- Feb 3, 2023
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.
- Jan 18, 2023
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.
- 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 15, 2022
By recursively integrating according to $x_{n+1}\!\left(t\right):=\int_{t_0}^tf\!\left(x_n\!\left(s\right),s\right)\,\mathrm ds+C$ from $x_0\!\left(t_0\right):=C$, 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.
- Nov 11, 2022
The Hölder mean of $\vec x$ with weights $\vec w$ and a parameter $p$ is defined as $M_{p,\vec w}\!\left(\vec x\right):=\left(\vec w\cdot\vec x^p\right)^{\frac 1p}$, 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.
- Nov 9, 2022
The function $\left(1-z\right)^{n+1}\sum_{k=1}^\infty k^nz^k$ is a polynomial of degree $n$ w.r.t. $z$, and the sum of its coefficients is $n!$. This turns out to be properties of Eulerian numbers.
- Nov 8, 2022
Suppose $P$ is a point on the circle $\odot C$. When is the sum of distances from $P$ to two edges of $\angle O$ extremal? It turns out to be related to angle bisectors (the intersections of $\odot C$ and the bisector of $\angle O$ or its adjacent supplementary angle are extremals), while the edge cases (at the intersections of $\odot C$ and edges of $\angle O$) are a little tricky: we need to use the bisectors to divide the plane into four quadrants, pick the two quadrants where the line intersecting $\odot C$ at $P$ lies, translate the region to make it center at $C$, and see whether $P$ is inside the translated region.
- 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…
- Jun 13, 2020
Joukowsky transformation of a circle centered at $\left(1,1\right)$ of radius $1$ is a curve resembling a heart.
- May 31, 2020
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.