Wikipedia:Reference desk/Archives/Mathematics/2021 January 14

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January 14

Economic forecasting

What kind of learning material would one suggest for forecasting economical time series of how fund will behave on the future? I'm saving money and interested to learn time series forecasting methods.— Preceding unsigned comment added by 2001:999:13:e2ee:6569:8f1d:f166:b163 (talk • contribs)

If you hope to achieve better results by having a better forecast, then my advice is to forget it. However many resources you put into that, there are organizations that have more resources and that have already placed their orders while you are still collecting the data. You may have had the experience of standing in line, in one line among several, and another line is moving much faster so you decide to switch lines, only to see that now your new line is stalled. Likewise, changing your portfolio because another composition seems bound to outperform the current one has about 50% chance of leading to an improvement. Most organizations active in forecasting economical time series have stopped using mathematical models (Fourier analysis, wavelets, system identification, ...) and switched to neural networks. Some links to give an idea: [1], [2], [3], [4], [5].

How is Fourier Series applied in Thin-shell structure?

From here I learnt that, Fourier Series applied in Thin-shell structure. How is that? Rizosome (talk) 05:57, 14 January 2021 (UTC)

The link to Thin-shell structure was misleading; I have replaced it with a link to Membrane theory of shells.  --Lambiam 10:19, 14 January 2021 (UTC)

nice approximation of pi

hello, what is this approximation of pi? I haven't found it on our approximations of pi page (I was looking for anything where it said 1 over sqrt(3)). Also, if it's not there in any of its forms (?), could someone more knowledgeable in programming and math and math notation add it to the page Aecho6Ee (talk) 13:46, 14 January 2021 (UTC)

here is an alternate formulation with better FP behaviour Aecho6Ee (talk) 13:49, 14 January 2021 (UTC)

I suspect this is an implementation of Liu Hui's π algorithm. The factor 6 and the use of 1/√3 fit with the first approximation of the circle being a hexagon, and it looks like bisection gives rise to a quadratic equation. And, obviously, each next bisection requires another doubling.  --Lambiam 15:31, 14 January 2021 (UTC)

Instead of producing increasingly better approximations by computing the circumferences of inscribed regular polygons, this process uses polygons circumscribed around the unit circle, taking half their circumferences, thereby approaching ${\tfrac {1}{2}}\cdot 2\pi$ from above. The numbers that are printed are the values of ${\tfrac {1}{2}}\cdot 12\cdot 2^{n}\tan {\frac {2\pi }{12\cdot 2^{n}}}$ for $n=1,2,3,...~.$ Since $\tan x=x+{\frac {x^{3}}{3}}+\cdots ,$ the error goes down each step with a factor of about $2\cdot {\tfrac {1}{2^{3}}}={\tfrac {1}{4}}.$  --Lambiam 22:07, 14 January 2021 (UTC)

Thank you so much! Aecho6Ee (talk) 09:50, 17 January 2021 (UTC)

Why do some people say 1 is a prime number??

Excuse me. Somebody made an edit at Talk:1 saying that it's a fact that 1 is a prime number. Do people really think this?? Georgia guy (talk) 14:17, 14 January 2021 (UTC)

@Georgia guy: Yes, some actually do. Those are who were told that a prime is a number which divides only by 1 and by itself. --CiaPan (talk) 14:23, 14 January 2021 (UTC)

They need to learn to be more flexible with understanding prime numbers. Georgia guy (talk) 14:25, 14 January 2021 (UTC)

See Prime_number#Primality_of_one. AndrewWTaylor (talk) 14:27, 14 January 2021 (UTC)

@Georgia guy: Apparently they don't think so. They think they know everything they need (at least everything about prime numbers). :) --CiaPan (talk) 14:33, 14 January 2021 (UTC)

Any other mathematics-related beliefs that some people have for similar reasons?? Georgia guy (talk) 14:36, 14 January 2021 (UTC)

A lot of people seem to think there's something unusual or mystical about the fact that "pi goes on forever" (i.e. has an infinite non-repeating decimal expansion, which of course is true for almost all real numbers.) AndrewWTaylor (talk) 14:45, 14 January 2021 (UTC)

Actually, for each non-integer if you only choose appropriate base (e.g. 1/2 in ternary). --CiaPan (talk) 14:55, 14 January 2021 (UTC)

