Ask HN: What did you learn last month? (July 2026) What did you learn recently? What concept finally "clicked" for you? |
Ask HN: What did you learn last month? (July 2026) What did you learn recently? What concept finally "clicked" for you? |
Two specific mistakes I did fix along the way: my landing page force redirected visitors to their browser language, which can hide the other language versions from crawlers. Google's own docs say offer, don't force. And my internal links pointed at page.html while my sitemap said /page, so Google was discovering every page at one address and being told the real one was another. This could have also contributed to the limbo my website is in (although Gemini says that that doesn't happen). Fingers crossed
Most modern frameworks feel like they have so much bloat and overhead. I wanted to make some simple things is a simple way, and it seemed like a good solution for that.
This idea seems like what I have been looking for.
1. https://github.com/jelly-org/ui/blob/main/src/components/res...
What finally clicked: Hermitian matrix, self-adjoint operators, quantum operators for energy and momentum, the Metric (for modeling spacetime curvature), etc.
Nature follows differential equations.
Differential equations are generally complex to solve but have properties that are helpful. E.g., a second order differential equations would have two independent constants of integration in the solution. If two have already been found, a third independent one won't be needed and you already know that you have a complete solution.
For linear equations, if you find enough orthogonal basis functions that are each a solution, even if by hook or crook, you would have found all solutions.
Some functions and operators yield a function with the same form as the original function. E.g., derivative of exponentiation, second derivative of sine, etc.
The differential equation can happen to be such that a function of the above type then cancels out from the equation. This simplifies the equation.
If the above process yields enough orthonormal basis functions, then we know we have the entire space of solutions.
When the operator happens have some properties, the above happens.
Euler's equation links exponential to sine and cosine. That makes complex numbers useful. Then exponentiation covers sine and cosine too.
Magnitude of complex numbers is the number multiplied by its complex conjugate.
The math proofs extend from real numbers (actually needed by Physics) to complex numbers by using complex conjugates.
Hermitian matrices and self-adjoint operators are special cases that bring orthogonal basis functions, real-valued solutions (in spite of using complex numbers), etc.
References:
Spectral Theorem
Sturm Liouville Theory
Just started their COBOL course now.