Wikipedia:Reference desk/Archives/Mathematics/2010 September 22

= September 22 =

quantifier depth examples

 * Arithmetic:
 * $$\Pi_1^0$$ - Goldbach's conjecture, Fermat's last theorem
 * $$\Sigma_2^0$$ - P=NP (conjectured to be false)
 * $$\Pi_2^0$$ - Twin prime conjecture, P ≠ NP conjecture
 * $$\Sigma_3^0$$ - ?
 * $$\Pi_3^0$$ - Waring's problem
 * higher - ?


 * Analysis
 * $$\Sigma_1^2$$ - Continuum hypothesis (assuming axiom of choice, see discussion)
 * $$\Pi_2^1$$ - Weak König's lemma
 * $$\Pi_2^2$$ - Continuum hypothesis (alternate formulation)


 * Algebra
 * $$\Delta^2_2$$ - Artinian rings are Noetherian

Can anyone add good, mathematically interesting entries to the above? Is there a list like this anywhere? Does the concept make any sense? Thanks.

66.127.54.226 (talk) 00:44, 22 September 2010 (UTC) 66.127.54.226 (talk) 06:49, 22 September 2010 (UTC)
 * You've got your ornamentation upside down. Goldbach's Conjecture and Fermat's Last Theorem are $$\Pi^0_1$$, for example.  The statement that a space is compact is $$\Pi^2_2$$.  Weak Konig's Lemma and Ramsey's Theorem for Pairs are $$\Pi^1_2$$  —Preceding unsigned comment added by 203.97.79.114 (talk) 03:48, 22 September 2010 (UTC)
 * Oops, you're right, I do did have the subscripts and superscripts reversed, now fixed. Thanks. 66.127.54.226 (talk) 06:49, 22 September 2010 (UTC)

Can anyone explain the OP's notation? Robinh (talk) 07:20, 22 September 2010 (UTC)
 * See arithmetic hierarchy and analytic hierarchy. I'm surprised about compactness being $$\Pi_2^2$$ and am not sure what that's called, since I thought "analysis" didn't include such formulas.  Quick version: $$\Pi_1^0$$ is sentences like (&forall;x)&phi;(x);  $$\Pi_2^0$$ is sentences like (&forall;x)(&exist;y)&phi;(x,y), etc, where the quantifiers range over natural numbers.  With a 1 in the superscript spot, the quantifiers can range over sets of naturals, with a 2 up there, they can range over sets of sets of naturals, etc. 66.127.54.226 (talk) 07:49, 22 September 2010 (UTC)
 * A cover is a set of sets. Hence you need third order quantifiers to discuss it.--203.97.79.114 (talk) 08:39, 22 September 2010 (UTC)
 * Topological spaces might be less than satisfying, since they're not first order structures, so let's look at rings. Artinian and Noetherian are $$\Pi^2_1$$ properties.  The theorem that Artinian implies Noetherian is thus $$\Delta^2_2$$.--203.97.79.114 (talk) 08:44, 22 September 2010 (UTC)
 * For that matter, the theorem that Noetherian is equivalent to every ideal being finitely generated is $$\Delta^2_2$$.--203.97.79.114 (talk) 08:49, 22 September 2010 (UTC)


 * The continuum hypothesis is much more complicated than $$\Pi^1_2$$. It's actually $$\Sigma^2_1$$.  It says "there is a wellordering of the reals whose every proper initial segment is countable".  Such a wellordering, if it exists, can be coded as a type-2 object (a set of reals), and verifying the rest of the claim involves quantifying only over type-1 objects.
 * By the way, the contributions from 203.97.79.114 seem to be assuming a rather different meaning for the notation. That's OK as long as it's specified.  The meaning I'm using is the standard one in descriptive set theory, and also agrees with the comments of 66.127.54.226.  By that interpretation, it doesn't make any sense to talk about the complexity of "compactness" in general, because the underlying space could involve objects of arbitrarily high type.  I think 203 is thinking of some model-theoretic sense.  Again, nothing wrong with that, but you ought to specify it; the descriptive-set-theoretic sense is the one more generally understood. --Trovatore (talk) 09:02, 22 September 2010 (UTC)
 * Woops. Too much time spent doing reverse math; in my head all structures are built on $$\omega$$.  Seen through that lens, I was using the same meaning.  Alternatively, my comments could be taken in the sense of model theory.--203.97.79.114 (talk) 09:23, 22 September 2010 (UTC)


 * So are you saying that classical analysis didn't discuss compactness? Because I'd always thought classical analysis embedded naturally in second order arithmetic.  That's why the $$\Pi_2^2$$ surprised me.  Thanks.   Also: why (from an arithmetic point of view) would the reals be expected to have a well-ordering?  I thought CH was usually stated as "every subset of the reals is either countable or equinumerous to the reals".  Which would be $$\Pi^2_2$$ come to think of it. 66.127.54.226 (talk) 10:34, 22 September 2010 (UTC)
 * Compactness in general is a notion from general topology, not from classical analysis. Certainly if you just want to talk about what subsets of Rn are compact, that's much simpler.
 * The "no intermediate cardinality" version of CH is indeed $$\Pi^2_2$$, but the more useful (and in my view more fundamental) $$2^{\aleph_0}=\aleph_1$$ version is simpler, namely $$\Sigma^2_1$$. Given the axiom of choice, the two are equivalent.  Without the axiom of choice you could have the "no intermediate cardinality" version without a lot of the consequences that we think of as following from CH. --Trovatore (talk) 17:14, 22 September 2010 (UTC)

Thanks, I've updated the list a little bit. What I'm really wondering though is whether collecting lists like this is interesting and if it has been done elsewhere. 71.141.90.138 (talk) 21:15, 22 September 2010 (UTC)

Instantaneous Acceleration/Uniform Circular Motion
I'm working on a homework problem and was wondering if I've nailed it:

The sweep-second hand of a clock is 3.1 cm long. What is the instantaneous acceleration (magnitude and direction) of the second hand's tip at the beginning of a 5.0 s interval?

