Ciro Santilli @cirosantilli 35

For the typical case of a linear form over ${\mathbb{R}}^n$, the form can be seen just as a row vector with n elements, the full form being specified by the value of each of the basis vectors.

Solution $Z$ for given $X$ and $Y$ of:where $e$ is the exponential map.

If we consider just real number, $Z=X+Y$, but when X and Y are non-commutative, things are not so simple.

Furthermore, TODO confirm it is possible that a solution does not exist at all if $X$ and $Y$ aren't sufficiently small.

This formula is likely the basis for the Lie group-Lie algebra correspondence. With it, we express the actual group operation in terms of the Lie algebra operations.

Notably, remember that a algebra over a field is just a vector space with one extra product operation defined.

Vector spaces are simple because all vector spaces of the same dimension on a given field are isomorphic, so besides the dimension, once we define a Lie bracket, we also define the corresponding Lie group.

Since a group is basically defined by what the group operation does to two arbitrary elements, once we have that defined via the Baker-Campbell-Hausdorff formula, we are basically done defining the group in terms of the algebra.

Integrable functions to the power $p$, usually and in this text assumed under the Lebesgue integral because: Lebesgue integral of $L^p$ is complete but Riemann isn't

The type of laser described at: Video "How Lasers Work by Scientized (2017)", notably youtu.be/_JOchLyNO_w?t=581. Mentioned at: youtu.be/_JOchLyNO_w?t=759 That point also mentions that 4-level lasers also exist and are more efficient. TODO dominance? Alternatives?

Bibliography:

For every continuous symmetry in the system (Lie group), there is a corresponding conservation law.

Furthermore, given the symmetry, we can calculate the derived conservation law, and vice versa.

As mentioned at buzzard.ups.edu/courses/2017spring/projects/schumann-lie-group-ups-434-2017.pdf, what the symmetry (Lie group) acts on (obviously?!) are the Lagrangian generalized coordinates. And from that, we immediately guess that manifolds are going to be important, because the generalized variables of the Lagrangian can trivially be Non-Euclidean geometry, e.g. the pendulum lives on an infinite cylinder.

If your kids are about to starve, fine, do it.

But otherwise, Ciro Santilli will not, ever, spend his time drilling programmer competition problems to join a company, life is too short for that.

Life is too short for that. Companies must either notice that you can make amazing open source software projects or contributions, and hire you for that, or they must fuck off.

Companies must either notice that you can make amazing projects or contributions, and hire you for that, or they must fuck off.

This is a good book. It is rather short, very direct, which is a good thing. At some points it is slightly too direct, but to a large extent it gets it right.

The main goal of the book is to basically to build the Standard Model Lagrangian from only initial symmetry considerations, notably the Poincaré group + internal symmetries.

The book doesn't really show how to extract numbers from that Lagrangian, but perhaps that can be pardoned, do one thing and do it well.

A phase of Fe-C characterized by the low ammount of carbon.

Describes perfect lossless waves on the surface of a string, or on a water surface.

Uniqueness: math.stackexchange.com/questions/1113622/uniqueness-of-solutions-to-the-wave-equation

As mentioned at: math.stackexchange.com/questions/579453/real-world-application-of-fourier-series/3729366#3729366 from solving partial differential equations with the Fourier series citing courses.maths.ox.ac.uk/node/view_material/1720, analogously to the heat equation, the wave linear equation can be be solved nicely with separation of variables.

Is the set of all n-by-y square matrices.

Because $GL(n)$ is a Lie group we can use Section "Lie algebra of a matrix Lie group".

For every matrix $x$ in the set of all n-by-y square matrices $M_n$, $e^x$ has inverse $e^{-}x$.

Note that this works even if $x$ is not invertible, and therefore not in $GL(n)$!

Therefore, the Lie algebra of $GL(n)$ is the entire $M_n$.

Full set of all possible special relativity symmetries:

In simple and concrete terms. Suppose you observe N particles following different trajectories in Spacetime.

There are two observers traveling at constant speed relative to each other, and so they see different trajectories for those particles:Note that the first two types of transformation are exactly the non-relativistic Galilean transformations.

The Poincare group is the set of all matrices such that such a relationship like this exists between two frames of reference.

17 of them.

"Water" is the name for both: