A disaster. More cars and less trains...
Bibliograpy:
- Losing Track by Channel 4 (1984), especially episode 5
- www.theguardian.com/uk/2013/mar/02/beeching-wrong-about-britains-railwaysToday the makeup of UK transport looks very different from the one envisaged by Dr Beeching. Rail passenger figures have almost doubled over the past 10 years; commuter trains are crammed; young people are deserting the car for the train; and Britain's railway bosses are struggling to meet soaring demands for seats. The legacy of Beeching - dug-up lines, sold-off track beds and demolished bridges - has only hindered plans to revitalise the network, revealing the dangers of having a single, inflexible vision when planning infrastructure."The crucial lesson to take from the Beeching anniversary is that you have to be flexible when planning transport infrastructure. Beeching was not," says Colin Divall, professor of rail history at York University. "Yes, many loss-making lines did need closing down, but nowhere near the number earmarked by Beeching, as we can now see with terrible hindsight."
See also: e-learning website.
After something broke on the website due to SQLite vs PostgreSQL inconsistencies and took me a day to figure it out, I finally decided to update the test system so that
OURBIGBOOK_POSTGRES=true npm test will run the tests on PostgreSQL.Originally, these were being run only on SQLite, which is the major use case for OurBigBook CLI, which came before the website.
But the website runs on PostgreSQL, so it is fundamental to test things in PostgreSQL as well.
We know that superfluidity happens more easily in bosons, and so electrons joins in Cooper pairs to form bosons, making a superfluid of Cooper pairs!
Isn't that awesome!
Ultimate explanation: math.stackexchange.com/questions/776039/intuition-behind-normal-subgroups/3732426#3732426
Only normal subgroups can be used to form quotient groups: their key definition is that they plus their cosets form a group.
Intuition:
One key intuition is that "a normal subgroup is the kernel" of a group homomorphism, and the normal subgroup plus cosets are isomorphic to the image of the isomorphism, which is what the fundamental theorem on homomorphisms says.
Therefore "there aren't that many group homomorphism", and a normal subgroup it is a concrete and natural way to uniquely represent that homomorphism.
The best way to think about the, is to always think first: what is the homomorphism? And then work out everything else from there.
Stack Overflow policy that you cannot harshly criticize the character of a politician Updated 2025-01-03 +Created 1970-01-01
Why. Why. Why is there no limit to how much I can help, but there is a limit to how many thanks I can get?
At most, limit it to a single answer to avoid highly publicized events, e.g. an answer being shared on Reddit. But across answers? It makes no sense.
The two ways main ways to overcome this limit are the 15 point answer accept reputation and bounties.
200 reputation per day works out 73k a year BTW.
The busy beaver game consists in finding, for a given , the turing machine with states that writes the largest possible number of 1's on a tape initially filled with 0's. In other words, computing the busy beaver function for a given .
There are only finitely many Turing machines with states, so we are certain that there exists such a maxium. Computing the Busy beaver function for a given then comes down to solving the halting problem for every single machine with states.
Some variant definitions define it as the number of time steps taken by the machine instead. Wikipedia talks about their relationship, but no patience right now.
The Busy Beaver problem is cool because it puts the halting problem in a more precise numerical light, e.g.:
- the Busy beaver function is the most obvious uncomputable function one can come up with starting from the halting problem
- the Busy beaver scale allows us to gauge the difficulty of proving certain (yet unproven!) mathematical conjectures
Correspond to the angular part of Laplace's equation in spherical coordinates after using separation of variables as shown at: en.wikipedia.org/wiki/Spherical_harmonics#Laplace's_spherical_harmonics
The inaugural that predicted the Josephson effect.
Published on Physics Letters, then a new journal, before they split into Physics Letters A and Physics Letters B. True Genius: The Life and Science of John Bardeen mentions that this choice was made rather than the more prestigious Physical Review Letters because they were not yet so confident about the results.
Digits 0 to 9, white on black background:
for i in `seq 0 9`; do convert -size 512x512 xc:black -pointsize 500 -gravity center -fill white -draw "text 0,0 \"$i\"" $i.png; done There are unlisted articles, also show them or only show them.