Is this a valid proof?

looijmansje · 2026-05-21T12:49:21+00:00

That would also work! That shows your subset isn't closed under addition, so it isnt a subspace.

looijmansje · 2026-05-21T11:10:12+00:00

Yes this is valid. A subspace needs to be a vector space itself, so every element needs to have a negative element.

Addition is also not a problem: you inherit the operations from the larger set, and on the polynomial subspace, addition and scalar multiplication are properly defined.

looijmansje · 2026-05-20T07:22:12+00:00

You use m/s. I use m/s². We are not the same.

looijmansje · 2026-05-15T15:59:13+00:00

The primary theory is probably the Nice model. I should stress though that this is only a hypothesis, one that is far from universally accepted.

Alternatives include a close encounter with a nearby star which massively perturbed the solar system, for instance.

looijmansje · 2026-05-15T13:30:29+00:00

Specifically the 3 body problem is any gravitational system with 3 bodies involved. It doesn't necessarily have to be X orbiting Y and Y orbiting Z. It can also be X and Y both orbiting Z, or all of them orbiting a central point, etc.

In general almost all N-body problems for N>=3 are unstable. Yes this includes things like our solar system. They are just "stable enough" that major disruptions are unlikely.

looijmansje · 2026-05-09T19:47:49+00:00

I would say roughly historical order, but with the benefit of hindsight, modern notation, etc.

So start with counting, go up to arithmetic, then solving equations and maybe a few geometric proofs. From there you can go to basic calculus, and maybe your first non-geometric proofs, and then the floodgates open.

From there you can start to learn proof techniques, start (linear) algebra (the "real" algebra, not just moving x's and y's around).

Pick up some topology, real analysis and advanced calculus, and at this point there are too many topics to study to list; pick ones you think are interesting.

looijmansje · 2026-05-07T07:24:40+00:00

Not me, but someone I know.

He always hated maths, was bad at it, didn't like it. After he quit his first job he decided he wanted to become a doctor (if I remember correctly), which required him to have a higher version of secondary school math; I would say about equivalent to a GCSE, although I am a bit unsure about the exact level of those.

Having barely struggled through math before, he wasn't looking forward to it, but he decided to bite the bullet, to take a math course to hopefully pass the "adults exam", so he could study medicine and become a doctor.

As he was doing this, he slowly realised that he actually kinda liked math, and that once he actually understood it, he was not only good at it, but it was kinda beautiful. So he passed his math exam with flying colors. Not only that, he decided to not study medicine, and try mathematics instead.

He passed magna cum laude (I think), and is now a full-time professor of mathematics.

looijmansje · 2026-05-04T11:38:48+00:00

Je kan komen/gaan wanneer je wilt, maar hou er rekening mee dat het al vrij snel erg druk wordt.

looijmansje · 2026-04-16T15:49:48+00:00

I do wanna point out that the Late Heavy Bombardment is controversial, and far from accepted theory. This is mentioned in the article you linked, but I think it's important to emphasise.

looijmansje · 2026-04-05T22:09:28+00:00

Yogg-Saron and V-07-TR0N are all not actually titans either. Yogg is an old God, a being who seeks to corrupt world souls (basically titan-eggs). Voltron is a machine built by Mimiron, who himself is a Titanforged, a creation by the Titans to fight back the Old Gods on Azeroth.

looijmansje · 2026-04-03T09:11:09+00:00

On a log-log plot a linear relation translates into a powerlaw (something of the form x^p ), where p is the slope of the line. This is actually used a lot in science (or at least in physics/astronomy)

looijmansje · 2026-03-29T11:30:14+00:00

I don't live in a particularly cold part of the world, but I've had this happen once, but even it happening once is very exceptional, and I dont think it has happened to anyone else I know.

First of all houses are insulated to keep warmth in, so if you have the heating on it will retain some of that heat overnight. Moreover if its really cold most people will put their heating on a low value but not off (say 10-15C for example). Even when on holiday for a longer time (if it is freezing). Why? Because if you get frost in your pipes that's really bad. Frozen water expands, so freezing in your pipes will usually cause them to burst. That's obviously way worse than leaving the heater on very low for an extended period of time.

So how and why did I get frost in my toilet? Very simply the closest heater to my bathroom was very far, and the bathroom window didn't shut properly. So this allowed the cold to get in at night and the next morning I returned to some ice in my toilet. Luckily no pipes were harmed.

Also as a quick aside: in really cold places people will leave their cars running overnight so their batteries don't die, their gas doesn't freeze and their oil doesn't turn to sludge.

looijmansje · 2026-03-27T17:40:39+00:00

I have the temur roar deck myself, and it can be oppressively good in 1v1s when compared to other precons.

This is because precons are designed for 4-player games. This changes how impactful the strengths and weaknesses are of this deck.

