Wikipedia:Reference desk/Archives/Mathematics/2010 December 6

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December 6

Continuity and uniform continuity

According to Spivak, a function can be continuous on an open interval or the real line without being uniformly continuous there. He further says that if the function is continuous on [a,b] it is uniformly continuous on [a,b]. Why is it that a function can be continuous on an open interval or the real line without being uniformly continuous there? He doesn't give a proof here as he usually does, and his explanation (in words) is rather vague. 24.92.78.167 (talk) 01:03, 6 December 2010 (UTC)

Try 1/x on (0,1), or x² on R. Algebraist 01:25, 6 December 2010 (UTC)

Yes, the point is that uniform continuity demands a 'one size fits all' definition for the limit, \delta cannot depend on x_0, as it can for regular continuity. Intuitively, any differentiable function on an open domain with an unbounded first derivative will fail to be uniformly continuous. Restricting to closed intervals changes the picture dramatically, and highlights the importance of compactness. SemanticMantis (talk) 15:32, 6 December 2010 (UTC)

Rigor of a wikibooks article

How rigorous (and correct) is http://en.wikibooks.org/wiki/Calculus/Choosing_delta , and the Calculus Wikibook in general? To me it does not look very correct, but I do not know calculus well enough to fix it myself. 24.92.78.167 (talk) 03:43, 6 December 2010 (UTC)

The writing is a little sloppy but on a quick glance, the content looks basically correct. Did you a spot a particular error? 67.117.130.143 (talk) 05:36, 6 December 2010 (UTC)

the uncountable real "numbers"

how can there be uncountable sets? on what basis are we to consider that the contents of such mathematical objects may be considered as divisible into individual elements on the same ontological footing as those objects uniquely definable (and hence countable)? 82.12.115.27 (talk) 04:12, 6 December 2010 (UTC)

There are uncountable sets because Cantor showed that no set is as large as its power set and so the power set of a countably infinite set is, by definition, not countable. CRGreathouse (t | c) 05:09, 6 December 2010 (UTC)

http://en.wikipedia.org/w/index.php?title=Wikipedia:Reference_desk/Mathematics&action=edit&section=17

To elaborate, the Cantor_diagonalization argument is actually fairly accessible. However, his proof by contradiction does not illuminate the ontological nature of the real numbers. For that, a more constructive approach would use Dedekind_cuts. In particular, each and every (uncountable) real number can be uniquely defined in this manner. Thus, the concepts of `uniquely defined' and `countable' are completely independent. Hope this helps. SemanticMantis (talk) 15:26, 6 December 2010 (UTC)

Most real numbers are not definable. -- Meni Rosenfeld (talk) 16:06, 6 December 2010 (UTC)

Thanks for the clarification, Meni, I was not familiar with that sense of `definable'. As per the article you cited, I should have said `each ... real number can be unambiguously described'. So, when we associate a real number with a limit of rationals or unambiguously describe it with a Dedekind cut, this does not satisfy the definition of `definable'... Can you explain (or give resources for) the motivation for a concept of definition that is more restrictive than unambiguous description? SemanticMantis (talk) 20:08, 6 December 2010 (UTC)

"Definable" can mean various things (depending on the language the definition is supposed to be in), but generally means definable by a finite string in a finite language. This fits with the intuition that a definition should be something you can actually say: it's not very satisfying to "define" a weird infinitary object with a definition which is itself a weird infinitary object. Algebraist 20:21, 6 December 2010 (UTC)

Ok, but if we define a real number by a Dedekind cut, that looks like a finite string to me. Moreover, I suspect that it is Well-defined. So a `typical' transcendental number is well-defined, but not definable... is the problem that the finite string I allude to is not described by a formal finite language? SemanticMantis (talk) 20:49, 6 December 2010 (UTC)

How is a Dedekind cut anything resembling a finite string? Algebraist 20:52, 6 December 2010 (UTC)

I simply meant that the expressions here [1] are written in finitely many symbols of the (pseudo?)language we commonly write math in. I realize now that I'm out of my depth in logic and formal language theory, but thanks for your answers. SemanticMantis (talk) 21:07, 6 December 2010 (UTC)

