sgdpk

I don't think Sheldon Axler wants to literally banish determinants, they are a useful tool.

But they do obscure the meaning in a lot of proofs. If you start using them immediately, it is also hard to motivate where they come from, what's the intuition behind them and how the students could arrive at the definition themselves.

And even if you use them and they produce short one-line proofs, you should also prove all the properties behind them first, which is where all the complexity is hiding.

The nice point about Linear Algebra Done Right is that it explains how Linear Algebra works without resorting to "determinant magic". In the end, you will also understand why determinants work so well in so many proofs.

In any case, his book is really suited for a second course in Linear Algebra, so maybe it is beneficial to start using determinants right away. Personally I understood them better with his book.

Disclaimer: I don't know Sheldon Axler, but I read his book.

Might be as a way to differentiate between page views and people reading the article. I think Quora does this. You get a separate counter for people who expand your post.

This does not surprise me much. There have been more than two or three occasions in the past two years where Facebook, Whatsapp and Instagram were having connection issues simultaneously, which for me is an indication of how the services are getting more integrated.

I think you are right, I am complaining mostly about the "front page of the internet" part. I like visiting other subs, and I usually find answers to Google searches there, because some communities are really good.

I am also interested in this... I used to like Reddit a lot, but I feel their push to reach a broader audience has pushed down the quality of the front-page subs. It is dominated by a sense of moral superiority: "Am I the Asshole", "Justice Served", Leopards ate my Face" are very common now.

I like Quora because of specific authors that write high-quality answers (mainly mathematics).

I will read this a bit later, but is the idea to extended the structure of the tangent space and make it a full geometric algebra?

Personally, an appealing part of GA is clarifying that the magnetic field is best viewed as an "oriented area", instead of a pseudo-vector [1], which is a concept that is frequently ill-defined. Usually, lecturers say that the magnetic field is essentially a vector, but to be careful that it flips directions when you mirror space. But how then can you check this by just looking at the usual three coordinates of the vector?

With GA, you associate the magnetic field with an oriented plane element. Oriented planes have this flip-mirroring property. Because space is 3D, you have exactly one vector to which this plane element is perpendicular, so you can identify the field with a vector. This is called the Hodge duality, and this particular form is accidental to 3D space.

If you need more convincing, the magnitude of the magnetic field is given by the cross-product, which calculates the area spanned by two vectors.

You could argue that you don't need GA for this, only to use the exterior product instead of the cross product. But by using an algebra that contains both vectors and oriented areas, you can effectively sum the electric and magnetic fields. This will give you pretty compact and beautiful equations, which is the essence of this article.

[1] https://en.wikipedia.org/wiki/Pseudovector

I always see his site mentioned. I am not convinced about this criticism. The author is not an expert on sleep or a neuroscientist. The most you can say is that Walker has not responded to this criticism.

He has a strong opinion that you can only develop perfect pitch while young, and I have to agree. His videos are great.

My favorite hill to die on! Thanks for writing!

Yes, that's it.

Indeed, this is why I wrote about normalization in my original answer :)

Indeed! I highly recommend following the author.

Sure, there is certainly a lot of jargon here which I could have explained better, but in the end one has to be familiar with some calculus and complex numbers to make sense of this topic. Depending on where you're from, this could be either late high-school to second/third year of university.

So by exponential I mean the exponential function, in the mathematician's sense: something which takes numbers as inputs and produces other numbers as outputs in a unique way. Already a complication: these can be complex numbers.

Then, the derivative of a function is its "instantaneous rate of change". How much does the function increase for small (infinitesimal, really) changes in input? Since this rate may be different for every point, we are defining a new function f': for every input of f, f' gives you the local rate of change. So there you go, the derivative takes a function f and produces a function f'. We call this an operator. Because the derivative has very nice properties, we call this a "linear" operator.

Finally, the "eigen"-things. These are really fancy words for the simple concept of a fixed-point: when you apply an operation to a certain thing X and just get that X back. So, the function x^2 has two fixed-points: 0 and 1. These would be "eigenpoints", but we don't call them that because we reserve the term "eigen" for linear operations, such as the derivative operator or matrix-multiplication.

The point of my answer is that the exponential function is elegantly understood as the eigenfunction of the derivative. In other words: which function is its own derivative? Which one tells you how it changes just by looking at it?

I find that the most elegant way to understand exponential-related things is by defining the exponential as the function satisfying f'=f (with suitable normalization). In other words, it is the "eigenfunction" of the derivative, which is a linear operator. This is a very natural thing to ask when studying derivatives, ODEs, etc.

With this definition, you can define the constants e and pi in a natural way. In particular, 2 * pi * i is the period of e^x. It then makes sense to analyse the function e^ix, with period 2*pi, and separate its real and imaginary parts. These are sure to be periodic real functions, and we can name them cos(x) and sin(x).

This Quora answer talks about this approach pretty well [1].

[1] https://qr.ae/pNy7qp

This is part of a monthly series of seminar at the interstection of Quantum Computation and High-Energy Physics.

Abstract: This talk has three parts. In part 1, I discuss the current status and near-term prospects for quantum computing and quantum simulation. In part 2, I emphasize the opportunity to advance our understanding of quantum field theory, high energy physics, and nuclear physics using quantum simulation platforms. In part 3, I describe some recent and ongoing work developing classical and quantum algorithms for simulating high-energy scattering in quantum field theory, particularly in one spatial dimension.

It's also consistent with the set notation, which makes it easier to remember. Dict keys already kind of work like a set, this extends the behavior to deal with the values as well.

Yes, that's the point. Of course it should be sensible and rounded.

If we were too literal with this definition, 12 would even change everyday because of the analemma [1].

[1] https://en.m.wikipedia.org/wiki/Analemma

My thoughts exactly. Keep 12pm at the time the sun is in its maximum and just do things earlier/later.

This is a nice summary of the basics of immunology and how it relates to SARS-CoV-2.

It's good to see a piece that shows how complex the science behind COVID is, in the middle of so many contradictory or over-simplified news.

In the book "Why we sleep?", Matthew Walker suggests something similar. That dream sleep is fundamental in making associations. In this case, generalizing and getting rid of overfitting. He talks a bit about problem solving when sleeping and how this leads to "a-ha" moments when waking up.

This means that the idea in this paper is already "out there", contrary to what the abstract states. But it's exciting to have a framework to talk about it quantitatively.

The principle the article is implicitly talking about is Landauer's principle [1]. It states that if information is lost in a system, it must be as heat dissipation.

[1] https://en.wikipedia.org/wiki/Landauer%27s_principle

I second Gonçalo M. Tavares

Indeed, I have an ad blocker. Thanks!

This is nice :)

I tried on Firefox and the thumbnails remain gray, I can only see them when I hover the title.

You can see that your constant is the solution to x=cos(x) (if the equality is true, you can replace the x on the right side with cos(x), and repeat this process).

The solution apparently has a name, Dottie constant [1].

[1] https://mathworld.wolfram.com/DottieNumber.html

This was discussed just two days ago:

https://news.ycombinator.com/item?id=23561018

Or even the Digamma [1] signal that appeared independently at the same time in two experiments at CERN.

[1] https://en.m.wikipedia.org/wiki/750_GeV_diphoton_excess

Spaced repetition [1], they send you questions over increasing time periods to consolidate what you learned in the long term.

[1] https://en.wikipedia.org/wiki/Spaced_repetition

I agree with this! I prefer reading Axler's 2nd edition because it is a lot less distracting.