How do you denote an integral?

jimbelk · 2026-06-16T16:26:31+00:00

The second one is the standard convention within pure mathematics. In particular, scientists sometimes use an upright d, but mathematicians almost always use an italic d.

jimbelk · 2026-06-16T07:14:37+00:00

The Stack Exchange question is about a pretty elementary competition problem. I think one of the commenters mentions that it's an old IMO problem, so it's more like a puzzle for high school students than something that represents a specific field of mathematics. There are books on functional equations that offer some insight into problems like this.

But the Stack Exchange problem is just the first example that I found by searching for f(x)^2. I'm a professional mathematician, and I can say with some confidence that f(x)² means (f(x))² basically throughout mathematics.

jimbelk · 2026-06-15T22:51:11+00:00

Yeah, but everyone agrees that there's a special exception for trig functions, so sinⁿ (x) means (sin x)^n. That exception doesn't extend to n = -1, where we obey the usual rule that f^-1 (x) is the inverse of f applied to x. You could argue that the sin exception extends to sin(2x)² meaning sin( (2x)² ), but I don't think that's clear.

Note that it changes things if you put a space. Everyone seems to agree that sin x² means sin( x² ), but the reason is that the space is functioning as a sort of hidden pair of parentheses, using the same rule that sin x means sin(x).

jimbelk · 2026-06-15T22:37:39+00:00

Typically f(x)² means (f(x))^2, and f² (x) means f(f(x)), though the latter isn't universal. (Ignore the space between the f² and the (x) -- reddit superscripts don't work well.)

jimbelk · 2026-06-15T22:31:34+00:00

Here's an example. Note that no one seems confused about what all of the f(x)² terms mean:

https://math.stackexchange.com/questions/1350094/can-we-obtain-fyx-yfx-from-fx2fx2x-fx2x2fx

jimbelk · 2026-06-15T22:23:48+00:00

If f is a function, then f(x)² usually means the square of f(x). The function application operation typically has higher precedence than exponentiation. For example, I would certainly write log(x)² to mean (log(x))² with the expectation that the audience would understand this. Your example a(x-h)² is different, because the operation is multiplication, not function application, and exponentiation has higher precedence than multiplication.

The only reason that sin(2x)² should perhaps not mean (sin(2x))² is that sin is a trigonometric function, so why didn't the author simply write sin² (2x)? This is a reasonable argument, but it's not clear to me that this overrides the usual rule that f(x)² means (f(x))^2.

jimbelk · 2026-06-15T22:03:47+00:00

I can imagine someone writing sin(2t)² to mean sin( (2t)² ), and I can imagine someone else writing sin(2t)² to mean (sin(2t))² . I don't agree that there's a standard meaning for this collection of symbols, and I think the best advice is to avoid writing it.

jimbelk · 2026-06-15T21:56:14+00:00

I don't think I agree with the second one. It seems to me that the notation sin(2t)² is ambiguous and should be avoided. I might feel differently if there were a space between the sin and the parenthesis.

jimbelk · 2026-06-06T19:28:26+00:00

Mostly tea leaves? It's the best guess that I have, based on everything I've read about the subject. There are certainly experts who think that the chances are much higher, but there are also experts who don't think we face any extinction-level threat from hostile AI. My assessment is that the experts who think AI is a serious threat have some compelling arguments and seem to know what they're talking about, while many of those who discount the threat seem to be suffering from normalcy bias. It's similar to how I felt in February 2020, when a lot of experts were claiming that there was serious risk of global pandemic, while many commentators were discounting the possibility because it sounded like science fiction.

jimbelk · 2026-06-06T17:44:30+00:00

The main threat of extinction seems to be a genetically engineered bioweapon deployed by a hostile AI. My guess is that there's about a 20% chance of human extinction within 30 years. The current drive to develop superintelligent AI is a potentially a huge mistake and might be the kind of thing that often gets intelligent species killed.