Right, the Dedekind cut corresponding to √2 can be encoded as a finite string as appears in the link. So can π and more exotic numbers like the real root of $x^{5}=x+1$. But the majority (as in, all but a countable subset) of real numbers do not have such a representation. This follows trivially from the fact that the number of finite strings over a finite alphabet is countable. It's impossible to give an explicit example of an undefinable number because giving it as an example requires a definition of it. But I'm not very versed in the philosophical implications of this, which appear to be what the OP is after. -- Meni Rosenfeld (talk) 08:43, 7 December 2010 (UTC)

this is the crux of my 'objection'. it is not aimed at cantor's diagonal argument in demonstration of uncountable entities (note: not sets), but at our importing the intuitive sense of 'individual constituents' of the entities in question. it is one thing to say that a countably-infinite entity (such as the set of natural numbers) 'contains members with individual ontological basis', but quite another to suggest the same applies for an uncountable entity such as the continuum, in which through finitary process only a sparse subset can be differentiated. it would occur to me that the Aristotelian laws of identity and excluded middle do not appertain to what remains of the reals after the removal of the its definable elements. as such, it is inappropriate to speak of this entity as comprising individual self-same elements. 82.12.115.27 (talk) 04:49, 7 December 2010 (UTC)

"uniquely definable (and hence countable)" doesn't make any sense at all. "Unique definability" in no way implies countability. Michael Hardy (talk) 16:53, 6 December 2010 (UTC)

this is incorrect, see Definable real number, specifically "However, most real numbers are not definable: the set of all definable numbers is countably infinite (because the set of all logical formulas is) while the set of real numbers is uncountably infinite (see Cantor's diagonal argument). As a result, most real numbers have no description (in the same sense of "most" as 'most real numbers are not rational')." 82.12.115.27 (talk) 04:32, 7 December 2010 (UTC)

See Richard's paradox.—Emil J. 14:57, 7 December 2010 (UTC)

Actually, now that I've seen it, there are serious issues with the definable real number article. Definable numbers do not form a set (an object of set theory), and a fortiori they cannot be shown countable in the sense of the theory itself, the statement is meaningless. It is only meaningful to say that in any given model of ZFC there are countably many definable reals, where "countably" refers to cardinality in the metatheory, external to the model. (And even if the definable reals of some model of ZFC happen to form a set in the model, there is no reason why the model should contain a function that assigns to each definable real the Gödel number of one of its defining formulas, hence the model may think that the set is uncountable.) Cantor's diagonal argument thus cannot be applied, to begin with it's perfectly possible that the whole model is countable. Indeed, if ZFC is consistent, then it also has a model where every set (and in particular, every real number) is definable (constructions of such models are mentioned above), hence it is not true that undefinable reals necessarily exist.—Emil J. 19:17, 7 December 2010 (UTC)

Well, there are serious issues, but the first one you mention is not really one of them. Of course the definable real numbers (whatever "definable" means) must form a set. The only way a collection of sets can fail to be a set is if it's a proper class. You can't, using the first-order language of set theory alone, say what you mean by "definable in the first-order language of set theory", but that doesn't mean the concept isn't well-specified. It's an error to think that every set-theoretic statement must be relativized to some model of ZFC. The von Neumann universe is intuitively well-specified, and suffices to give meaning to set-theoretic statements in and of itself. --Trovatore (talk) 07:20, 8 December 2010 (UTC)

You are confusing two meanings of a "set". The concept of definable real numbers cannot be expressed in set theory, it can't be used from within the theory itself. It only makes sense if you take a model of set theory (and I mean a real model in the sense of model theory, with set as a domain, not a class interpretation which set theorists also call "models"), and consider reals definable in that model. Then the set of such reals is a set in the metatheory (it is a subset of the domain of the model), but in general it is not a set in the model itself (there is no object in the model whose elements, according to the elementhood predicate of the model, is the set of definable elements of the model). It's an error to think that every set-theoretic statement must be relativized to some model of ZFC, but in this particular case, there is no way around it. The concept is not well-specified otherwise, just as the concept of "the set of all sets that are not elements of themselves" is not. There are many ways how a "collection of sets" can fail to be a set, and while being a proper class is one of them, it's not the only one. Another one is not being a class in the first place, i.e., not being definable.—Emil J. 13:25, 8 December 2010 (UTC)

No, you're dragging in unnecessary complications, and also introducing category errors when you talk about "sets in theories" (or metatheories). Sets don't exist in theories; theories are syntactic objects and don't have any sets inside.