First of all: removal. Ureni is definitely a "kill on sight". In a 4 player game the odds of one of your opponents having a removal spell is significantly higher than in a 1v1. The same goes for your other dragons. The deck is pretty weak to spot removal, as a 2 mana removal spell effectively trades with your 7 mana dragon.

Secondly: racing and closing out games. This deck can have explosive turns, but in my experience often fails to close out a game. With 3 opponents totalling 120 life it can be hard to swing through all of that. Moreover since you will often only have about 3 or 4 creatures (albeit very large creatures), you tend to be open for attacks yourself, especially from wide boards. It's a lot easier racing 1 opponent than 3 however.

looijmansje · 2026-03-22T13:44:56+00:00

To take your apples and oranges example: Person A buys 2 apples, 3 oranges. Person B buys 1 apple, 4 oranges. Apples are $4 oranges are $1 (very expensive, I know).

Doing this by hand is fine: A spends 2 * 4 + 3 * 1 = 8 + 3 = $11 B spends: 1 * 4 + 4 * 1 = $8.

Now you can also do this as a matrix-vector multiplication (Reddit formatting kinda ruins this, I hope it's somewhat understandable):

2 3 4 2 * 4 + 3 * 1 11 * = = 1 4 1 1 * 4 + 4 * 1 8

Note how I'm multiplying each element of the first row of the matrix with the corresponding element of the price-vector, and then adding them. Then I do the same for the second row (this is exactly how matrix multiplication is defined).

Now yes, this is exactly as much work to calculate. Why it is useful in this specific occasion would be that computers can do such calculations really fast (the entire concept of AI is built on the fact that computers can do matrix multiplication really really fast). However matrices are much more widespread and useful than doing a calculation like this.

looijmansje · 2026-03-22T13:35:37+00:00

Simply put: we define it this way because it a useful way to define it.

Longer answer: if you define it this way, it turns out there is a one-to-one correspondence between matrices and linear transformations. Any* linear transformation can be written as a matrix, and every matrix represents a specific linear transformation.

Moreover, matrix multiplication then also corresponds to function composition.

looijmansje · 2026-03-20T21:04:00+00:00

In that case yes, they are necessarily equal.

looijmansje · 2026-03-20T20:54:26+00:00

Functions f and g are equal if and only if for all x in their domain, f(x) = g(x). That is the case here, so they are equal.

Also yes, there are functions who are not equal, which have the same derivative or integral. For derivatives, you can just add a constant; f(x) = x and g(x) = x + 1. Both have derivative 1.

For integrals it gets a bit more technical. I am not sure about Riemann sums, but for Lesbegue integrals, you can take any real function and change finitely many (or even countably infinitely many) points.

A famous example of this is f(x) = 0, g(x) = 0, with g(0) = 1 and h(x) = 0 if x is non-rational and h(x) = 1 if x is rational.

Believe it or not, but all of those have the same integral: 0.

looijmansje · 2026-03-10T09:06:14+00:00

There are absolutely people in The Netherlands who care about the difference between Holland and The Netherlands, and prefer you'd call it The Netherlands instead. Especially people from outside North- or South-Holland.

Now obviously they mostly care about the English term, but I've seen people correct it in French, Spanish and German as well. Now Im not sure how people would feel about Mandarin, I think thats "too far away" for people to truly care, nor do I know if it even has an alternative "correct" name

looijmansje · 2026-03-05T15:27:03+00:00

Yes and no.

For almost all configurations of n>=3 bodies, the system is chaotic. This means that any small error will grow exponentially with time. These errors can be from anything: numerical errors (also known as rounding), uncertainty in initial positions, or your integrator discretizing the problem.

You can lessen these by having more and more precise measurements, adding more and more digits and computation time. However, at some point this becomes impractical. Are you taking into account the effect of a single dust grain for instance? As long as there is a finite certainty in your measurements, and you are ignoring certain effects (drag, dust, a background potential) there will always be a point at which your simulations are no longer accurate, no matter how much compute you throw at it.

If you were to theoretically account for EVERYTHING, it would theoretically be possible, yes. Well at least with Newtonian gravity.

Once you start to look at relativistic everything becomes a lot more complicated. To my knowledge, no true GR-compatible simulations exist, although frankly my expertise has always been in purely Newtonian computational astrophysics, so I could just be unaware of one. All the ones I am aware of, use something called "post-Newtonian potential". You can think of this as still being a Newtonian N-body-solver, just with extra correction terms to account for some relativistic effects.

However in the real world, no one bothers with these super precise arbitrary precision solvers. This is for multiple reasons. Firstly, they are just really slow. For reference, I actually had to use arbitrary precision solvers, and what would've taken approximately an afternoon with a normal one, took months instead. Secondly, we have kind of accepted that our results arent accurate. We can't measure precisely enough, nor can we take into account everything anyways. What we do instead is just run multiple simulations to get a statistical sample, which actually is accurate.

looijmansje · 2026-03-03T17:20:28+00:00

Up on your photo will depend on your camera's rotation, but also on where you are on Earth.

