In my (old) field, this was not really true, but maybe it’s true in some other fields.
But it is true that if all these grants are cut, it will have a big effect on research output due to killing off early career researchers. It’s clear that the US Government doesn’t give a shit that it’s going to ruin the country’s research credentials.
The succinct way of defining the topology (on NxN) is the product topology of the discrete topology(/ies). Maybe that’s the discreteness you’re feeling?
Axiom of Choice is not regarded as a big deal by most set theorists, but it’s interesting when it comes up. The diagonalisation proof that there are undetermined games uses choice to well-order the set of strategies, so it’s actually the other way around: without choice it is consistent (assuming consistency of some other stuff) that all games in this formulation are determined. This is called the Axiom of Determinacy.
The axioms in question are power set and replacement: to prove full Borel determinacy you need to apply the power set axiom infinitely many times (using the replacement axiom). These two axioms are what gives the ZFC axioms their power, really.
Set theorists nowadays I don’t think debate about axioms per se. Set theory nowadays is at once somewhat pluralistic and somewhat settled (paradoxically). I’ll explain a little: set theorists are basically agreed that the ZFC axioms are natural, intuitively “true” (many set theorists would not put scare-quotes there, but I would), powerful enough to do all ordinary mathematics and more, and probably consistent. They also generally agree that many large-cardinal axioms are natural and probably consistent, though there is a wide variation in whether people think they are “true”; there is not nearly so much intuition that such huge objects could exist. This is different than our intuition behind the axiom of infinity, because that axiom is actually needed to do some ordinary mathematics (though you can do without it for a lot!)
The Projective Hierarchy continues the stratification of the Borel Hierarchy even further. If you assume infinitely many woodin cardinals, then you can prove Projective Determinacy. I have heard Tony Martin being quoted as saying that “if Projective Determinacy were found to be inconsistent” (and hence infinitely many Woodin cardinals is inconsistent) “then I’d be having serious doubts about [the axiom of] Replacement.” This gives you a flavour of how people think about the relationship between these concepts.
Steam already has an FPS monitor, right? I think that’s enough for 95% of everything (and it’s very unobtrusive).
I wonder if this is the best they can come up with in response to the airbase attacks?
As far as I can work out about this USA, this is not true. It is certainly not true where I am from. It may be true in the case of postdoctoral researchers (but not always), i.e. relatively junior researchers who don’t yet have a permanent position. But a permanent position is just that - it’s like a permanent job, and you’re paid a salary by the university that gave you that position. You will typically also need to apply for grants in order to pay for things like:
I did two postdocs during my time in academia and both were grant funded (one awarded to me, one awarded to a more senior researcher who then took me on as a postdoc). I also applied to one postdoc position I remember which although fixed term, not permanent, was paid for by the university. I worked with many permanent staff who had salaries from the university as well as grants for other things.
As far as I can tell in the USA the only real difference is that your salary may only be for the 9 teaching months, not the full academic year, and you’re expected to top up those 3 months if you want to be paid a proper wage.
This is not really “mildly infuriating” and you should make the link point to the article, not to a screenshot from it.
As a field of study, it’s the study of two-player games of perfect information (so think chess, not football or poker) in which each player may make countably many moves (you can also look at uncountable-length games but it’s not common). I’ll give you more detail than I would a child :P
Each player takes turns to move. You can encode the moves they make as coming from some set - for example they might just play numbers. The rules of the game are imposed by a winning set, which is a set of countable-length sequences of moves, and we say that player I wins if the infinite sequence of her first move followed by player II’s first move followed by her second move, etc, is in the winning set. Otherwise player II wins. (There are no draws, which technically means chess falls outside the scope of this setup, but it turns out not to be a big deal)
(This allows you to encode what moves are allowed by the rules - you just say that any sequence which contains a move where that player broke a rule is a loss for that player, regardless of what comes afterwards.)
Each winning set defines a different game. The property of determinacy is a property of sets of infinite sequences which says that there is a winning strategy for either player. A strategy is just a function which takes the finite sequence of moves up to that point in the game and tells the player (the player for whom the strategy is) what to do. A winning strategy is one which, if followed, always results in a win for that player.
If we modify the rules of noughts and crosses (tic-tac-toe) so that draws are arbitrarily decided to give a win to player I, we know that this (finite) game has a winning strategy. In fact, any finite game has a winning strategy (or, if there are draws, this means there is a non-losing strategy). The outline of the proof is that if player I does not have a strategy to get to one of the (finitely many) winning states, then we can find a strategy for player II which avoids those winning states. (Remember, winning states are winning for I).
So, which games are determined? Are all games determined? Well, it’s actually easy (through a diagonalisation argument, same as proving uncountability of the reals) that not all infinite (countable-length, that is) games played with natural numbers (as moves) are determined. But you can create a way of categorising the sets of countable sequences of natural numbers (i.e. the possible winning sets) by a kind of complexity. This is the basis of descriptive set theory. It starts with topology: you can define basic open sets in this space as those sets consist of all infinite sequences which share a common finite prefix. Closed sets are the complements of open sets, as usual. But then you can define a hierarchy of complexity where the next level are countable unions of closed sets, then the next level are countable unions of complements of countable unions of complements of open sets. (An introduction to descriptive set theory will say more about this).
