Ulysses’ trip


  • Drawing a heart using Joukowsky transformation

    Joukowsky transformation of a circle centered at $\left(1,1\right)$ of radius $1$ is a curve resembling a heart.

  • Generalization of Euler–Lagrange equation

    We may generalize Euler–Lagrange equation to higher dimensional optimization problems: find a function defined inside a region to extremize a functional defined as an integral over that region, with the constraint that the value of the function is fixed on the boundary of the region.

  • View of the world (physically rather than philosophically)

    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?

  • Simulating a mechanical system using rpg_core.js

    Continuing my last work of simulating a mechanical system using RGSS3, I made a new version using rpg_core.js, the game scripting system shipped with RPG Maker MV. This version is live on web!

  • Simulating a mechanical system using RGSS3

    Hamiltonian mechanics gives us a good way to simulate mechanical systems as long as we can get its Hamiltonian and its initial conditions. I implemented this simulation in RGSS3, the game scripting system shipped with RPG Maker VX Ace.

  • Writing a DSL with commands split by space

    DSL means domain-specific language. Ruby is a powerful script language in terms of building DSLs (as sublanguages of Ruby). In this article, I implemented my idea of a DSL with commands split by space. For example, you may just write a b c to run the commands a, b, and c one after another! This trick is heavily applied in my project alda-rb. How do I achieve this?

  • Running Jekyll myself

    I have got tired of letting GitHub Pages to run Jekyll for me. I cannot use custom plugins! I decide to run Jekyll myself and push the built result to GitHub repo. I wrote a script to do this.

  • Algebraic structure of chemicals

    Regarding balancing chemical equations, actually we are trying to find a non-trivial linear combination of some chemicals to get zero. The interesting thing is that the coefficients can only be integers ($\mathbb Z$), which is not a field, so the algebraic structure of chemicals is not linear space. They actually live in a free $\mathbb Z$-module. See how I formalize this idea in mathematical language.

  • The concentration change of gas in reversible reactions

    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.

  • Snippet: O, ’tis too true!

    O, ’tis too true!

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