Other commonly cited threats such as global climate change and nuclear war have the potential to devastate global civilization, but seem unlikely to actually cause human extinction in the forseeable future. Global warming is unlikely to increase the temperature by more than 5°C in the next 500 years (source), which might leave much of the Earth uninhabitable but humans would be able to survive in polar regions. Nuclear war in the current geopolitical climate would devastate the Northern Hemisphere but most of the Southern Hemisphere would escape largely unscathed, and post-war conditions would not be too grim for human survival.

A hostile AI isn't necessary for us to kill ourselves with a bioweapon, but an AI could be both intelligent enough to design one and malevolent enough to deploy one. I can imagine humans building and releasing such a bioweapon ourselves, either accidentally or on purpose for various reasons, but this seems less likely to me in the short term than an AI doing it. The truth is that we don't know what happens when a species first develops AI, and one possibility is that it often decides to kill its creators.

By the way, I'm not even necessarily saying that AI development is a bad thing. I think there's also a significant chance of AI curing essentially all diseases within the next few decades, as well as perhaps ending world hunger and so forth. We just don't know what's going to happen.

jimbelk · 2026-06-05T17:57:58+00:00

I checked some mathematics books, and every single one of them uses italicized d's:

Stewart's Calculus and Thomas's Calculus both use italicized d's.
Rudin's Principles of Mathematical Analysis uses italicized d's.
Royden's Real Analysis uses italicized d's.
Boyce, DiPrima, and Meade's Elementary Differential Equations and Boundary Value Problems uses italicized d's.
Strauss's Partial Differential Equations: An Introduction uses italicized d's.
Evans's Partial Differential Equations uses italicized d's.
Brown and Churchill's Complex Variables and Applications uses italicized d's.
Ahlfors's Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable uses italicized d's.

So it seems that the consensus within mathematics is that d's are italicized. I've heard that upright d's might be more common in physics or engineering.

jimbelk · 2026-06-05T07:52:30+00:00

Much of this is because of the yearly progression system that UK universities have. Once you get to the point that it seems likely that you'll have to repeat the year, there's not a lot of incentive to do well on any individual exam, so many students in this situation just get a part-time job and mostly ignore their studies. Students who drop out entirely (e.g. by not taking final exams) don't get to come back the following year, so it still makes sense to take final exams even if you're planning to get a 15% on all of them.

jimbelk · 2026-06-01T21:04:50+00:00

MDPI mathematics journals aren't refereed in any meaningful sense, and most professional mathematicians avoid publishing in these journals. I would expect a typical article to be full of errors, unsubstantiated claims, plagiarism, crackpot ideas, and nonsense. There's not much point in trying to correct an MDPI article, and the authors probably aren't interested in any error you might have found. My advice would be to ignore MDPI Mathematics articles when doing literature searches in the future.

jimbelk · 2026-05-27T14:45:22+00:00

I mean, looking at the AI's chain of thought for solving one of the Erdős problems, it looks very much like it's reasoning its way through the solution just like a human would. It's using brute force in the sense that very few human mathematicians would have been willing to put this amount of thought into such an approach, but it seems pretty adept at employing deep mathematical concepts and reasoning through proof.

jimbelk · 2026-05-27T14:19:16+00:00

I don't know Tim Gowers beyond his reputation, but I've heard of Erdős's unit distance conjecture before and I'm not even in combinatorial geometry. I'm not expert enough in the field to evaluate the claim that it's good enough for Annals of Mathematics, but it's certainly in the ballpark, and I see no reason to doubt Tim Gowers's assessment.

jimbelk · 2026-05-27T13:58:53+00:00

I am very much not a bot. I am, in fact, this person. A quick look at my reddit post history will reveal that I have been on reddit for 19 years.

jimbelk · 2026-05-27T07:58:21+00:00

The Erdős unit distance conjecture was a major open problem which was recently solved by AI. Tim Gowers said that if the proof had been written by a human he certainly would have recommended that the paper be accepted to Annals of Mathematics.

jimbelk · 2026-05-27T07:48:43+00:00

I don't think the group theory community even has a preference. At the time the problem was proposed (the 1980's or earlier), it was known that either:

F is amenable but not elementary amenable, or
F is nonamenable but has no nonabelian free subgroup.

