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July 13

Efficient numerical integration

Runge kutta methods provide an excellent solution to systems of the form ${\dot {x}}_{i}=f_{i}\left(\{x\}\right)$ but does anyone know of a good numerical recipe for solving $\sum _{j}g_{ij}\left(\{x\}\right){\dot {x}}_{j}=f_{i}\left(\{x\}\right)$ without having to invert the matrix $g_{ij}$ at every time step? — Preceding unsigned comment added by 84.92.32.38 (talk) 14:24, 13 July 2014 (UTC)[reply]

This isn't a huge help but I think really you only need to solve the equations for the

{\dot {x}}_{j}

, which is somewhat easier that finding the inverse of the matrix. Suppose there were a more efficient method for generating a solution. Then you could apply it the case where the g's are constant, and from the solution you could determine the

{\dot {x}}_{j}

, which would then give you a more efficient method for sets of linear equations. So, and it would nice if someone checked my reasoning here, finding the solution of the differential equation must be at least as hard as solving systems of linear equations. Perhaps there is some good news though in that there are iterative methods for solving linear equations. So you could use the solution at point k as the starting point for the iteration at point k+1 and since presumably the solutions are close you may only need one or two iterations to get an accurate result. That assumes that the g is smooth and well conditioned. Of course everything here assumes g is nonsingular.--RDBury (talk) 16:11, 13 July 2014 (UTC)[reply]

I need a little help

How do you do this problem? 5`7-6 — Preceding unsigned comment added by 66.87.65.238 (talk) 23:49, 13 July 2014 (UTC)[reply]

What does that operator between 5 and 7 mean?--Jasper Deng (talk) 03:28, 14 July 2014 (UTC)[reply]

Now that is an interesting question! I can't find any real use of it as a mathematical operator. `#Use_in_programming mentions use in LaTeX for quotations, and also mentions use as command substitution in e.g. Bash shell. None of those make sense here though... SemanticMantis (talk) 15:18, 14 July 2014 (UTC)[reply]

Could be a mistake for an uparrow, meaning 5 to the power 7 (minus 6, which for all I know is meant to be combined with the 7 rather than deducted last).→86.146.61.61 (talk) 18:56, 14 July 2014 (UTC) g[reply]

exactly what does operator mean

Is that supposed to be a multiplication sign, division sign, or something else ? StuRat (talk) 00:54, 15 July 2014 (UTC)[reply]

We want to know what the apostrophe in your problem means. What is the context of your problem? Katie R (talk) 12:49, 15 July 2014 (UTC)[reply]

For clarification, and reference apostrophe is this: ' -- grave accent or backtick is this: ` (which is what is in the original problem, but the two are often confused. There is also the Prime (symbol), which can mean a few things, but none are relevant here. (I also thought 5`7-6 might be the number of a problem in some book, but we'd need to know what the problem is...) SemanticMantis (talk) 15:17, 15 July 2014 (UTC)[reply]

Don't spend time on this. Somebody is just having a little fun watching everybody scratching their heads. Even the most inexperienced poster in the whole world would realize that the problem is totally incomprehensible. YohanN7 (talk) 15:25, 15 July 2014 (UTC)[reply]

(Duly noted. But I can't be trolled. I WP:AFG, and only post when I'm having fun ;) SemanticMantis (talk) 15:37, 15 July 2014 (UTC)[reply]

July 14

July 15

count of MxN legal counterchanged grids?

