
🔬 Science Algebra
  1. Emmy Noether, despite facing discrimination as a woman in academia, made significant contributions to mathematics and physics.
  2. Noether's work in invariant theory and abstract algebra, along with her collaborations, influenced the development of advanced algebraic tools used in treating quantum formalism.
  3. Noether played a mentorship role in shaping the career of another influential female mathematician, Grete Hermann, who made important contributions to the foundations of quantum mechanics.