This one’s diehard. Every two norms on a finite dimensional vector space are equivalent: that is to say, each one can be rescaled by a constant to bound the other from both above and below. A consequence of this is that all norms induce the same topology. In infinite dimension, this is not true anymore. But so far I have only seen examples of pair of norms where one beats the other on the whole space....
Some years ago I studied Japanese for some months, out of sheer desire to consume anime in its raw form. The Japanese writing system is one of those systems that are so inefficient one eventually begins to love them. They’re a mess, but dually offer a sadistic pleasure for those who master them. Other than two “alphabets”, hiragana and katakana, which really aren’t a problem, Japanese uses kanji, repurposed Chinese characters....
A rotating free rigid body can be analyzed with the so called Poinsot construction: Conservation of moment of inertia and energy give two ellipsoids, intersecting in another ellipsoid Angular momentum moves in a precession but nonetheless is at every moment normal to the ellipsoid Thus, from the point of view of the plane normal to angular momentum, the ellipsoid rolls without slipping along a curve called herpoloid Or something like that....
Studying the Fourier series decomposition of some periodic signals, I found some had, in addition of having only sine waves without cosine contribution, due to being odd, only sine waves of odd frequences. Is there any kind of physical significance to this property?
To explain this cryptic drawing this tweet by John Baez might help. So is this all Representation Theory in Physics is about?
Periodically driving the fixed endpoint of a pendulum with sufficiently high frequency can lead to new stable configurations or to stable points turning into unstable ones. The most unexpected result is the stabilization of the inverted pendulum. Kapitza’s calculations show that the periodic force can be associated with an extra potential that changes the shape of the classical sinusoidal one. Two simple realizations correspond to fixing the endpoint’s oscillation to the vertical and horizontal axis....