- Dec 25, 2022
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!
- Nov 6, 2022
By using power series, we can prove that the problem of solving linear homogeneous ODE with constant coefficients can be reduced to the problem of solving a polynomial with those coefficients. This article illustrates this point in detail, but it uses a very awful notation…
- May 14, 2020
The view of the world… Physically! In this article, I tried to use mathematical language to describe models of the physical world. A view of the world should include: a space (actually spacetime) with some mathematical structure on it (whose points are events in the world), a symmetry principle describing the symmetry of the world, and a motion law to describe the physics and dynamics of the world. This article proposed models for Galilean, Einsteinian, and even Aristotelian worlds. Can you come up with even other worlds?
- Apr 12, 2020
A reversible elementary reaction takes place inside a closed, highly thermally conductive container of constant volume, whose reactants are all gases. Given the reaction equations and the reaction rate constants, a natural question to ask is how the concentration of each gas changes w.r.t. time. In this article, I will answer this question by proposing a general approach to solve it.
- Mar 27, 2020
This article explores the concept which I call the frequency assignment, which is a mapping from $N$ (the set of notes) to $\mathbb R^+$ (the set of frequencies). Concepts such as octaves, intervals, and equal temperaments are introduced.
- Feb 9, 2020
Monkey-patching is a powerful tool in programming. In this article, I used techniques of Ruby metaprogramming to define a series of methods def_after, def_before, etc. to help monkey-patching. They look graciously in that we can use it to shorten the codes for monkey-patching (avoiding aliasing and repeating codes).
- Jan 25, 2020
In this article, I introduce the barycentric coordinates: it is an elegant way to represent geometric shapes related to a simplex. By using it, given a simplex, we can construct a hyperellipsoid with the properties: its surface passes every vertex of the simplex, and its tangent hyperplane at each vertex is parallel to the hyperplane containing all other vertices.
- Jan 6, 2020
In this article, I try exploring an idea: using complex numbers to combine pairs of canonical variables into complex variables: $\mathbf c:=\alpha\mathbf q+\mathrm i\beta\mathbf p$. It turns out that we can write canonical equations $\frac{\mathrm d\mathbf c}{\mathrm dt}=-2\mathrm i\alpha\beta\frac{\partial\mathcal H}{\partial\mathbf c^*}$, Poisson brackets $\left\{f,g\right\}=-2\mathrm i\alpha\beta \left(\frac{\partial f}{\partial\mathbf c}\cdot \frac{\partial g}{\partial\mathbf c^*}- \frac{\partial f}{\partial\mathbf c^*}\cdot \frac{\partial g}{\partial\mathbf c}\right)$, and canonical transformations $\frac{\partial\mathbf c^*}{\partial\mathbf c’^*}= \frac{\partial\mathbf c’}{\partial\mathbf c}, \frac{\partial\mathbf c}{\partial\mathbf c’^*}= -\frac{\partial\mathbf c’}{\partial\mathbf c^*}$ in these complex numbers. Finally, I show two examples of using them in real problems: a free particle, and a harmonic oscillator.