Math: Fleeting Thoughts

[Qoppa]

If coding gets one, we should too.

I'm really picky about notation... It bothers me to no end when someone uses notation I dislike (for example, - instead of \ for set difference). Recently, I spent a while trying to decide if I should use [imath](a)[/imath] for the ideal generated by a, and [imath]\langle a \rangle[/imath] for the subgroup generated by a, or if I should use [imath]\langle a \rangle[/imath] for both. I eventually opted to distinguish the two.

Also, how the deuce do you pronounce "noetherian"?

[Macbi]

If coding gets one, we should too.

I'm really picky about notation... It bothers me to no end when someone uses notation I dislike (for example, - instead of \ for set difference). Recently, I spent a while trying to decide if I should use [imath](a)[/imath] for the ideal generated by a, and [imath]\langle a \rangle[/imath] for the subgroup generated by a, or if I should use [imath]\langle a \rangle[/imath] for both. I eventually opted to distinguish the two.

One of our teachers mangles notation doing induction. Say we're proving a general formula for a summation, he'll use P(n) to mean the value of the sum of the first n terms, and the statement we're trying to prove.

Also, how the deuce do you pronounce "noetherian"?

Can't help you there.

Is there a theorem like Cantor–Bernstein–Schroeder for surjections?

[t0rajir0u]

Also, how the deuce do you pronounce "noetherian"?

My professor just says noh-thearian.

Is there a theorem like Cantor–Bernstein–Schroeder for surjections?

Yep. It's essentially the same.

[skeptical scientist]

Also, how the deuce do you pronounce "noetherian"?

I've heard "nēʹthîr-ē-ən," "nōʹthîr-ē-ən," and "nə-thîrʹē-ən," but I'm not sure which (if any) is correct. It's named after Emmy Noether, which Wikipedia says to pronounce "nøːtɐ." For comparison, that "øː" is the same vowel sound as the 'ö'/'oe' in "Gödel" and "Goethe," so the 'oe' in "Emmy Noether" is pronounced the same way. However, I don't think that carries over to the pronunciation of "noetherian," since the pronunciation has probably been anglicized. I think I'd go with "nə-thîrʹē-ən" as the closest English approximation.

[Buttons]

Let's talk about mathematical eponymous adjectives! Pretty much all the ones I can think of are of the form name+ian. What exceptions do you know? My favorite I just learned yesterday: the eponymous adjective for MacMahon is Mahonian! Beats platonic any day, if you ask me.

[Qoppa]

The adjectival form of Riemann is.... Riemann. He's so badass his name doesn't need to be adjectified. Actually, Lebesgue too. And Fourier. It seems that there are a good number of mathematicians who's name as an adjective is the same.

[stephentyrone]

The adjectival form of Riemann is.... Riemann.

Like a Riemannian Manifold? Oh wait...

[skeptical scientist]

The adjectival form of Riemann is.... Riemann.

Like a Riemannian Manifold? Oh wait...

I'm reminded of the episode of Star Trek: TNG when Westley Crusher mentioned that he'd been studying "Rieanaman tensor fields." That quote made my soul hurt.

[Qoppa]

But we have Riemann integral, and Riemann sum!

[jestingrabbit]

The adjectival form of Riemann is.... Riemann.

Like a Riemannian Manifold? Oh wait...

I'm reminded of the episode of Star Trek: TNG when Westley Crusher mentioned that he'd been studying "Rieanaman tensor fields." That quote made my soul hurt.

The last thing I saw him in was a really crappy movie called book of days. It sucked. In huge sucking bouts of suckiness. A real suck fest. Suckerpalusa. It's a bad, bad movie.

But we have Riemann integral, and Riemann sum!

I'll give you the second but not the first.

FT: Means on amenable groups - largely ignored, kinda useless.

[auteur52]

But we have Riemann integral, and Riemann sum!

It's pretty common for multiple forms of a name to be used as adjectives. Just like Gaussian distribution vs Gauss sum or Jacobian vs Jacobi sum, etc.

[Rachel!]

We have Riemann numbers, too! I, II, III... yeah I'm not so hip on math, but I enjoy puns.

My fleeting thought is this: I am mortally offended that my physics teacher taught me to mispronounce Euler's name (as yoo-ler) for so long that I can't change the way I say it (I've heard it's supposed to be oy-ler. Even though I know it's wrong.

wahhh!

[Cytoplasm]

This is perfect!

Alright, maths helpn'z por favor?

So I am trying to find volume of the M&Ms in this baby bottle:

Spoiler:

I think it's about 8" tall and the diameter is about 2".

This is what I've been trying to far:

For volume of the cylinder part I used: V=1^2x4Pi; V=4Pi; I'm not quite sure what 4pi is without a calculator.
Then I thought of taking the sort of cone shape by doing:
V= 1/3Pir^2h, about

So... V= 1/3pi x 1^2 x 4; (1/3)4pi about equal to 1.33333..pi?

Then add the two..I must win these M&Ms with the power of math..I must! Anyone want to conquer with me? Muhahaha..

Edit: I ended up getting this:
11.6165186405129211.6165186405129211.6165186405129211.6165186405129211.61651864051292. I fail...