For an M by N grid of 0s and ls (of which there are 2^(M*N)), define a legal counterchanged grid if it can be formed by inverting a series of entire rows and/or columns. It would seem to me the entire grid can be derived from the values of a single row and a single column and as such, the number of legal grids would be 2^(M+N-1), is this correct?Naraht (talk) 16:35, 15 July 2014 (UTC)[reply]

Just trying to understand the Q here. Are you saying you can invert each column and row as many times as you want, in any order, or only once ? StuRat (talk) 18:14, 15 July 2014 (UTC)[reply]

As many times in any order.Naraht (talk) 18:54, 15 July 2014 (UTC)[reply]

Yes, that sounds correct to me, assuming I understand your description. If a row is defined, then it will have to be either inverted or not when it is copied to each other row, and the decision is based on the values in the column you've defined. There are other configurations that would uniquely identify a grid, not just a row and column, but intuitively it seems that M+N-1 is the minimum number of values that must be fixed. Katie R (talk) 18:46, 15 July 2014 (UTC)[reply]

The question of how many possible sets of M+N-1 entries uniquely define a valid counterchange is a different, also interesting question.Naraht (talk) 18:54, 15 July 2014 (UTC)[reply]

Apparently the answer to this is OEIS: A072590. --RDBury (talk) 21:03, 16 July 2014 (UTC)[reply]

Nice find! It certainly does seem like this problem transforms beautifully. Each spanning tree uniquely describes a chain of reasoning to deduce the M+N-1 entries in a specific row and column, and each chain of reasoning uniquely identifies the set of entries that it starts from. Katie R (talk) 14:04, 17 July 2014 (UTC)[reply]

July 16

July 17

Large Recursive Ordinals and Large Cardinals

Let me preface this by saying: I'm just beginning, as in yesterday, to introduce myself to Ordinal analysis, and that this question comes more from a musing/stray thought than it does anything else. That said is there any sort of expected symmetry between large cardinal properties and large recursive ordinal properties - in other words, LCA some large cardinal axiom, does the structure of LCA tell us something about the nature of the proof theoretic ordinal of ZFC + LCA? Moreover, can we, from the relations of the various types of large cardinals, infer anything about the relations between their proof theory ordinals when added to ZFC? I apologize, in advance, if this question is not the most clear, or if it is stupidly glossing over something obvious. Thanks for any help/insight:-)Phoenixia1177 (talk) 06:58, 17 July 2014 (UTC)[reply]

It's not my field of mathematical logic, (except, or course, I knew ε₀ for PA), but the article ordinal analysis states: "Most theories capable of describing the power set of the natural numbers have proof theoretic ordinals that are so large that no explicit combinatorial description has yet been given." Most large cardinals require power set in their definition. — Arthur Rubin (talk) 07:23, 17 July 2014 (UTC)[reply]

Thank you for the reply - I realize that we have no description of these ordinals, but are we able to say anything about them otherwise? For partial analogy, as far as I know, we have no way of describing large cardinals directly, but we can say that if a Huge cardinal and a Supercompact cardinal exist, then the first huge is smaller than the first supercompact. In this case it need not be size comparisons, but is there anything that could potentially be said? --addendum: another analogy: we can't directly describe the Absolute Galois Group of the Rationals, but we can say things about the group itself; is there something that might be said/expected to be said about the overall collection of ordinals that would result? I realize that this is vague, my brain just keeps nagging me that there is something interesting here.Phoenixia1177 (talk) 07:38, 17 July 2014 (UTC)[reply]

This is a little off the track from your question, but the example that you mention is kind of an interesting one, because although the first huge cardinal comes before the first supercompact, the existence of a huge cardinal is a much stronger axiom, in terms of consistency strength, than the existence of a supercompact.

There are other anomalies like this, but as far as I'm aware, they're all based on LCAs that are not "local" — that is, you can't say that a cardinal is supercompact just because there's some sufficiently large α such that V_α thinks the cardinal is supercompact.

Non-local LCAs are a bit problematic because they rely on knowing stuff about the whole universe. In Woodin's work around 2000, he gave an abstract definition of large cardinal (not widely adopted, but interesting) that simply made locality part of the definition. I don't know of any counterexample to the claim that, when you restrict attention to local LCAs, the LCA with the larger first witness always has larger consistency strength. (However, a properly stronger LCA might have the same first witness as the weaker one.)

