When taking a penalty kick in soccer, the kicker must chose left or right.
And before he kicks, the goalkeeper must also decide left or right, because there is no time to see where the ball is going.
Because the kicker is right footed however, he kicker kicks better to one side than the other. So we have four probabilities:
  • goal kick left keeper jumps left
  • goal kick right keeper jumps right
  • goal kick left keeper jumps right. Note that it is possible that this won't be a goal, even though the keeper is nowhere near the ball, as the ball might just miss the goal by a bit.
  • kick right and keeper jumps left. Analogous to above
PC game Updated 2025-07-16
Paulo Freire Updated 2025-07-16
en.wikipedia.org/wiki/Paulo_Freire:
During his childhood and adolescence, Freire ended up four grades behind, and his social life revolved around playing pick-up football with other poor children, from whom he claims to have learned a great deal. These experiences would shape his concerns for the poor and would help to construct his particular educational viewpoint. Freire stated that poverty and hunger severely affected his ability to learn. These experiences influenced his decision to dedicate his life to improving the lives of the poor: "I didn't understand anything because of my hunger. I wasn't dumb. It wasn't lack of interest. My social condition didn't allow me to have an education. Experience showed me once again the relationship between social class and knowledge"
OMG so nice.
Pauli-X gate Updated 2025-07-16
The quantum NOT gate swaps the state of and , i.e. it maps:
As a result, this gate also inverts the probability of measuring 0 or 1, e.g.
Paul Allen Updated 2025-07-16
Particle creation and annihilation Updated 2025-07-16
Predicted by the Dirac equation.
We've likely known since forever that photons are created: just turn on a light and see gazillion of them come out!
Photon creation is easy because photons are massless, so there is not minimum energy to create them.
The creation of other particles is much rarer however, and took longer to be discovered, one notable milestone being the discovery of the positron.
In the case of the electron, we need to start with at least enough energy for the mass of the electron positron pair. This requires a photon with wavelength in the picometer range, which is not common in the thermal radiation of daily life.
This notation is not so common in basic mathematics, but it is so incredibly convenient, especially with Einstein notation as shown at Section "Einstein notation for partial derivatives":
This notation is similar to partial label partial derivative notation, but it uses indices instead of labels such as , , etc.
Analytic continuation Updated 2025-07-16
visualizing the Riemann hypothesis and analytic continuation by 3Blue1Brown (2016) is a good quick visual non-mathematical introduction is to it.
The key question is: how can this continuation be unique since we are defining the function outside of its original domain?
The answer is: due to the identity theorem.
Lysozyme Updated 2025-07-16
Breaks up peptidoglycan present in the bacterial cell wall, which is thicker in Gram-positive bacteria, which is what this enzyme seems to target.
Part of the inate immune system.
It is present on basically everything that mammals and birds excrete, and it kills bacteria, both of which are reasons why it was discovered relatively early on.
COVID-19 Updated 2025-07-16
COVID happens in two stages:
  • viral infection
  • inflammatory phase, where the body takes over, and sometimes harms itself. It seems that people are not generally contagious at this point?
This distinction is one of the reasons why separating the virus name (SARS-CoV-2) from the disease makes sense: the disease is much broader than the viral infection.
Synopsys Updated 2025-07-16
C (programming language) Updated 2025-07-16
It gets the job done, but cannot make a large codebase DRY without insanity.
As of 2020, C is like Latin, and we are in the Middle Ages, where it has become a lingua franca.
It is interesting to note how late C appeared: 1972, compared e.g. to Fortran which is from 1957. This is basically because C was a "systems programming language", i.e. with focus on pointer manipulation, and because early computers were so weak, there was no operating system or many software layers in the early days. Fortran however was a numerical language, and it ran directly on bare metal, an application that existed before systems programming.
Examples under c.

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