Plancherel theorem Updated +Created
Some sources say that this is just the part that says that the norm of a function is the same as the norm of its Fourier transform.
Others say that this theorem actually says that the Fourier transform is bijective.
The comment at math.stackexchange.com/questions/446870/bijectiveness-injectiveness-and-surjectiveness-of-fourier-transformation-define/1235725#1235725 may be of interest, it says that the bijection statement is an easy consequence from the norm one, thus the confusion.
Markua Updated +Created
In leanpub you write your book in a markdown variant they call Markua, marketed as "markdown for books".
TODO is there a reference implementation that runs locally for HTML output? Or the only reference implementation is closed under leanpub?
Muon Updated +Created
Lie algebra of Updated +Created
This is a good first concrete example of a Lie algebra. Shown at Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 4.2 "How to linearize a Lie Group" has an example.
We can use use the following parametrization of the special linear group on variables , and :
Every element with this parametrization has determinant 1:
Furthermore, any element can be reached, because by independently settting , and , , and can have any value, and once those three are set, is fixed by the determinant.
To find the elements of the Lie algebra, we evaluate the derivative on each parameter at 0:
Remembering that the Lie bracket of a matrix Lie group is really simple, we can then observe the following Lie bracket relations between them:
One key thing to note is that the specific matrices , and are not really fundamental: we could easily have had different matrices if we had chosen any other parametrization of the group.
TODO confirm: however, no matter which parametrization we choose, the Lie bracket relations between the three elements would always be the same, since it is the number of elements, and the definition of the Lie bracket, that is truly fundamental.
Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 4.2 "How to linearize a Lie Group" then calculates the exponential map of the vector as:
with:
TODO now the natural question is: can we cover the entire Lie group with this exponential? Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 7 "EXPonentiation" explains why not.
Linux distribution Updated +Created
Fashion MNIST Updated +Created
Same style as MNIST, but with clothes. Designed to be much harder, and more representative of modern applications, while still retaining the low resolution of MNIST for simplicity of training.
Squeezed state of light Updated +Created
Feijoada Updated +Created
d'Alembert operator in Einstein notation Updated +Created
Given the function :
the operator can be written in Planck units as:
often written without function arguments as:
Note how this looks just like the Laplacian in Einstein notation, since the d'Alembert operator is just a generalization of the laplace operator to Minkowski space.
Softcover (LaTeX) Updated +Created
LaTeX subset that output nicely to HTML.
Too insane though due to LaTeX roots, better just move to newer HTML-first markups like OurBigBook Markup or markdown.
Fusion ignition Updated +Created
Meter Updated +Created
Lie bracket of a matrix Lie group Updated +Created
This makes it clear how the Lie bracket can be seen as a "measure of non-commutativity"
Because the Lie bracket has to be a bilinear map, all we need to do to specify it uniquely is to specify how it acts on every pair of some basis of the Lie algebra.
Then, together with the Baker-Campbell-Hausdorff formula and the Lie group-Lie algebra correspondence, this forms an exceptionally compact description of a Lie group.
One parameter subgroup Updated +Created
The one parameter subgroup of a Lie group for a given element of its Lie algebra is a subgroup of given by:
Intuitively, is a direction, and is how far we move along a given direction. This intuition is especially vivid in for example in the case of the Lie algebra of , the rotation group.
One parameter subgroups can be seen as the continuous analogue to the cycle of an element of a group.
University of Bristol Updated +Created
University of Manchester Updated +Created
Mt. Gox Updated +Created
The first Bitcoin exchange. Coded as a hack, and they didn't manage to fix the hacks as the site evolved in a major way, which led to massive hacks.
Their creation is clearly visible on the archive history of bitcoin.org: web.archive.org/web/20100701000000*/bitcoin.org which started having massively more archives since Mt. Gox opened.
Video 1.
One Mistake Brought Down This FBI Most Wanted Hacker by Crumb (2023)
Source. Good overview of Mt. Gox.
Wikidata Updated +Created
Marijuana Updated +Created
Piezoelectric motor Updated +Created

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