It's true that a model in general need not have an object whose elements dot dot dot as you say. But nevertheless there must be such an object in the von Neumann universe. Here's why: The von Neumann universe is constructed by iterating the true, absolute, powerset operation. That is, it collects absolutely all sub-collections of a given set, all the ones that could ever Platonistically exist, not just the ones that are specifiable in some fixed language. --Trovatore (talk) 17:00, 8 December 2010 (UTC)

I'm not dragging in any complications, on the contrary, you are ignoring problems that directly lead to paradoxes when ignored. What you are saying about the von Neumann universe only makes sense if you take a hard-line fundamentalist Platonist position. It's a grave error to impose such skewed views on a Wikipedia article, especially without any warning to the reader.—Emil J. 17:15, 8 December 2010 (UTC)

I have always found Trovatore to be a strong advocate of the "hard-line fundamentalist Platonist position". That's not a criticism or a compliment, it's just a confirmation. — Carl (CBM · talk) 17:27, 8 December 2010 (UTC)

The issue of definable numbers came up on MathOverflow recently, where this answer about them was quite critical of the treatment of definable numbers in our article. We should probably discuss the article on its talk page (Talk:Definable number) rather than here.

The underlying disagreement is about whether, as Trovatore says, "It's an error to think that every set-theoretic statement must be relativized to some model of ZFC." Trovatore's concern is that there is a natural-language meaning of "definable real number" which is lost in formalization. The opposing concern is that the statement "A set exists which contains every real number that is definable, and no other elements" is not expressible in the language of set theory. — Carl (CBM · talk) 17:16, 8 December 2010 (UTC)

Golden Ratio

Could someone recommend a good book or text on the golden ration? Preferably for someone who has some background in math. 74.15.138.27 (talk) 08:32, 6 December 2010 (UTC)

Have you looked at the article yet? It looks pretty good, and has a long list of references and further reading at the bottom.81.98.38.48 (talk) 17:11, 6 December 2010 (UTC)

The further reading section seems quacky -- a lot of books on divine geometry and dienergy...the article itself isn't very informative imo, it gives results without proofs or going in depth. 74.15.138.27 (talk) 17:31, 7 December 2010 (UTC)

∑ra^r

What is $\sum _{r=0}^{n}r\times a^{r}$ in terms of $a$ and $n$  ? --220.253.244.70 (talk) 10:59, 6 December 2010 (UTC)

Summations of the form $\sum _{n}n^{k}x^{n}\$, k an integer, can be simplified using repeated integration or differentiation with respect to x. See if you can make progress from there. Eric. 82.139.80.218 (talk) 11:18, 6 December 2010 (UTC)

Start with

$\sum _{r=0}^{n}ra^{r}=0+a+2a^{2}+3a^{3}+\dots +na^{n}.$

Then, grouping terms, you get

$(a+a^{2}+a^{3}+\dots +a^{n})+(a^{2}+a^{3}+\dots +a^{n})+(a^{3}+\dots +a^{n})+\dots +a^{n}.$

Using the familiar conversion (for |a| < 1)

$a+ar+ar^{2}+\dots +ar^{n-1}={\frac {a(1-r^{n})}{1-r}},$

you should be able to derive a useful expression for the sum. —Anonymous Dissident^Talk 11:36, 6 December 2010 (UTC)

${\begin{aligned}\sum _{r=0}^{n}ra^{r}=a\sum _{r=0}^{n}ra^{r-1}=a\sum _{r=0}^{n}{\frac {d}{da}}a^{r}=a{\frac {d}{da}}\sum _{r=0}^{n}a^{r}=a{\frac {d}{da}}\,{\frac {1}{1-a}}={\frac {a}{(1-a)^{2}}}.\end{aligned}}$

The interchange of the summation and differentiation is justified because power-series are well-behaved in that particular way in the interior of their region of convergence. Michael Hardy (talk) 16:52, 6 December 2010 (UTC)

The OP asked about a finite sum, so the interchangeability is trivial. -- Meni Rosenfeld (talk) 18:04, 6 December 2010 (UTC)

Error above: I hadn't actually noticed it was a finite sum, so the first step that doesn't have the summation sign needs to be altered accordingly. Michael Hardy (talk) 21:54, 7 December 2010 (UTC)