[Office_Shredder]

Did you divide by the volume of an M&M?\

You might want to find the height and diameter and assume it's a box shape instead because they don't fit together perfectly

[Sir_Elderberry]

I'm reminded of the episode of Star Trek: TNG when Westley Crusher mentioned that he'd been studying "Rieanaman tensor fields." That quote made my soul hurt.

"The Outrageous Okona." Just, uh, so you know.

[btgreat]

If you are really a bamf, they put your name in lower case, e.g. abelian groups.

Exception to the -ian rule: Galois. Galois theory, connections, etc. use just his name, and I've never seen anyone try (and likely fail) to use the term "Galoisian."

[auteur52]

If you are really a bamf, they put your name in lower case, e.g. abelian groups.

Exception to the -ian rule: Galois. Galois theory, connections, etc. use just his name, and I've never seen anyone try (and likely fail) to use the term "Galoisian."

It's pretty hard to do that with any name that ends in a vowel (sound). That rules out most French names.

I've always heard that "you're most respected if your name is no longer capitalized", e.g., abelian, noetherian, but I haven't seen that many more cases than that. Just because we capitalize Galois, Riemann, Gauss, Cauchy, etc. does not mean we respect them any less...

[Buttons]

Yeah, I've always been skeptical of that rule. Just going quickly through the lowercase eponymous adjectives on Wikipedia, we've got chauvinistic, daedal, draconian, epicurian, gargantuan, herculean, hermaphroditic, hermetic, macadamized, manueline, martial, masochistic, maudlin, narcissistic, ohmic, onanistic, parkinsonian, plutonic, protean, quixotic, ritzy, sadistic, satanic, stentorian, terpsichorean, thespian, and thrasonical.

The data are... unconvincing.

[Balsamic]

Despite studying CS and achieving decent results, I've always shunned away from mathematics. Frankly, I have never been very adept at it. Of late, I've started to look into functional programming... That triggered me to get out the old notes on lambda calculus and look at them in a new light. Subsequently, I've started my foray into all things mathy and I feel a whole new world opening up to me. I feel my mind expanding and I don't want it to stop.

[Something Awesome]

Regarding adjective forms of names: It's always amused me to call a set "Zornable" if it satisfies the hypothesis of Zorn's Lemma. I'm also sometimes overly dramatic/graphic when I write proofs: "Since we have shown [imath]A[/imath] to be Zornable, we smack it with Zorn's Lemma and out pops a maximal element [imath]\hat{a} \in A[/imath]!"

[t0rajir0u]

noetherian

Do people really not capitalize noetherian? I guess I never noticed that. What about artinian?

[auteur52]

Do people really not capitalize noetherian? I guess I never noticed that. What about artinian?

Well, Noetherian is capitalized in most of the literature I've seen, but I've seen it not capitalized elsewhere (often in online sources). Artinian is also capitalized in some of the major literature (e.g., Atiyah/MacDonald, Dummit/Foote) but not capitalized in Lang. Apparently the non-capitalization thing is a lot more common in French, but I don't know from first-hand experience.

[MHD]

I wonder how many iterations I should use to calculate sine and arcsine to 28 digits precision...

[stephentyrone]

I wonder how many iterations I should use to calculate sine and arcsine to 28 digits precision...

Depends. How many iterations of what algorithm?

Although, if you only need 28 digits, you really shouldn't be using an iterative algorithm at all, and there are plenty of free libraries that can do this for you.

[Qoppa]

Random brain block or something...
Can we combine a proof by (strong) induction with a proof by contradiction? That is, can we prove a base case, make an inductive hypothesis, and then contradict the inductive hypothesis? Ah hell, a sketch of my proof might help. I'm proving that successive Fibonacci numbers have no common divisor. I assume that this is true for all [imath]n < k[/imath]. Then I assume that [imath]F_k[/imath] and[imath]F_{k-1}[/imath] have a common divisor [imath]d[/imath] and show that this means [imath]d|F_{k-2}[/imath], a contradiction. Does that work? I'm pretty sure it does, but for whatever reason I reread my proof after finishing it and decided that didn't quite look right.

[jestingrabbit]

Does that work?

Yes.

[wolfemancs]

Back to the original posts topic of what annoys us in math, this is one of my biggest pet peeves ever (yeah, I know, pretty lame).

Math books that number sections 1.1, 1.2, 1.3. . . . .1.10, 1.11 . . .

It's a MATH book. . . they should know that 1.1 and 1.10 are the same number.

I'd be perfectly happy if it was 1-1, 1-2, 1-3. . . 1-10. . .
or 1.01, 1.02, 1.03. . . . 1.10 . . .
or lots of other things, just not the one thing that happens the most often.

[Talith]

Back to the original posts topic of what annoys us in math, this is one of my biggest pet peeves ever (yeah, I know, pretty lame).

Math books that number sections 1.1, 1.2, 1.3. . . . .1.10, 1.11 . . .

It's a MATH book. . . they should know that 1.1 and 1.10 are the same number.

