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- WolframAlpha - computational knowledge engine
- HackMD - free online notes tool supporting Markdown language
- Desmos Scientific Calculator - a very handy calculator for everyday usage
- AutoDraw - a simple tool to draw

- Get link
- Other Apps

Vibrating string without damping is represented by the following differential equation: $$ \begin{cases} \begin{array}{l@{\ }l@{\ }l} \frac{{{\partial ^2}u}}{{\partial {t^2}}} = {a^2}\frac{{{\partial ^2}u}}{{\partial {x^2}}} , & \hspace{0.25in} 0 \leqslant x \leqslant L , 0 \leqslant t \leqslant \infty , & \hspace{0.25in} \text{string equation} ; \\ u\left( {0,t} \right) = u\left( {L,t} \right) = 0 , & \hspace{0.25in} 0 \leqslant t \leqslant \infty , & \hspace{0.25in} \text{boundary conditions} ; \\ u\left( {x,0} \right) = f\left( x \right) , \frac{{\partial u}}{{\partial t}}\left( {x,0} \right) = 0 , & \hspace{0.25in} 0 \leqslant x \leqslant L , & \hspace{0.25in} \text{initial conditions}; \end{array} \end{cases} \tag{1} $$ Splitting the string equation into two coupled equations We need to transform the equation: $$ \frac{{{\partial ^2}u}}{{\partial {t^2}}} = {a^2}\frac{{{\partial ^2}u}}{{\parti

This article is based on the Train-platform paradox simlation available at https://train.tdworakowski.com . The paradox If you consider two relativistic phenomena which are length contraction and time dilation , the special theory of relativity may seem inconsistent. Imagine a train 100 meters long is passing a platform 100 meters long traveling at 90% of the speed of light . According to the theory, for the observer on the platform the train is shortened and the time inside it elapses more slowly. But for observer inside the train the length of the train is normal, time elapses normally, however the platform is shortened and the time on the platform elapses more slowly. How is it possible that both of these facts coexist? To answer this question we need to understand the third relativistic phenomenon which is relativity of simultaneity . If we consider these three phenomena together, the theory becomes consistent. You can play with the simulation to confirm that the theory

Using this trick you can speed up a lot your work at command line in Linux (and not only). Pressing Ctrl + R allows you to quickly find a previously used command. How to use it? In command line terminal press Ctrl + R . Start typing some part of command, for example `pul`: (reverse-i-search)`pul': git pull If you want to find a next match, press Ctrl + R again (repeat until successful). If you want to edit the current match press LEFT or RIGHT and edit the command. If you want to exit without running any command, press Ctrl + G . If command you need is displayed, press ENTER to run it. Useful shortcuts These shortcuts are not related to the bash history tool (Ctrl+R), but may be useful if you edit a found command. Ctrl + LEFT/RIGHT - jump over the words. Ctrl + A - move the cursor to the beginning of the line. Ctrl + E - move the cursor to the end of the line. Ctrl + U - remove the content from the be

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