sgdpk

Second Axler's book! (probably as a second exposure after a first course, to really understand what's going on in Linear Algebra)

We have been pretty well informed of Musk's shenanigans in Portugal. His sieg heil spread pretty quickly.

Ben Felix recounts Buffett's answer to this in 2019:

https://youtu.be/32SnTUtdsOI?t=557

Cool :) I am a scientist, so having an easier way to parse the abstracts would be most welcome. Keep up the good work.

It looks great. But it's missing one critical feature of "fast-food" apps like TikTok: the content is not easily digestible. Which is understandable, because scientific papers are dense.

Maybe a good idea would be to parse the abstract through an LLM to make it more understandable (maybe caching the results so it's not expensive)? Maybe also using some standard style, like starting with a couple of "dumbed-down" sentences of the article for the non-expert, and progressively explaining better.

Good read. It is also refreshing to read how Schrödinger came up with his equation even if it was not clear how to interpret it.

He was attempting to formalize de Broglie's "particles as waves" concept, which, according to the article, "could obtain the quantization rules of Niels Bohr and Sommerfeld by demanding that an integer number of waves should be fitted along a stationary orbit."

Schrödinger's equation put that claim on firm mathematical grounds. It gave correct predictions. But just what this new "wave function" was remained up to interpretation.

I don't know about this book, but I highly recommend "Linear Algebra Done Right" by the same author. It is a very clear presentation of Linear Algebra. Although I would recommend it for someone who already took a first course on it.

Thank you for replying! The project seems promising, I'll follow with interest. Hope it goes well :)

Does this handle elimination of variables in a polynomial system (using an eliminating order for Groebner basis)?

This is a visualization of what the states of a qubit can look like in two different physical systems (particle spin / photon polarization). Despite their differences, they can be described using the same mathematical object: a unit vector, here represented on the famous Bloch sphere.

I have worked on this aspect on quantum computation [1]. The main problem is that reversibility is a feature of isolated quantum systems. In practice, they are not isolated.

Why? It's not just because of small interactions with the environment that we cannot control. It's that even the apparatuses that we use to control/drive the logical instructions (lasers, electrical transmission lines) should be taken into account if the computer is to be considered isolated. But usually they aren't, and this leads to inevitable losses of reversiblity in the data register.

In other words, unitary (reversible) operations do not come for free.

I think that in quantum computers it is more likely that energy-efficiency will come from some sort of algorithmic advantage.

[1] https://arxiv.org/abs/2210.10470

I am facing the same problem right now. I am abroad but cannot install foreign apps. Have you (or them) found any solution to this in the meantime?

To be fair, I was taught exactly what this paper claims in my Physics degree. Although I was also taught what they call the "myth" in other classes.

When systems (think "automated" systems like computer programs, mathematical axioms, formal systems, etc, where conclusions can be drawn/calculated "mechanically" from a few starting points) get large enough, they gain the ability to become self-referential. That is, they become expressive enough to encode statements about themselves. A hallmark of this are "incompleteness theorems" like those of Godel or the Turing halting problem.

The book argues that these "strange loops" (of a system onto itself) are behind the emergence of intelligence and consciousness, because physical matter itself gives rise to human intelligence, albeit being a mechanical system.

My view is that there is a landscape of mathematical truths, but we can only explore/discover precisely those things that our imagination allows.

In other words, we discover what we can invent.

I think you mean table spoon, which is often enough dressing for one person, in my experience.

Yet another fact discovered by von Neumann.

I am becoming convinced that sugar, although not exactly a drug to which we could apply the concepts of addiction and withdrawal (like tobacco, alcohol, or opioids), nevertheless creates habits that are really hard to break.

Notice how we train animals with treats: sniffing drugs, attacking robbers, etc. It's a powerful behaviour modulator.

Our brains are no different. In a pavlovian way, we reinforce behaviours that give us a sugary reward. But in our case, the feedback loops are really short. Instead of doing a difficult task, our behaviour to get the reward is simply going to the kitchen and opening a snack.

Do you ever get the feeling, after a meal, that you feel like having "a little something", like a sweet? To me, that's like a learned pavlovian behaviour. And every time we cave, we reinforce that automatic response. It can be anxiety-inducing not to do the behaviour.

The question in my mind is: how do we break this pattern? Because it's easy to do it once, but it's statistically hard to keep it up many times. You will slip up and reinforce the behaviour again.

I am becoming increasingly convinced that we need to change the environment around us, that is, regulate the amount of added sugar in foods.

Good point. I tried asking ChatGPT to write the numbers from 1 to 10 in 1 sec intervals and it couldn't.