Also, the order of consistency strength on LCAs seems to be the same as how far up the Wadge hierarchy they prove regularity properties for sets of reals (again with possible collisions — for example, weak large cardinal axioms, those consistent with V=L, don't prove any regularity properties beyond ZFC itself). --Trovatore (talk) 11:32, 17 July 2014 (UTC)[reply]

Your "slight tangent" answers a question at least as interesting to me as my original, doubly so since I had no idea about the Wadge Hierarchy - thank you:-) Do you know of any good introductions to Woodin's work, or where to start with it? For some reason, I feel like there is some sort of connection between all the various hierarchies in computation, logic, and the transfinite, like they're all just instances of some underlying thing - like the difference between classical and modern algebraic geometry. Anyways, I digress. Thank you again for the interesting ideas:-)Phoenixia1177 (talk) 16:55, 17 July 2014 (UTC)[reply]

Don't know if this is exactly what you are looking for, but it is Woodin and it is around 2000:

Woodin, W. Hugh (2001a). "The Continuum Hypothesis, Part I" (PDF). Notices of the AMS. 48 (6): 567–576.
Woodin, W. Hugh (2001b). "The Continuum Hypothesis, Part II" (PDF). Notices of the AMS. 48 (7): 681–690.

YohanN7 (talk) 20:37, 17 July 2014 (UTC)[reply]

Yes, that's it exactly. Anyone reading it should be aware that not all of it has held up perfectly in the intervening years. A result that Woodin thought he knew (that there's a limit to the large cardinals you can have in HOD) fell through. I don't know how much of that paper depends on that. --Trovatore (talk) 21:29, 17 July 2014 (UTC)[reply]

July 18

Awk

Is there any valid reason why Awk (number) redirects to Large numbers? There's no explanation in the article. Rojomoke (talk) 09:58, 18 July 2014 (UTC)[reply]

I'd guess because awk is able to handle fairly long numbers, it holds them in double precision float. But that's not a good reason for having that link so it should just be deleted. Dmcq (talk) 10:07, 18 July 2014 (UTC)[reply]

Double precision float cannot compute N - (N - 1) correctly if N is Avogadro's number, so I'd say that awk doesn't deserve the redirect. (Neither is temporary real aka extended precision aka long double.) It would take a 79-bit significand to handle the "mol" version of Avogadro. Double and long double are still inadequate with 52 and 64 bits. - ¡Ouch! (^{hurt me} / _{more pain}) 12:00, 18 July 2014 (UTC)[reply]

I found the redirect while looking for the AWK article, but I was wondering if there was a separate mathematical meaning to the term. Rojomoke (talk) 12:28, 18 July 2014 (UTC)[reply]

Just seconding Dmcq, after creating the redirect the same editor made a bunch of test edits, so it appears to me that the redirect was one as well, which would mean a speedy delete is in order. I can't find any mathematical meaning for awk.--RDBury (talk) 18:16, 18 July 2014 (UTC)[reply]

In the future, this sort of notice would be better placed at Wikipedia talk:WikiProject Mathematics. --Trovatore (talk) 18:21, 18 July 2014 (UTC) [reply]

It wasn't a notice, it was a question. I wonder if the poster was thinking of the Ackermann numbers, a sequence which rapidly gets into very large numbers. I think I've seen the word abbreviated as "ack", which might be confused with "awk". --50.100.189.160 (talk) 19:32, 18 July 2014 (UTC)[reply]

It was a question, but not about math. It was more of a "what do we do about this page" sort of question, or at least that's my take on it. Those are more wikiproject-like. I don't want to belabor this; there was no real harm done. I just want people to be aware that the wikiproject exists, and that this is the sort of thing it's there for. --Trovatore (talk) 20:10, 18 July 2014 (UTC)[reply]

July 19

Random walk on polygon

What is the easiest way to show that, for a symmetrical random walk starting on a vertex of a polygon, the expected value of the number of steps to visit all n vertices is n(n-1)/2? And how would one derive the expected value of the number of steps to cover all n edges? →86.146.61.61 (talk) 12:30, 19 July 2014 (UTC)[reply]