How exactly does Knuth's Up-Arrow notation work?
I've done some research, and found this on Wikipedia. \begin{matrix}a\uparrow b=a^{b}=&\underbrace {a\times a\times \dots \times a} \\&b{\mbo...
22 Apr 2026 1 READS
Archive: News
Integrate $e^{-ax}$ and $xe^{-ax}$?
I'm making exercises about integration but I don't really get it. How do you solve these two integrals from 0 to +infinity? $\int Ae^{-ax}\,dx$ ...
22 Apr 2026 1 READS
Laplace transform of $t^2 y''$
So I need to take the Laplace transform of the following function: $$ Eqn. 1: \\\frac{1}{2}\mathcal{L}{\{t^2 \ddot y\}} $$ I've found that $$ Eqn. 2:...
22 Apr 2026 1 READS
What is the connection between calculus of variations and PDEs?
I have heard of calculus of variations and I would like to know how it is related with partial differential equations. What type of equations (hyperbolic,...
22 Apr 2026 1 READS
What is definition of a periodic function?
Periodic functions are defined over the domain of a function,say it is $[a,b]$. I A Maron defines a periodic function as such:The function $f(x)$ is calle...
22 Apr 2026 1 READS
How to sketch the curve of parametric equations
I have the following parametric equations: $$x = \sin\theta,\qquad y = \cos\theta,\qquad 0 \leq \theta \leq \pi.$$ The corresponding Cartesian equation is...
22 Apr 2026 1 READS
What is a homology of chain complex?
First of all I don't know much about homological algebra, and algebraic topology I just took a class using Kinsey book (Topology of Surfaces) which i...
22 Apr 2026 1 READS
Reverse engineering a PID controllers parameters
In an attempt to copy an existing PID controller, I wish to identify the active PID controllers parameters. I found this formula for discrete PID controll...
22 Apr 2026 0 READS
Easier way to show different forms of Jacobi's formula for the derivative of the determinant
Jacobi's formula for the derivative of the determinant of a matrix A is $$ {\displaystyle {\frac {d}{dt}}\det A=\det A\;\mathrm {tr} \left(A^{-1}{\fr...
22 Apr 2026 1 READS
Divergence of power series
In my lectures we considered a power series and used the ratio test for absolute convergence to find the radius or interval of convergence. However it was...
22 Apr 2026 1 READS