I calculated that the average velocity would be .325 cm/s since change in r=2piR/12 R=3.1cm  and change in t is 5 s. since a=(v^2)/R for uniform circ motion, I ended up with a=.034 cm/s^2.

I'm thinking that I've missed something, and I'm also completely stymied on the direction of the acceleration vector although I know that it points towards the center.209.6.54.248 (talk) 00:45, 22 September 2010 (UTC)
 * Your value for the magnitude |a| is correct. For the direction, and to help see what is going on, try writing the vector for the position of the tip (x,y) as a function of the time t.  Then find dx/dt and dy/dt.  Use a few trig identities to interpret the result geometrically. 66.127.54.226 (talk) 01:13, 22 September 2010 (UTC)
 * You don't need the 5 s number at all (its speed never changes, after all). You can just say that it moves $$2\pi R=19.48~\text{cm}$$ per minute which equals your 3.25 mm/s.  (I was confused for a moment by your "2piR/12 R", but I realize now there are two separate equations.)  However, to get the direction you need to know which direction it's pointing (that is, where in the current minute are we?) because (as you said) the acceleration is toward the center.  --Tardis (talk) 03:40, 22 September 2010 (UTC)

While this is the intended solution, note that unlike velocity, acceleration is an absolute quantity; you can ask "what is the acceleration of the tip" instead of "what is the acceleration of the tip w.r.t. the clock". Then because the acceleration due the rotation of the Earth is much larger (unless the clock is located close to the North or South Pole), you'll get a different answer for the two cases. Count Iblis (talk) 01:41, 23 September 2010 (UTC)

Determining a volunteering shift
Hello, fellow WP:EDIANS! I've been tasked with making a volunteering schedule for the State Fair of Texas exhibit we'll have. Here's the breakout: I have no real idea how to even start. A little help?! Thanks! :D. — Duncan What I Do / What I Say 01:46, 22 September 2010 (UTC)
 * Volunteering hours run from 10 AM - 7 PM (9 hours)
 * We will have to work three shifts at different stations for the same number of hours at each one
 * We need to take a 1-hour lunch break
 * Edit I think I got it. 2:40 x 3 = 8 hours (our working hours). — Duncan What I Do / What I Say 01:59, 22 September 2010 (UTC)
 * I very rarely post on the Mathematics desk because I don't consider myself much of a mathematician, but it would seem that we'd also need to know (A) how many total volunteers you need to schedule, (B) how many different stations there are that need volunteers, and (C) how many volunteers you need at a time at each station.  Kingsfold  (Quack quack!)  14:11, 22 September 2010 (UTC)
 * I don't think your task is possible if you insist on complete 2h 40 min sessions and also on a lunch break (unless the whole fair closes down for lunch). You need to schedule some split shifts (e.g. 2h then lunch then 40 minutes) and you also need to stagger your lunch breaks.  I suggest that you set up a board with the required staffing, and use colored strips of paper to represent the volunteers.  As Kingsfold mentioned, we need more information if you need more help.  You could use timetabling software, but a large board and colored paper is usually quicker and simpler unless you have dozens of volunteers (in which case, use numbered paper strips). By the way, are you insisting on exactly 2h 40 min at each station each day?  I would work in complete hours, and just average out the balance between stations over several days.    D b f i r s   22:14, 22 September 2010 (UTC)

defining a function
Let r, R be fixed natural numbers. Suppose we have a function $$f:N\rightarrow \{1,2\cdots r\}$$. My book says we can now define a function $$g:N\rightarrow \{1,2\cdots r^R\}$$ by $$g(\alpha)=g(\beta) \iff f(i\alpha)=f(i\beta)\forall 1\le i\le R$$. It is not clear to me how this defines a function g. Can someone explain. Thanks-Shahab (talk) 07:43, 22 September 2010 (UTC)
 * What do you mean by rR? 66.127.54.226 (talk) 07:51, 22 September 2010 (UTC)
 * This would be a lot easier if the ranges were $$\{0,1,\dots, r-1\}$$ and $$\{0, 1,\dots, r^R-1\}$$, respectively, so I'm going to pretend they are. Define $$g(\alpha) = \sum_{i < R} f(i\alpha)r^i.$$  Then $$g$$ is as desired.  Effectively we're treating the elements of $$\{0,1,\dots, r^R-1\}$$ as the $$R$$-tuples with entries from $$\{0,1,\dots,r-1\}$$--203.97.79.114 (talk) 09:07, 22 September 2010 (UTC)

Zero polynomial
What is the degree of zero polynomial ? —Preceding unsigned comment added by Rishiredbacksr (talk • contribs) 14:17, 22 September 2010 (UTC)
 * See degree of a polynomial. Algebraist 14:55, 22 September 2010 (UTC)