1/2 is rational. A rational number's decimal expansion will terminate or repeat. Georgia guy (talk) 14:58, 14 January 2021 (UTC)

...and if they repeat they do not terminate. --CiaPan (talk) 15:01, 14 January 2021 (UTC)

...and AndrewWTaylor explicitly said non-repeating.--202.74.195.81 (talk) 03:32, 15 January 2021 (UTC)

A "terminating" decimal expansion actually repeats; it just keeps repeating 0. --Trovatore (talk) 05:31, 15 January 2021 (UTC)

I have encountered many misguided beliefs, such as that π exactly equals 22⁄7, or that diagonal arguments such as used in the halting problem are invalid. The belief that it is a fact that 1 is a prime number is of a different nature; it reflects an inability to realize that the meanings of terms depend on their definitions and are not in themselves facts. In Belgian education the number 0 is (by definition) positive – perhaps reflecting a generally positive Belgian outlook on life. As late as 1911 we can still find in an encyclopedia: "The number 1 is usually included amongst the primes".  --Lambiam 15:53, 14 January 2021 (UTC)

This is really just a question of definition. See prime number#Primality of one. If you included 1 as a prime number, you'd get a slightly simpler, and perhaps aesthetically slightly more pleasing, definition. But you'd have to pay for it by writing over and over again something like let p be a prime number other than one.

As Lambiam notes above, usage has not always been uniform. Today, with negligible exceptions, it is; if you use "prime number" to include 1, you will simply not be understood. This is not a mathematical question, just one of terminology. --Trovatore (talk) 19:32, 14 January 2021 (UTC)

Yep. When I was in high school, it was made very clear to us that 0 is not a natural number; but as you will see if you follow the link, this is far from universal usage today. In other words, you may have been taught one definition of a mathematical concept but that doesn't make it the only correct one. --142.112.149.107 (talk) 01:58, 15 January 2021 (UTC)

Let's be clear — there's only one "correct" use of "prime number" in current mathematical English, and it excludes the number one. Calling one a prime number is "incorrect". But it's incorrect as a matter of language and convention, not as a matter of mathematics. And if you're reading sufficiently old texts, you have to keep in mind that in the past there were different usages.

This is somewhat different from "natural number", which is used both ways by current mathematicians. --Trovatore (talk) 02:31, 15 January 2021 (UTC)

Thanks, I should have made that clearer. --142.112.149.107 (talk) 06:03, 15 January 2021 (UTC)

I watched a video recently on youtube (I have no link - I think it was a numberphile video) that discussed this. The unreferenced plot was that sometimes things work more cleanly if 1 is treated as a prime and sometimes they work more cleanly if it is treated as a non-prime. The video suggested that it used to be the case that 1 was treated as a prime by the majority of mathematicians but over the last ...insert a number of years... the consensus has become that 1 should be a non-prime. -- SGBailey (talk) 19:57, 15 January 2021 (UTC)

Many sloppy definitions of prime numbers, e.g. on random websites, fail to say "greater than one". The authors are probably unaware that their definition includes 1 or that the standard definition doesn't. PrimeHunter (talk) 20:19, 15 January 2021 (UTC)

@PrimeHunter: The 'greater than 1' condition looks somewhat arbitrary and very specific. A more general condition with the same implications is 'has exactly two divisors'. This excludes the unit pretty easily without pointing a finger at it . CiaPan (talk) 16:21, 16 January 2021 (UTC)

I have seen that definition but don't like it. It's not why they are important, and "exactly two" looks somewhat arbitrary and very specific to me. It doesn't sound fundamental, and the word "prime" is associated with first, not two. PrimeHunter (talk) 19:24, 16 January 2021 (UTC)

If one leaves aside the divisibility of all numbers by 1 as trivial, primes could be defined as numbers with exactly one non-trivial divisor. But then, the divisibility of a number by itself is just as trivial as the 1 case. So, how about numbers with no non-trivial divisors. But that gets us away from "prime". Just thinking aloud here, apparently ... -- Jack of Oz ^{[pleasantries]} 21:22, 16 January 2021 (UTC)

