renyicircle

>what's the meaning of '+' there?

Mathematicians often reuse the same notation for different things where it's clear (for the mathematicians) what the implied meaning is. That means however that non-mathematicians can get confused.

The plus here is the operation that we call "the sum of complex numbers", where the numbers are a and bi. We can work towards this by first defining complex numbers as a tuple of two real numbers with particular sum and multiplication laws for these tuples. So (a, b) + (c, d) = (a + b, c + d) where the left side plus is for complex numbers and the right side plus is for real numbers. Similarly for multiplication, (a, b) * (c, d) = (ac - bd, bc + ad).

We associate (1,0) with the real number 1, and similarly for tuples of the form (a,0). The association is what allows us to see the real numbers as a subset of the complex numbers rather than being some unrelated set. It makes sense because the set of tuples of the form (a,0) with the multiplication laws of complex numbers behaves the same way as the real numbers and is in 1:1 correspondence with them. This association is used every time we write (a,0) as a(1,0) or just "a". We also write (0,b) as b(0,1) since (b,0)*(0,1) = (0,b), from the multiplication law. A general tuple (a, b) then can be written as (a,0) + (0,b) or, as what we've said is the same thing, a + b(0,1). Now we have simplified notation for all of these things except (0,1). It follows from the multiplication law that (0,1)*(0,1) = -(1,0) = -1. The complex number whose square is -1 is denoted as i. All these simplifications allow you to write (a,b) as a + bi.

> More fundamentally what's addition and multiplication? Is there a continuity between them, like an operator that represents both addition and multiplication meeting them half-way?

I'm not personally aware of any concept of continuity between these operations. In abstract algebra, operations are fundamental to an algebraic structure. The field of real numbers or the field of complex numbers are algebraic structures defined by their elements, the binary operation called addition, the binary operation called multiplication, and the laws that these operations satisfy. For example, both are commutative and associative, but the distribution law shows that they're different: a*(b+c) = a*b + a*c, but not a+b*c = (a+b)*(a+c). Also, a+0 = a, but a*1 = a, so the operations have different special elements that "fix" the other element when an operation is applied. It may be interesting to think what a "half-way operator" would have to be like with respect to these laws.

>What's the significance of square and square root? Aren't they just some exponent values (2, 1/2) which have no particular significance on the real line?

The idea of complex numbers and the more general idea of field extensions, which describes the process of obtaining complex numbers from reals, came from the study of polynomials. Polynomials can only have terms of integer degree. They satisfy many nice properties and quite rich mathematical structures arise from them which would not be possible with general non-integer degrees. Second degree polynomials with real coefficients, like x^2 + 1, are the lowest-degree polynomials such that they may not have real roots, and x^2 + 1 is the simplest example, but any other polynomial can be used to produce the complex numbers as well, if has a factor like x^2 + a. This is where the 2 comes from. It's not significant from a real line perspective (2 is just a number among the infinite reals) but it's significant from an algebraic one.

Thanks for pointing this out, I hadn't noticed that. I'd expect a large customer to be persuaded by Danfoss to upgrade within a year, especially before publishing the details of the vulnerability, so you're probably right.

This is a great find, would be nice for it to be mentioned in the original article. It all worked out just like the title says. I guess someone didn't upgrade their firmware in time.

I didn't expect to spend so much time today thinking about fridges. I have so many questions. If it's all so connected and monitored and notifies of issues, couldn't it notify anyone that the settings are weird, like continuous 50 degree Celsius defrosting? How much access did the hackers have, apart from temperature controls? Could nobody intervene quickly before the damage was done? Was there a way to "unplug" the fridges or they would just keep frying, so they had to get people across all the stores doing manual resets?

It's probably a small drop compared to the overall amounts of wasted food. I guess the hope is that it will teach some people important lessons, like "your refrigerator shouldn't be connected to the Internet, and if it absolutely has to, take some time to consider security".

A rogue knife could potentially stab someone if it was secretly a people-stabbing robot which was unknowingly activated, causing it to move towards people to stab. What AI companies are doing is handling this robot that they know can potentially stab people, and trying to convince it nicely not to activate because they don't know what the activation trigger is and how to remove it. "Rogue" doesn't really make sense in this context.

Thank you for the info.

> I fear most Russians will just turn on Ukraine

In which ways does the situation where "most Russians turn on Ukraine" differ from the current reality?

Damn, this would have been a good argument for the US to fund Ukrainian long-distance operations, had Russia not been so far behind in the AI race already.

A student's letter from the post says

> will math academia be big enough and accessible enough for anyone with sheer passion (despite not being the brightest mind in the field) to pursue a career in, or will it inevitably shrink such that it will only really be accessible to the brightest minds?

Could someone with a recent math PhD experience tell me if it this hasn't been so for years? Competition is incredibly high because people have to compete with every "brightest mind" from all around the world. PhDs are overproduced relative to the number of permanent positions. It's not a question that you only start to think about after you've seen LLMs do mathematics, and if that's the case the student should be made fully aware of what he's getting into. I don't want to call the original post "dishonest" but I wish this was addressed here without the LLM angle. However, it was written by a tenured professor, so he did say the only thing he realistically could.

Well, adding more Lean verifications was something they were planning to do from the start, so it's no surprise that they did. The withdrawals are more interesting in that regard. I don't really understand why they didn't only put out the results that had Lean verification already. This way they could avoid the withdrawal thing, because these papers didn't have Lean, and also add whole new papers after they had Lean code for them, which would be the title of the post then.

The 6 additions are Lean verifications for already published pre-prints, not new results.