The hottest Algebra Substack posts right now

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.
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.
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.

The Plang programming language offers great flexibility, allowing for simplified and personalized app development without the need for extensive programming.
FoodDiaryApp in Plang demonstrates how job-related tasks can be efficiently accomplished with minimal user input.
Plang allows for the creation of variables within user input, showcasing a unique approach to designing personalized applications with user-defined logic.

There is a rich algebra of orders involving operations like addition and multiplication.
The disjoint sum operation creates a combined order without interactions between the two parts.
The ordered sum operation combines two orders by placing one above the other, creating new orders with distinct properties.

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.
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.
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.

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Japanese multiplication method visually illustrates how multiplication works.
Understanding the algebra behind multiplication enhances long multiplication skills.
The Japanese method teaches the 'why' behind multiplication, not just the 'how'.

Learning complex mathematical theories like Galois Theory can be accessible and engaging through practical problem-solving and exploring related topics.
Understanding the concepts and applications of field extensions and automorphisms is crucial in grasping the fundamental ideas of Galois Theory.
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.

When working with polynomial equations with complex roots, Galois theory provides a powerful tool to understand and solve them.
Field extensions, such as adding roots of numbers to the rational field, play a key role in finding all roots of a polynomial.
Galois theory reveals the importance of group theory in understanding the symmetries and roots of polynomial equations.

26 is a unique natural number sandwiched between a square and a cube, a discovery by Fermat, a notable French mathematician.
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.
Understanding algebraic structures like groups, rings, and unique factorization plays a crucial role in various fields, from cryptography to machine learning.

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.
The book emphasizes computational aspects, presents algorithms, and discusses complexity analysis, making it a valuable resource for cryptography applications.
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.

Emmy Noether, despite facing discrimination as a woman in academia, made significant contributions to mathematics and physics.
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.
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.