Archive of posts in category “math”

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• Jan 3, 2026

  Determining the stabilizability of the abelian sandpile model

  The abelian sandpile model is a cellular automaton where each cell is a sandpile that can topple to put grains on neighboring cells when having enough grains. It is stable if none of the sandpiles can topple. A natural question to ask is whether it can become stable eventually. In this article, I will show that one can determine the stabilizibility usiung integer linear programming for the abelian sandpile model on an arbitrary finite graph.

  /math #graph theory #linear programming #long paper
• Apr 11, 2025

  Eigenfunctions of the Laplacian on an annulus with homogeneous Neumann boundary condition

  Several features of the eigenfunctions of the Laplacian on an annulus with homogeneous Neumann boundary condition are discussed. The distribution of the eigenvalues is discussed in detail, making use of a phase angle function called $\tht$. The limiting cases of a disk and a circle are discussed.

  /math #mathematical physics #pde #ode #long paper
• Oct 10, 2024

  Some understanding of Grassmann numbers out of intuition

  I was briefly introduced to Grassmann numbers when I studied quantum field theory. I then had the natural question of how we can formally define them. In this article, I went with my intuition and tried to answer this question.

  /math #quantum field theory #linear algebra #abstract algebra
• Nov 9, 2023

  Rotational symmetry of plane lattices as a simple example of algebraic number theory

  For a plane lattice, there is only a finite number of different rotational symmetries that are compatible with the discrete translational symmetry. For example, the 5-fold rotational symmetry is not one of them. Why is that? It turns out that whether an $m$-fold symmetry is compatible with translational symmetry is the same as whether $\varphi(m)\le2$.

  /math #complex #condensed matter physics #algebraic number theory #lattice #mathematical physics
• May 9, 2023

  The distribution when indistinguishable balls are put into boxes

  Suppose there are $n$ distinguishable boxes and $k$ indistinguishable balls. Now, we randomly put the balls into the boxes. For each of the boxes, what is the probability that it contains $m$ balls? This is a simple combanitorics problem that can be solved by the stars and bars method. It turns out that in the limit $n,k\to\infty$ with $k/n$ fixed, the distribution tends to be a geometric distribution.

  /math #probability #combinatorics #ruby
• Feb 3, 2023

  Introducing bra–ket notation to math learners

  Bra–ket notation is a good-looking notation! I am sad that it is not generally taught in math courses. Let me introduce it to you.

  /math #from zhihu #quantum mechanics
• Jan 18, 2023

  This is what will happen after you get a haircut

  Denote the length distribution of one’s hair to be $f(l,t)$, where $l$ is hair length, and $t$ is time. Considering that each hair may be lost naturally from time to time (there is a probability of $\lambda\,\mathrm dt$ for each hair to be lost within time range from $t$ to $t+\mathrm dt$) and then restart growing from zero length, how will the length distribution of hair evolve with time? It turns out that we may model it with a first-order PDE.

  /math #calculus #probability #pde
• Dec 25, 2022

  The longest all-$1$ substring of a random bit string

  Given your probability of breaking the combo at each note, what is the probability distribution of your max combo in the rhythm game chart? I considered the problem seriously!

  /math #long paper #rhythm game #algorithm #probability #stochastic process
• Nov 15, 2022

  Solving ODE by recursive integration

  By recursively integrating according to $x_{n+1}\!\left(t\right)\coloneqq\int_{t_0}^tf\!\left(x_n\!\left(s\right),s\right)\,\mathrm ds+C$ from $x_0\!\left(t_0\right)\coloneqq C$, we can get the solution of the ODE $x'\!\left(t\right)=f\!\left(x\!\left(t\right),t\right)$ with initial conditions $x\!\left(t_0\right)=C$ as the limit of the sequence of functions.

  /math #ode #calculus
• Nov 11, 2022

  Hölder means inequality

  The Hölder mean of $\vec x$ with weights $\vec w$ and a parameter $p$ is defined as $M_{p,\vec w}\!\left(\vec x\right)\coloneqq\left(\vec w\cdot\vec x^p\right)^{\frac 1p}$, and the value at $p=-\infty,0,+\infty$ are defined by the limits. We can prove using Jensen’s inquality that the Hölder mean increases as $p$ increases. This property can be used to prove HM-GM-AM-QM inequalities.

  /math #calculus #from zhihu

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