Ciro Santilli @cirosantilli 37

The author seems to have uploaded the entire book by chapters at: www.physics.drexel.edu/~bob/LieGroups.html

And the author is the cutest: www.physics.drexel.edu/~bob/Personal.html.

Overview:

A linear map is a function $f:V_1(F)\to V_2(F)$ where $V_1(F)$ and $V_2(F)$ are two vector spaces over underlying fields $F$ such that:

A common case is $F=\mathbb{R}$, $V_1={\mathbb{R}}_m$ and $V_2={\mathbb{R}}_n$.

One thing that makes such functions particularly simple is that they can be fully specified by specifyin how they act on all possible combinations of input basis vectors: they are therefore specified by only a finite number of elements of $F$.

Every linear map in finite dimension can be represented by a matrix, the points of the domain being represented as vectors.

As such, when we say "linear map", we can think of a generalization of matrix multiplication that makes sense in infinite dimensional spaces like Hilbert spaces, since calling such infinite dimensional maps "matrices" is stretching it a bit, since we would need to specify infinitely many rows and columns.

The prototypical building block of infinite dimensional linear map is the derivative. In that case, the vectors being operated upon are functions, which cannot therefore be specified by a finite number of parameters, e.g.

For example, the left side of the time-independent Schrödinger equation is a linear map. And the time-independent Schrödinger equation can be seen as a eigenvalue problem.

WellSync, if you are gonna useSync this wonky language thing inSync one place, you might as well useSync it everywhereSync and make it more decent. See also: how to convert async to sync in JavaScript.

Their CLI debugger is a bit crap compared to GDB, basic functionality is either lacking or too verbose:Documentation at: nodejs.org/dist/latest-v16.x/docs/api/debugger.html

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.

Setting: you are sending bits through a communication channel, each bit has a random probability of getting flipped, and so you use some error correction code to achieve some minimal error, at the expense of longer messages.

This theorem sets an upper bound on how efficient you can be in your encoding, for any encoding.

The next big question, which the theorem does not cover is how to construct codes that reach or approach the limit. Important such codes include:

But besides this, there is also the practical consideration of if you can encode/decode fast enough to keep up with the coded bandwidth given your hardware capabilities.

news.mit.edu/2010/gallager-codes-0121 explains how turbo codes were first reached without a very good mathematical proof behind them, but were still revolutionary in experimental performance, e.g. turbo codes were used in 3G/4G.

But this motivated researchers to find other such algorithms that they would be able to prove things about, and so they rediscovered the much earlier low-density parity-check code, which had been published in the 60's but was forgotten, partially because it was computationally expensive.

Sample usage:This produces simple one liners for each request.

What you likely want is the -V option which fully disassembles each frame much as you can do in the GUI Wireshark:

Ciro Santilli can accept closed source on server products more easily than offline, because the servers have to be paid for somehow (by stealing your private data).

Convergence: math.stackexchange.com/questions/629630/simple-proof-euler-mascheroni-gamma-constant

We define it as a linear map where the domain is the same as the image, i.e. an endofunction.

Examples:

A polynomial of degree 1, i.e. of form $ax+b$.

As of 2020, no one knows how to build the major desktop distros fully from source into the ISO, and especially so in a reproducible build way. Everything is done in build servers somewhere with complicated layers of prebuilds. It's crap.

See also: cirosantilli.com/linux-kernel-module-cheat/#linux-distro-choice

Yes:

No:

The first quantum mechanics theories developed.

Their most popular formulation has been the Schrödinger equation.

Local symmetries appear to be a synonym to internal symmetry, see description at: Section "Internal and spacetime symmetries".

As mentioned at Quote , local symmetries map to forces in the Standard Model.

Appears to be a synonym for: gauge symmetry.

A local symmetry is a transformation that you apply a different transformation for each point, instead of a single transformation for every point.

TODO what's the point of a local symmetry?

Bibliography:

One major difference between the elliptic curve over a finite field or the elliptic curve over the rational numbers the elliptic curve over the real numbers is that not every possible $x$ generates a member of the curve.

This is because on the Equation "Definition of the elliptic curves" we see that given an $x$, we calculate $x^3+ax+b$, which always produces an element $y^2$.

But then we are not necessarily able to find an $y$ for the $y^2$, because not all fields are not quadratically closed fields.

For example: with $a=1$ and $b=1$, taking $x=1$ gives:and therefore there is no $y\in \mathbb{Q}$ that satisfies the equation. So $x=1$ is not on the curve if we consider this elliptic curve over the rational numbers.

That $x$ would also not belong to Elliptic curve over the finite field ${\mathbb{F}}_4$, because doing everything $\mathrm{mod}4$ we have:Therefore, there is no element $y\in {\mathbb{F}}_4$ such that $y\times y=2$ or $y\times y=3$, i.e. $2$ and $3$ don't have a multiplicative inverse.

For the real numbers, it would work however, because the real numbers are a quadratically closed field, and $\sqrt{3}\in \mathbb{R}$.

For this reason, it is not necessarily trivial to determine the number of elements of an elliptic curve.