It’s quite easy to prove that all open sets and all closed sets in this hierarchy of complexity are determined. It’s a little harder to prove that the second level is determined, and harder still to prove that the third level is. Eventually a guy named Tony Martin (D. A. Martin) proved that all Borel sets in this hierarchy are determined. If you know your analysis, the Borel sets are exactly what you’re thinking: they’re the sets formed by all arbitrary countable unions, intersections and complements of open sets.
The interesting thing about this proof was that it needed a huge amount of set theoretic “power”. Most ordinary mathematics like analysis doesn’t need all the axioms of set theory, but this needed a massive chunk of them. This makes it interesting to set theorists because it tells us something about the relationship between something quite concrete: complexity of sets and strategies for easily-defined games on the one hand, and something quite abstract: the axioms of set theory. This pattern continues higher up: more determinacy can be proved if you assume even stronger axioms, going beyond what is typically included in set theory.
How much sugar do you add?
How long do you boil it for? (You don’t know; you have to monitor the temperature)
How long does putting it all in jars take you? (ages)
It’s not difficult, but it is time consuming and not trivial.
Making jam is not trivial but it I think that makes it rewarding! My dad has made jam and marmalade for as long as I’ve known and it’s always an event. My parents have hundreds of jars (for some reason my dad calls them bottles? Only in a jam context though!) and every so often he cooks up a giant pot of jam with an old-fashioned sugar thermometer, testing the batch on a piece of baking paper, then bottling everything up. He often did it with my sister, who now also makes her own jam.
He labels all the jars, and we’ve opened jars that were… I dunno, a decade old I’m sure, and they were totally fine. So they will definitely keep for a long time!
I have a PhD in mathematics (set theory). It would be pretty tough to explain to a child. Specifically my PhD is in determinacy, which is way easier to explain than most branches of set theory, but you do need a decent understanding of infinity to really get anywhere.
I’m sure Netanyahu will be calling the IDF anti-semitic in a few hours.
I mean the UK has 6% of its energy over the year come from solar, and 30% from wind, and installations are only accelerating, so this amount of installed solar is far from unrealistic.
Installing storage approximately doubles the LCOE of solar energy, so this is also feasible from a cost point of view as we get rid of dispatchable gas turbines.
Basically if we solve storage, we can get rid of nuclear, but not before.
I am not saying we should get rid of nuclear. I am saying we should keep some nuclear, also once we have got less gas and more storage. Does this resolve some things in this discussion for you?
Not really enough information. I will assume that by “installed solar power” you mean peak generation when the sun is shining, and that instead of peak use, you mean 40% of average use, i.e. let’s suppose that at an average moment the country consumes 100GW and, if the sun is shining, generates 40GW from solar.
Assume further the sun is up for half the year and the sky is clear for half the year, meaning the total amount of your yearly electricity you can generate with solar is 10% assuming typical weather. Then you would be able to reliably power the country with a combination of nuclear totalling 90% of average use (90GW) and enough storage that you can ride out cloudy periods.
No, COVID is very low in the US - just as in the early days when there was little testing available, you can track what the disease is doing by looking at hospital admissions data.
I did mention people who are particularly vulnerable, and if that includes you then the rest of what I said doesn’t apply to you. I find it’s more efficient to talk about the likely case and mention the exception in a single message than to first ask “are you immunocompromised” and then reply separately.
I would go a step further and have creative grants to people. It would work in a way similar to the BBC and similar broadcasters, where a body gets government money and then picks creative projects it thinks are worthwhile, with a remit that goes beyond the lowest common denominator. UBI ensures that this system doesn’t have a monopoly on creative output.
You say “maybe” like it’s some unknowable quantity, but we can see what is going on in Hungary and that it’s getting increasingly authoritarian, and we can see Russian attempts to influence western democracies through bot farms and the like. It’s not unknowable; the evidence is there.
There are companies, big companies even, that kill off underperforming projects after less than a year, and most don’t take years to agree projects.
I don’t understand what you’re asking, sorry.
I dunno about the MAGAs, but I think at this point, with COVID very low in the USA, most people are wearing masks because it sends a signal about where they stand on COVID, not for their own protection. Obviously there are some people who are particularly vulnerable, who can’t have the vaccine for example, for whom mask-wearing will remain important forever (and would likely have been beneficial before the pandemic) but I don’t believe that’s the majority.
The reason is that mask wearing in my home country went down to zero after most people were vaccinated and the virus became uncommon. But here mask-wearing was never politicised, so without an anti-mask group to oppose it never became a symbol for those who did wear them.
It wasn’t though, was it. The IDF are not Jews in general; they are multi-ethnic and are the armed forces of a country at war. Would a chant of “death to the Russian Armed Forces” be Russophobic? “Death to the Wehrmacht” for anti-German during World War 2? “Death to Hamas” for Islamophobia?
Identification of the armed forces of a state with a state is a sign of fascism, and the identification of the state with an ethnic group is a sign of extreme nationalism - though admittedly that is less the case with Israel and Jewish people.
Chanting “death, death to the IDF” is violent and inappropriate at a music festival. “Fuck the IDF” would’ve been fine though.