Here is a definition that is too difficult for a standard introduction, but (IMO) not arbitrary or artificial. The primes are the least set of positive integers such that all positive integers can be expressed as the product of a multiset of primes. It takes some effort to show that this is indeed a definition; it requires establishing that there exist sets that generate all positive integers (trivial – the whole set does the job), and furthermore that among these sets there is a least one – not difficult, but also not trivial.  --Lambiam 21:48, 16 January 2021 (UTC)

@JackofOz: the relevant concept is that of a proper divisor, which helpfully avoids the problem of which divisions are trivial. I think the definition "a prime number is a number whose only proper divisor is 1" is rather natural and does a good job of obscuring the subtlety. --JBL (talk) 02:48, 17 January 2021 (UTC)

I like it. So may it be written, so may it be done. :) -- Jack of Oz ^{[pleasantries]} 02:53, 17 January 2021 (UTC)

This definition is equivalent to "a prime number is a number that has exactly one proper divisor" (since, if it has a proper divisor at all, 1 is a proper divisor). It should be clear, though, that this is all restricted to positive integers.  --Lambiam 11:35, 17 January 2021 (UTC)

I agree with PrimeHunter that the "exactly two divisors" version is hacky and not illuminative of what's going on. I don't think "proper divisors" is really much better.

Here's how I might try to get at the real idea. Given a commutative ring an integral domain R, restrict to the part of it where multiplication is always irreversible; abc is always different from a. That boils down to excluding 0 and the units (invertible elements), but putting it this way, in my opinion, gets at the idea better.

Then within that semigroup, the primes are the elements that are not the product of two elements.

I'm curious whether there's a standard name for the multiplicative semigroup derived in this way from a commutative ring. --Trovatore (talk) 02:11, 18 January 2021 (UTC)

The additive structure is irrelevant; you don't need a ring. Only the multiplicative monoid of positive integers plays a role. The primes are the elements that have only a trivial factorization. This definition makes sense in any monoid (with a suitable definition of "factorization"; the German term Zerlegung, also used for factorization,^[6] does not by itself imply a multiplication operation). I have not considered whether this definition of "prime" has interesting applications for other monoids, For a freely generated monoid they are the generators, but that is boring.  --Lambiam 12:57, 18 January 2021 (UTC)

You don't need the additive structure to make sense of the definition, but the "only a trivial factorization" definition has the same problem; it includes the multiplicative identity as a prime.

What I'm saying is, throw away the multiplicative identity, so that you have, what, an "anti-group" or something? a semigroup in which multiplication is always irreversible. In that structure, the primes are the elements that have no factorization. --Trovatore (talk) 18:00, 18 January 2021 (UTC)

I agree that rings or integral domains are the correct context to generalize, but then the relevant idea is that of a prime ideal, not your ad hoc construction. (Also properness is a key defining property of prime ideals, so it is totally reasonable that it should appear as a defining property of prime numbers.) --JBL (talk) 14:29, 20 January 2021 (UTC)

That just kicks the can down the road. Why are prime ideals defined to exclude the whole ring? You can reasonably say it's because it's a posteriori more useful, but you could have done that from the start for the prime numbers themselves. --Trovatore (talk) 16:37, 20 January 2021 (UTC)

Whereas you argue that we should prefer an explanation based on a construction whose significance is questionable both a priori and a posteriori? (No, I know you don't actually argue this. But the right answer to the question "why did the question of the primality of 1 become settled around the time it did?" is "because people discovered the appropriate context to place the question in", and that context is rings + prime ideals.) --JBL (talk) 16:52, 20 January 2021 (UTC)

Hmm, possibly. That's a historiographical claim that would need to be backed up.

But the construction I mentioned is not really totally out of the blue. My understanding is that the ancient Greeks, for example, didn't consider 1 to be a "number" at all, and from some point of view it makes sense to say that if you multiply something, it has to get bigger. That's where the irreversibility comes from. Then the primes are things that you can't get by multiplying.

On another note, can we do something about WP's terminology with regard to ideals (and filters) in general? Most articles seem to be written on the assumption that the empty set is an ideal an ideal can contain the entire set as an element, which adds nothing useful to the theory but complicates the statements of most of the interesting results. We surely need to recognize that this usage exists, but I don't think the articles should be written using it. --Trovatore (talk) 17:04, 20 January 2021 (UTC)

Probably this video: [7]  --Lambiam 20:21, 15 January 2021 (UTC)