Edit: Oops, I read the question the other way around. Humans can easily emulate ChatGPT's behaviour in this scenario -- just say that, as an LLM, you cannot do the task.

Inspired by this, in my PhD we actually used a quantum computer to do classical logic to investigate if that can lead to energy savings [1]. Quantum machines are (in principle) reversible, so they may avoid Landauer's principle.

However, there are subtle energy costs that you can hit before getting to Landauer's. The most interesting to me is that the qubits can become entangled with the wires that control them! This reduces the quality of information, and one way around it is to use a lot of energy [2].

[1] https://arxiv.org/abs/2210.10470

[2] https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.89...

So, for your first questions, you can definitely entangle as little or as many qubits as you want. In practice, it gets harder the more qubits you want to entangle. But a state like the GHZ state can entangle all qubits in your system.

How do you entangle? You can just let two particles interact, so they "mix" their information. Example: you send laser light (photon qubits) to an atom (another qubits). After a short period, there is a probability that the atom absorbed photons, but that probability is not 100%. You just entangled light with an atom. Only measurement can give you information on whether the photons were absorbed or not.

In practice, each platform has its own entangling mechanism. Usually, it entangles only 2 qubits. Many-qubit entanglement can be achieved by pairwise entangling AB, BC, CD, etc.

The first practical example of qubit entangling operation was the Cirac-Zoller gate, you can check it out.

Regarding your second question, you can actually measure just part of the system, and measure the rest later. It will give you a partial collapse. It's called "tracing out", the quantum analogue of marginalization in probability.

I also just withdrew EUR (12h ago) and it arrived relatively fast.

I still see these guys in Lisbon's main train station trying to get new customers. They also seem to do some sort of face scan for new customers.

As someone commented, this example is not so mysterious. It's just a change of variables to make the argument of exp() dimensionless.

Here's a good example of the power of dimensional analysis: how small should the Earth be to collapse into a black hole?

Knowing nothing about the problem, you know it should involve at least two things: the mass of the Earth, M, and the gravitational constant G.

Since F=ma and F=GMm/d^2, we know that GM has units of distance^2 * acceleration (check that). This is equal to distance * (distance/time)^2.

We want a radius, which is a distance. And we almost have it! At least if we can get rid of the (distance/time)^2 factor. But that's a velocity^2! Now, what's a velocity that should be natural in questions about black holes and general relativity? Why the speed of light c, of course.

So, we can guess that the answer is GM/c^2.

Now compare this with the real answer: https://en.m.wikipedia.org/wiki/Schwarzschild_radius

The video you sent is by the last author of the paper, so it seems legit.

I have to second this. It's very well written and presents a clear view of what Linear Algebra is. Although it might be best used as a second book in Linear Algebra (depending on your preparation).

I don't know exactly from what angle you're looking at this, so let me explain through trigonometry.

We know that triangles may be displaced, rotated, flipped and scaled while still looking the same. We have a word for this: we say two triangles are congruent when they are the same up to these operations.

This means that there is something intrinsically invariant about triangles. Can we find it? Actually yes! If the sides of a triangle have lengths A, B, C, then the ratios A/B, B/C, etc. are invariant. That is, if you make a triangle twice as big, all sides will multiply by 2, so A/B becomes (2A/2B)=A/B -- it's the same!

So, we can come up with names for these invariant ratios. They're most useful for right triangles. To give names, we need to pick any of the two smaller angles as a reference, call it "a". Now, let C be the largest side, A be the side opposite to the angle and B the adjacent side. Then, the ratio A/C can be called sin(a), B/C can be called cos(a) and A/B can be called tan(a).

Since sin(a), cos(a) and tan(a) are ratios, they only depend on the angle, not how big your triangle is. But if you know some side of the triangle, then you can know all of the other using these values. So sin, cos and than really capture the uniqueness I was talking about!

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Now the applications. I memorized these as the "divide by C" rule.

The Pythagorean theorem says thar A^2 + B^2 = C^2.

Divide by C^2, and you get sin(a)^2 + cos(a)^2 = 1.

Divide by cos(a)^2, and you get tan(a)^2 + 1 = sec(a)^2.

These are all the Pythagorean theorem on disguise.

This is a great example! I was fortunate enough that my father explained it to me when I was in school. An equation is like a balanced scale: whatever you do to one side, you must do to the other, to keep things balanced.

I still remember that "click".

The Selfish Gene is great. The portions on Evolutionary Stable Strategies are particularly interesting. It shows how Darwinism can lead to a "steady state", where a population contains individuals with competing characteristics (selflessness VS selfishness), and why this not at odds with selection.

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.