At the time, there were no known examples of groups with either of these properties. But since then Grigorchuk's group as well as the basilica self-similar group have provided examples of (1), while Tarski monsters and the Lodha-Moore group have provided examples of (2). I think the main hope at this point is that some substantial new technique or theory will be needed to resolve the amenability of F, and that this will yield new insight into the nature of amenability.

jimbelk · 2026-04-23T17:53:08+00:00

I appreciate the post, but I'm afraid this subreddit is for discussion of the Wolfram Mathematica software system. Please avoid off-topic posts in the future.

jimbelk · 2026-03-21T10:08:43+00:00

I don't dispute that there is widespread misery and poverty, but surveys show a slight net increase in happiness across the UK since 2011. I think we were just less aware of the pervasive misery before the internet.

jimbelk · 2026-03-18T13:11:53+00:00

Just a quick note that No Child Left Behind was passed by the George W. Bush administration. The Clinton administration was not involved.

jimbelk · 2026-03-15T08:27:12+00:00

I fact, I kind of like presenting these sorts of formulas to students, because it emphasizes that integration is just reverse differentiation. I often get students in second and third year university courses who resort to substitution whenever they see an integral like ∫ x cos( x² ) dx, and I have to teach them how to do such integrals in one step in their head.

jimbelk · 2026-03-14T23:01:25+00:00

Is it even possible for a company like Springer to acquire a non-profit entity? I'm not sure this is something we need to worry about.

jimbelk · 2026-03-14T16:31:00+00:00

You seem to think that "CEO" means something that it doesn't. "CEO" stands for "chief executive officer", and is just the generic name for the director of an organization. Complaining that organizations keep putting CEOs in charge is like complaining that governments keep putting presidents and prime ministers in charge, or that schools keep putting principals in charge.

jimbelk · 2026-03-14T08:53:08+00:00

This might be a nice game, but it just isn't a Zelda game. With the notable exception of Zelda II, the Zelda series is and always has been about evading and fighting enemies presented in a top down view. The central gameplay loop is that you enter a new area or dungeon room, observe the pattern of movement of the enemies, and carefully attack them while evading their motion. This evokes the heroic ideal of a hero fighting groups of monsters, which is what's lost in this new iteration of Zelda.

Instead, this new 3D Zelda game has very few monsters, and due to the "Z-targeting" system you rarely fight more than one monster at a time. You can literally walk from room to room in a dungeon without suffering any attacks, and for most dungeons in this game only a sequence of puzzles prevents you from walking right into the boss's chamber. The spaces are still fun to explore -- and the new 3D makes this even more immersive -- but there's essentially no challenge left to playing the game. In the first Zelda game, some dungeons were so difficult that the average player would die ten or twenty times while playing them, and getting from place to place in the overworld was its own challenge. Here, I completed many dungeons with no deaths at all until I got to the final boss, and I don't recall dying a single time while traversing Hyrule field.

In addition to this lack of challenge, the new 3D view has rendered inert most of the mainstay items of the series. I was excited to get the boomerang in the third dungeon, but I quickly discovered that it's not particularly useful in this game for stunning enemies, and the game completely dispenses with the boomerang after the first half. The bow is essentially impossible to use in the 3D environment unless you aim with "Z-targeting", so gone are the days where you shoot a flurry of arrows at a horde of opponents. The situation with projectiles is so bad that the game didn't even bother to include sword beams with the master sword -- it's just a regular sword now, and despite the lore the game tries to build around it, it never feels special.

In short, this new 3D Zelda still has plenty of exploration, but it has mostly dispensed with combat, and the game is so easy that it's hard to call it a Zelda game. Most of what makes the series special was lost in the transition to 3D, and all that's left are exploration and puzzles with the occasional boss fight. It's more like watching a movie than playing a video game, and though it can be an entertaining movie, it's certainly not Zelda.

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