Wikipedia:Reference desk/Archives/Mathematics/2009 July 3

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July 3

Development of calculus

Why did calculus develop as late as it did? I would have thought that, given how easy it is to derive the derivatives of most functions and how elementary questions like "what's the rate of change of this quantity?" are, calculus should have been "discovered" thousands of years ago. I certainly pondered what were basically calculus problems long before I knew about calculus; I'm surprised that the ancient Greek mathematicians didn't do the same. --99.237.234.104 (talk) 07:58, 3 July 2009 (UTC)

As can be seen here, some of the basic concepts of differential and integral calculus floated around for millennia; it's just only in the past few hundred years that rigorous axioms and a solid branch of mathematics (calculus) have emerged. —Anonymous Dissident^Talk 10:49, 3 July 2009 (UTC)

(e/c) They did (Archimedes for instance). They did not develop the general theory, which is probably related to the fact that they did not think about functions in the algebraic way we are accustomed to. — Emil J. 10:55, 3 July 2009 (UTC)

I knew that certain aspects of calculus were around for a long time, but I'm surprised that they didn't develop further, considering the utility of calculus in solving mathematical problems. The complexity of what Archimedes and other mathematicians of the time managed to derive--like the surface area and volume of a sphere--is amazing; I'd never understand how they came upon such complicated proofs. That's why I'm surprised that nobody developed something as simple and intuitive as the basic concepts of calculus. --99.237.234.104 (talk) 11:19, 3 July 2009 (UTC)

A systematic development of calculus can't get very far without modern algebraic notation and the synthesis of geometry and algebra in analytic geometry. Diophantus started the development of algebraic notation, but even apparently simple advances like plus, minus and equals signs didn't emerge until the 16th century. Descartes didn't develop the key insight that an equation describes a curve and vice versa until the early 17th century. Major paradigm shifts like calculus affect our perspective so much that it can be difficult to appreciate how difficult they were to develop in the first instance. Gandalf61 (talk) 13:44, 3 July 2009 (UTC)

Discovering calculus is a harder problem than it appears to be because when you learn calculus, you can see only the easy part. The easy part is to answer all the questions after you know the questions. Michael Hardy (talk) 23:47, 3 July 2009 (UTC)

However, recall that the Hellenistic science has been abruptly stopped by the Roman expansion on the Mediterranean. For the story of the Hellenistic science I suggest Lucio Russo's wonderful book "The forgotten revolution".--pma (talk) 06:36, 4 July 2009 (UTC)

Method to solve 3-variable equation

x=(y/(z^y))

If I know x and z, what's the best way to obtain a value for y? I assume it can't be solved algebraically because I end up with y and log(y) in any rearrangement which can't be combined in a single term to make it y=f(x,z). If Newton's method is to be used then I'm not sure how to differentiate it or apply the method in this case. 86.163.186.102 (talk) 08:55, 3 July 2009 (UTC)

You can express the solution as $y={\frac {W(-x\,\log z)}{-\log z}}$, where W is the Lambert W function, if that's any help. — Emil J. 11:02, 3 July 2009 (UTC)

(ec) I don't have the time to explain this with much depth right now, but for any integer n, $y=-{\frac {W_{n}(-x\log(z))}{\log(z)}}$ for your case. This is assuming a non-zero log(z). W_n(x) is the Lambert W function. Hope this helps. I expect someone will be able to come along shortly and give something more elaborate and explained. —Anonymous Dissident^Talk 11:05, 3 July 2009 (UTC)

If you want to solve it numerically using Newton's method, you need to express the equation as $f(y)=0$. In your case $f(y)=x-{\frac {y}{z^{y}}}$ is one way, but $f(y)=xz^{y}-y$ is simpler. To differentiate it, you need to treat x and z as constants and use the normal rules of differentiation. The result is $f'(y)=xz^{y}\log z-1$, and the Newton iteration is $y_{n+1}=y_{n}-{\frac {xz^{y_{n}}-y_{n}}{xz^{y_{n}}\log z-1}}$. -- Meni Rosenfeld (talk) 11:37, 3 July 2009 (UTC)

Thanks everyone. Since using W is iterative too but is more complicated I just used Newton's method with your f(y) and f'(y) and it's working great. 86.163.186.102 (talk) 11:56, 3 July 2009 (UTC)

Mininmum Sample size for correlation study

I know of a formular for determining the minimum sample size for a prevalence study, which most research studies use. Is there any formular to determine minimum sample size for a correlation study, like a correlation between birth weight and fetal cord leptin concentration-a research?Tunmisadej (talk) 19:43, 3 July 2009 (UTC)

If you're doing homework or trying to replicate established procedures, look at the testbook/published literature. Otherwise, I can't think of a formula off-hand, but in general the best way to approach these sorts of problems is to work backwards. Figure out what sort of confidence level you want to have (0.05, 0.01, 0.001, etc.) in your results, and then using worst-case estimates for the various other parameters apply the equation you're going to use to analyze the completed study. You can then predict how the confidence level varies for various sample sizes. The sample size you want is the one that will give you at least the desired confidence level. A little extra work will get you information on how sensitive your confidence interval is to fluctuations in the various parameters. -- 76.201.158.47 (talk) 21:00, 4 July 2009 (UTC)

There is no formula, indeed there cannot be, as that would imply the existence of a threshold at which the correlation test began to work, and above which it did not work any better. What you probably want is a power calculation for correlation test.