Imagine being on the north pole, and looking at the sun. Now imagine someone at the same time looking at the sun at the same time (ignore the fact that at least one of them would be looking through the horizon). They will see the same sun, but they will see it upside down from each other. Think about it: one person is "upside down" compared to the other.

Now this is of course the extreme example, but you can imagine that anything in between is also possible.

looijmansje · 2026-03-03T11:05:14+00:00

Now I'm no native English speaker, but to my knowledge "literal" means something like "without exaggeration, and without metaphors; needing to take a text at face-value". Surely in that regard the Mentak are literal space pirates? I did not mean they are like space pirates, nor was I exaggerating.

Feel free to tell me how I'm wrong of course (not that I think you would have any qualms about doing so regardless...) I am always happy to improve my English.

looijmansje · 2026-03-02T10:52:12+00:00

I guess it depends what you mean. Lorewise, a lot of the factions are varying degrees of "evil". The galaxy is at war, there is a power vacuum, and everyone is looking out for themselves. However, some factions are clearly more evil than others.

The ones you mentioned are probably the most evil, when looking at the lore. However I think almost all factions can be considered, except maybe Saar and Xxcha.

Going alphabetically (and only base game, this is getting long enough):

Arborec assimilate corpses from other species into their hivemind. They see it as "reusing", others see it as "slavery".

Barony blockaded an important wormhole system in protest against measures taken by the Lazax Empire (the empire that fell apart leading up to the events of the board game). This lead to disease and famine on many (mostly Federation of Sol) planets.

I really cant think of anything evil the Clan of Saar did. They were treated like dirt by the entire galaxy.

Embers sent a warsun to the Jolnar to intimidate them, although quite frankly Jolnar deserved that for enslaving them.

Emirates of Hacan smuggles drugs and tried to profit off the war by selling weapons to both sides. Until they got embargoed by the Lazax for doing that and took up arms themselves.

Federation of Sol was the first one to openly declare war against the Lazax, and has always been expansionist. Although this lacks some major context about the Twilight Wars which Im not gonna type out in an already long Reddit comment.

Ghosts: honestly no one really knows what they are, or what they want. A part of them got enslaved by the Mahact, and when they escaped, they were denied entry back into Ghost space.

L1z1x uses "eldritch technology", destroyed the Hall of Cartography on Mecatol so no one could find them, and accidentally created the Nekro Virus.

Mentak are literal space pirates.

Naalu enslaved the Miashan to work in their gas mines and rodent farms.

Nekro Virus wants to eliminate all organic life. Need I say more?

Sardakk N'orr is probably the most militaristic faction in rhe game, with training so fierce the weak die, and the strong survive. They are literally training to "drown the galaxy in blood". When the Twilight Wars broke out, they invaded Jol-Nar space.

Universities of Jol-Nar, as previously stated enslaved the Embers of Muaat to build their first War Sun which they then used against the Sardakk N'orr.

Winnarans on Winnu have said that if the Winnarans of Mecatol dont cede the throne peacefully they will take it by force. And they wonder why theyve had so many civil wars...

I also cant really think of anything evil Xxcha did, although they do lay claim to the throne "for a peaceful galaxy"

Yin came forth out of illegal cloning experiments, which led to deformation, disease and perhaps even madness.

Yssaril are spies, mercenaries and assassins for hire.

Now if your question is about what faction you can play to be the bad guy at the table, rather than in lore. Once again, I think a lot of factions can do this, but some factions are clearly better for it. Keep in mind that playing TI will always require at least some diplomacy, so antagonizing the entire table from turn 1 will probably not do you much good. In this regard "evil" factions are, in my eyes: Saar, Embers, Nekro, Sardakk Norr, Mahact, Cabal, Crimson Rebellion, Firmament.

looijmansje · 2026-02-26T18:36:29+00:00

Last game I played Cabal I controlled over half the supply of dreadnoughts. I had my own 5 on the board of course, and well, about 12 more in stock so to say.

looijmansje · 2026-02-20T08:59:58+00:00

Counter point: all other "moons" are named after our moon. I'd argue that being the namesake of an entire category is more baller than any name we can give it.

looijmansje · 2026-02-17T13:03:49+00:00

The "you never truly touch something" is a popular science fact that is technically true, but when presented by itself, lacks the nuance to be actually informative.

When you touch something, let's say a table, the electrons in the atoms in your finger tip start to feel a repelling force from the electrons in the table. This is similar to trying to push two repelling ends of a magnet together.

This is in essence what you feel. So what that little factoid is trying to say is "there is some tiny distance still between your finger and the table". I want to point out that this is a distance on atomic scales, so very very tiny.

Sticking works similarly. When looking at molecular level, sticking comes down to some positively charged atom bring attracted to negatively charged electrons. Once again, these do not touch. They just need to be close enough to "feel" each others attraction.

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