Ulysses' trip

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May 14, 2025, 14:58:53

After using computers for all so many years, for the first time, the Insert key was actually useful when I was editing a file. I always thought the only reason it exists is for causing small headaches when my dumb fingers accidentally touch it.

The key was used when I was editing a string in a binary file, and I wanted to make sure that the offsets of all the contents afterward were not affected.

Articles

Nov 16, 2022
Finding curvature radius physically

Sometimes the curvature radius of a curve can be found by using physical methods although it seems that you must use calculus to find it. In this article, the curvature radius of the curve $x^2=2py$ at the point where the curvature radius is smallest is found by using physical methods without using calculus (with only high school knowledge). The answer is that the smallest curvature is exactly $p$ , and the point with smallest curvature is the vertex.
- Categories: physics
- Tags: from zhihu
Nov 16, 2022
A whirling point charge

In the vacuum, inside a fixed ring of radius $R$ with fixed charge $Q$ uniformly distributed, there is a point charge with charge $q$ and mass $m$ moving in the plane of the ring due tue the electrostatic force. It moves in the small region around the center of the ring, and the motion is periodic along a closed curve. The area of the region enclosed by the curve is $S$ . Denote the distance from the center to the point charge as $r$ , and $r\ll R$ . Find the magnetic induction $B$ at the center of the ring.
- Categories: physics
- Tags: electrodynamics, from zhihu
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.
- Categories: math
- Tags: ode, calculus
Nov 13, 2022
An example of non-uniform elements: heavy elastic rope

To illustrate the concept about non-uniform elements, we study a simple problem: suppose a uniform heavy elastic rope has mass $m$ , original length $L_0$ , and stiffness $k$ , and find the mass distribution and length of it when hung vertically. We can use the element method to solve this problem, but the elements are non-uniform in terms of length. The elements add up to get the total length $L=\frac{mg}{2k}+L_0$ .
- Categories: physics
- Tags: from zhihu
Nov 12, 2022
Kinetic energy, momentum, and angular momentum of rigid bodies

In this article, we will find that the inertia matrix naturally appears when we calculate the kinetic energy $T$ or the angular momentum $\mathbf M$ of a rigid body. Then, we introduce the concept of principal inertia $\mathbf J_{\mathrm{pri}}$ . We also study how the inertia matrix changes under translations and rotations and how those transformations may lead to conclusions that can help us simplify the calculation of inertia matrices.
- Categories: physics
- Tags: rigid body, linear algebra, classical mechanics, from zhihu
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.
- Categories: math
- Tags: calculus, from zhihu
Nov 9, 2022
A polynomial whose sum of coefficients is a factorial

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.
- Categories: math
- Tags: combinatorics, number sequence, from zhihu
Nov 9, 2022
The image of a circular object through a thin lens

The image of a circle with radius $r$ and centered at $C\left(-2f,0\right)$ through a thin lens at $x=0$ with focal length $f$ and centered at $O\left(0,0\right)$ is a conic section with the focus being $\left(2f,0\right)$ , the directrix being line $x=f$ , and the eccentricity being $\frac rf$ .
- Categories: physics
- Tags: geometrical optics, from zhihu
Nov 8, 2022
The point on the circle farthest to two lines

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.
- Categories: math
- Tags: elementary geometry, trigonometry, from zhihu
Nov 8, 2022
Typesetting math in emails

After some comparison among solutions, I use KaTeX to typeset math in my emails.
- Categories: programming
- Tags: jekyll, ruby, tex

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