Archive of posts in category “math”
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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. -
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 . The limiting cases of a disk and a circle are discussed. -
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. -
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 -fold symmetry is compatible with translational symmetry is the same as whether . -
The distribution when indistinguishable balls are put into boxes
Suppose there are distinguishable boxes and indistinguishable balls. Now, we randomly put the balls into the boxes. For each of the boxes, what is the probability that it contains balls? This is a simple combanitorics problem that can be solved by the stars and bars method. It turns out that in the limit with fixed, the distribution tends to be a geometric distribution. -
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. -
This is what will happen after you get a haircut
Denote the length distribution of one’s hair to be , where is hair length, and is time. Considering that each hair may be lost naturally from time to time (there is a probability of for each hair to be lost within time range from to ) 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. -
The longest all- 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! -
Solving ODE by recursive integration
By recursively integrating according to from , we can get the solution of the ODE
with initial conditions as the limit of the sequence of functions. -
Hölder means inequality
The Hölder mean of with weights and a parameter is defined as , and the value at are defined by the limits. We can prove using Jensen’s inquality that the Hölder mean increases as increases. This property can be used to prove HM-GM-AM-QM inequalities.
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