This tells you how many subjects you will need to test in order to have a certain likelihood (80% is common) to detect an effect of a certain size (which you have to provide). If you were looking for a correlation you think will be around 0.3 in magnitude, and want to find it 80% of the time it is really there, you would need around 85 subjects:

Using R.

 > library(pwr)
 > pwr.r.test(r=0.3,power=0.80,sig.level=0.05,alternative="two.sided")
             n = 84.7
             r = 0.3
     sig.level = 0.05
         power = 0.8
   alternative = two.sided

Tim bates (talk) 11:58, 11 July 2009 (UTC)

Connected component

How does one know the existence of a connected component (i.e., maximal connected subset) in a topological space? Put in another way, does one need Zorn's lemma to show the existence? (Our connected space doesn't address this question, maybe it should unless this is a trivial matter.) Somehow related question: the article says: "The components in general need not be open"; true, but this pathology almost never happens in practice, I believe. What guarantees that connected components are open? -- Taku (talk) 23:38, 3 July 2009 (UTC)

The connected component of the point x is simply the union of all connected subsets containing x. It's not hard to show that this set is connected. Alternatively, the connected components are the equivalence classes of the relation x~y iff there is a connected set containing both x and y. Neither of these characterizations require heavy machinery, and they certainly don't require choice.

In the rationals, the connected components are not open. Surely Q doesn't count as a pathology? Anyway, the most obvious condition that forces components to be open is that there be only finitely many of them. Algebraist 00:14, 4 July 2009 (UTC)

Yeah, taking union. Why didn't I think of that. Anyway, thank you. You answered a lot more than I asked :) -- Taku (talk) 01:34, 4 July 2009 (UTC)

A kind of a followup-question. This is probably trivial too, but how do we know that there is a connected set containing given a point x at all? (Obvious, say, in a metric space, but in general) -- Taku (talk) 01:50, 4 July 2009 (UTC)

{x} Algebraist 02:02, 4 July 2009 (UTC)

Right :) -- Taku (talk) 02:25, 4 July 2009 (UTC)

Probably the most marvellously succinct answer I've ever seen given on a refdesk, Algebraist :D Maelin (Talk | Contribs) 03:22, 4 July 2009 (UTC)

Let me expand slightly on Algebraist's (sufficient) response. In a topological space, the components are always closed since the closure of a connected set is connected. Thus Algebraist's remark is justified. Secondly, the perhaps more general requirement for the components of a space to be open, is that the space be locally connected. Local connectedness is both a necessary and sufficient requirement for the components of any open subspace of the space in question to be open.

Similarly, there may be only one connected set containing a point. For instance, in the space of rationals (as mentioned above), there exists no connected subspace having more than one point. Thus each point is contained in precisely one connected subspace.

With regards to the necessity of the axiom of choice/Zorn's lemma to justify the existence of connected components, I think that Taku was comparing the situation with that in ring theory. More specifically, the truth of the idea that any proper ideal in a ring is contained in at least one (proper) maximal ideal requires the application of Zorn's lemma.

To conclude, I think that the word "pathology" has no specific meaning and depends on the context in which it is used. For instance, I do not consider any space to be pathological (because I am a point-set topologist :)), but on the other hand, people may consider the Bug-eyed line to be a pathological example of a manifold. Some people may even consider the Vitali set to be pathological. To take a completely different view point, pathological examples are often the most interesting. In particular, if a theorem does not hold unless one rules out specific pathologies, it is probably less natural than a theorem which holds for a class, including every pathology. --PST 04:40, 4 July 2009 (UTC)

Ah, that's very nice (and complete solution): locally connected-ness is necessary and sufficient. Thanks a lot. (It's just so much quicker to simply ask than going through topology books myself. Not to mention that the library is closed for the July 4-th). As for Zorn's lemma, actually, I was thinking of cases of irreducible components, since I thought I read somewhere that one uses Zorn's lemma to construct a maximal irreducible subset (i.e., irreducible component]). As for "pathology", I used the term because to me connected components that are not open are counterintuitive. I guess one could say a topological space that is not locally connected is not intuitive: In fact, locally connected has this: "In fact the openness of components is so natural that one must be sure to keep in mind that it is not true in general". -- Taku (talk) 12:51, 4 July 2009 (UTC)

Careful: local connectedness is not necessary for openness of components. It's necessary for openness of components of every open set. Algebraist 15:03, 4 July 2009 (UTC)

Right :) so not exactly the complete solution, then. -- Taku (talk) 02:00, 5 July 2009 (UTC)

Actually, if you read my response carefully, I noted this - "Local connectedness is both a necessary and sufficient requirement for the components of any open subspace of the space in question to be open." Perhaps Taku did not notice the wording because it is easy to miss.

Yes, I know. Where do you think I cribbed it from? Algebraist 02:44, 5 July 2009 (UTC)

Taku's statement, "so not exactly the complete solution, then" suggested that he believed that I had not correctly noted the equivalence in my post. I understood that you were addressing Taku (and not me), and I then addressed Taku to clarify the confusion. --PST 11:42, 5 July 2009 (UTC)