I'd be perfectly happy if it was 1-1, 1-2, 1-3. . . 1-10. . .
or 1.01, 1.02, 1.03. . . . 1.10 . . .
or lots of other things, just not the one thing that happens the most often.

I used to think that way, but that I realised the decimal was actually being used as an index tool rather than a partitioning one. It probably would have been more logical to use a colon or semi colon, but then that might be seen as too biblical .

[the tree]

I forgot which thing is a Laplace transform and which thing is a Fourier transform, so I went to check and now I'm more confused. I swear I used to be quite comfortable with both of these.

[AllSaintsDay]

Fleeting thought I had yesterday:
Is [imath]\chi_\mathbb{Q}[/imath] (i.e., the characteristic function of the rationals) periodic? If so, what is its period?

[stephentyrone]

Is [imath]\chi_\mathbb{Q}[/imath] (i.e., the characteristic function of the rationals) periodic? If so, what is its period?

Yes. Every rational number is a period of [imath]\chi_\mathbb{Q}[/imath].

[AllSaintsDay]

Is [imath]\chi_\mathbb{Q}[/imath] (i.e., the characteristic function of the rationals) periodic? If so, what is its period?

Yes. Every rational number is a period of [imath]\chi_\mathbb{Q}[/imath].

I meant the more common definition of period, also called fundamental period.

[Ended]

Is [imath]\chi_\mathbb{Q}[/imath] (i.e., the characteristic function of the rationals) periodic? If so, what is its period?

Yes. Every rational number is a period of [imath]\chi_\mathbb{Q}[/imath].

I meant the more common definition of period, also called fundamental period.

Since all rational numbers are periods, there is no smallest period and therefore no fundamental period.

[Tac-Tics]

Just use whatever notation is best for getting the job done.

If that job is to teach students, consistency is a good thing. If not, then whatever works.

[AllSaintsDay]

Just use whatever notation is best for getting the job done.

If that job is to teach students, consistency is a good thing. If not, then whatever works.

Eh, there isn't really a job to get done. Just something that made me go "Huh." I won't bring this up while teaching periodic functions. Heck, I wouldn't even bring up the idea of whether constant functions are periodic, which has the same basic pathology as this, but less of a feel of weirdness.

Although, it does seems to me that calling this a period 0 function (by extending fundamental period from "smallest positive period" to "inf of positive periods") is at least as justified as saying that R has characteristic 0. (In the sense of justifying the definitions.)

[antonfire]

We say R has characteristic 0 for a pretty good reason. The characteristic of a field k is really just the kernel of the unique ring homomorphism Z->k, which is a prime ideal. When k is, say, R, it's the ideal generated by 0. When k is, say, Z_p, it's the ideal generated by p.


The same idea applies to periodic functions. I'd say a function f is periodic with period p if f(x) = f(x+p) for all x. Then the set of periods of a function is an additive subgroup of R. In the case of nice functions like sin and cos, this subgroup is generated by one element (2pi, in this case), so it's easily described.

In some cases, you can't describe it with just one number, and in that case we shouldn't try to assign a number to it. Saying that the period is zero would be trying to push the notation of "the period of a function" to do something it simply can't. Indeed, if you interpret things reasonably, "the fundamental period of f is 0" should just mean that the subgroup is the group generated by 0, i.e. f is not periodic at all.

By the way, for continuous functions into T₁ spaces the corresponding subgroup or R is closed. (It's the intersection over all x of { y-x | y in R; f(y)=f(x) }, each of which is closed.) So either the group is all of R, 0, or it's generated by one element (the inf of the positive ones). So for nonconstant continuous functions into nice spaces, it is good enough to talk about "the period" of a function to describe its symmetries.

[You, sir, name?]

Because I had nothing better to do, I derived a really awkward variant of calculus. Boils down to the formulation [math]F(x+dx) = F(x)Q_F^{dx}(x)[/math] which gives [math]\ln Q_F(x) = F^{-1}\frac{dF(x)}{dx}[/math]

You can create a really disturbing analog to integration based on a product instead of a sum with it. But as I said, it's awkward formulation. It's [imath]Q_F[/imath] generally undefined wherever [imath]F(x) = 0[/imath], which is sort of a nuisance.

That is all.

[Dason]

I was playing around the other day and I needed to remember the formula for [imath]\sum_{i=1}^k i^3[/imath] and I couldn't recall it off the top of my head. But I remembered working out a general method for figuring out [imath]\sum_{i=1}^k i^n[/imath] with a professor during my undergrad years but I could not for the life of me recall how we did it. I recall we inferred a general form where the sum would end up being a polynomial of order one greater than what we're summing and I think we used induction from there but I can't recall the details. Anybody know what I'm talking about and have any hints?

Note: I figured I'd revive this dead thread instead of creating an entirely new one just because I like the coding: fleeting thoughts and the general subforum's fleeting thoughts boards and wish this particular thread was a little more active.

[Tirian]

My copy of Conway's The Book of Numbers seems to have wandered off my shelf, but he derived some sort of polynomial with the Bernoulli numbers worked in. His point was to one-up the legend of Euler's youth and show how you could sum up the first hundred tenth powers in under a minute by hand.

[Talith]

I think that was Gauss, not Euler .