Pseudo-Riemannian Manifold of Random Ideas
A blog where I share some of my personal research and random thoughts on maths and physics
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Additional thoughts on: Random walks in one dimension
The math behind Sisyphos. After I wrote the blog post about discrete random walks in 1D and their connection to Bessel functions, I started wondering about their continuous counterpart. In particular, one question that got stuck in my mind was what happens if you have discrete steps in one direction but continuous motion in the…
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Random thoughts on: Random walks in one dimension
A wild Bessel function has appeared! I have to tell you something: I really like Bessel functions. All of them. Plain or modified, of first or second kind. I am particularly amazed at how often they appear in completely unrelated contexts. For example, they can be used to describe the vibrations of a drum or,…
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Applications of the shift operator
„And then a step to the right“ In the last post I promised some more examples of using shift operators, so here we are. To begin, consider the simple trigonometric integral equation . We can isolate the function using the shift operator and arrive at , from which the formal solution follows: . To proceed,…
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Fantastic shift operators and where to find them
„It’s just a jump to the left“ Everyone reading this should be familiar with the Taylor expansion of a function . It allows us to write as a series of powers of around some value using only the local information in the neighborhood around , . In particular, by Taylor’s theorem, the coefficients of the…
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What is operational calculus and what should it be?*
Just multiply the differential. Most people who have taken some sort of undergraduate math course in college are probably familiar with the following situation. You are given an ordinary differential equation of type and want to find the function that solves it. What can you do? Well, just multiply both sides by and integrate the…
Pseudo-Riemannian Manifold of Random Ideas 