Rotation of solids (calculus I HW)

I'm having trouble figuring out what the upper and lower limits of my integral should be when rotating graphs. For example, when finding the area y = sqrt(x), bound by y=2 and x=0, and rotated about y=2, will my interval run from 0 to 2 (that is from x=0 to x=2) or from 2 to 4 (y=2 to y=4)? Is there a sure-fire way of finding these upper and lower bounds? Also, is the area of this 2Pi? Thank you.161.165.196.84 (talk) 12:24, 6 December 2010 (UTC)

The best way to approach these problems is to visualise what's happening, either mentally or by drawing a picture. Intuitively, you're stacking infinitesimally thin cylinders (of radius 2 − √x) for 0 ≤ x ≤ 4. The volume is calculated via an integral which sums these infinitesimal volumes. The key is to figure out whether dx or dy represents the height of the cylinders. The bounds fall out naturally after this part. —Anonymous Dissident^Talk 13:31, 6 December 2010 (UTC)

The OP asked about the surface area, which is somewhat harder to visualize. -- Meni Rosenfeld (talk) 13:53, 6 December 2010 (UTC)

By the way, the relevant articles are Surface of revolution and Solid of revolution. -- Meni Rosenfeld (talk) 14:08, 6 December 2010 (UTC)

Characters on homology groups of simplicial complexes

I,ve been given this problem and have made some progress with it but am still slightly confused over the statement of it:

Essentially I need to find a simplicial complex (http://en.wikipedia.org/wiki/Simplicial_complex) such that the alternating sum of the characters for the homology groups $\sum (-1)^{i}h_{i}$, where $h_{i}$ is a character for the homology group $H_{i}$ (http://en.wikipedia.org/wiki/Simplicial_homology, better explained http://www.math.hmc.edu/%7Esu/pcmi/projects/simplicial_module/simplicial_pcmi.pdf) of the characters gives the character

$\chi (g)=|\{(x,y)\in G|g=xyx^{-1}y^{-1}\}|$

To explain in more detail:

is there a natural looking way to write this character as a difference of two permutation characters for some classes of groups? This would follow from the simplicial complex statement by taking the permutation characters on even/odd dimensional simplices and taking the difference of those characters.

I think I have stated the problem correctly, but please ask if you think I am unclear as I don't quite understand it. My main issues are:

Surely you cannot make a simplicial complex with homology groups isomorphic to any group? If not how to the characters on these groups relate to other groups like $S_{n}$ or $SL(2,q)$ for example.

As regards to the statement on permutation characters, how can you have a permutation character on a homology group, when the group is not acting on anything? Is there a natural way to see this as acting on the simplicial complex?

Sorry for the long post, If you think you may be able to help with even just a tiny part of my question it would be a great help.

Many thanks —Preceding unsigned comment added by 137.222.137.35 (talk) 12:53, 6 December 2010 (UTC)

I don't think I understand your question. What is the definition of "a character for the homology group $H_{i}$"? And is there some kind of action of a group on the simplicial complex involved in this? Aenar (talk) 17:39, 8 December 2010 (UTC)

To elaborate: Assuming you have an action of a group on a simplicial complex X you will get a representation of the group on $H_{i}(X)$ if you are looking at homology with coefficients in some field ($H_{i}(X)$ usually means homology with $\mathbb {Z}$-coefficients, where you don't get a representation). So in this case you will get a character for each $H_{i}(X)$. Is this the right setup for your problem? Aenar (talk) 17:56, 8 December 2010 (UTC)

1/N

When N is a positive integer, am I correct in thinking that 1/N must have a repeating pattern of at most N digits in any base? (They obviously aren't the same, since 1/11 = 0.090909... in base 10 (a 2 digit repeat) whilst in base 11 it is 0.100000... a 1 digit repeat.) -- SGBailey (talk) 16:54, 6 December 2010 (UTC)

Yes. Let b be the base, and write N = rs, where r | b^m for some m, and gcd(s,b) = 1. Then $b\in (\mathbb {Z} /s\mathbb {Z} )^{\times }$, hence $b^{k}\equiv 1{\pmod {s}}$ for some k dividing $|(\mathbb {Z} /s\mathbb {Z} )^{\times }|=\varphi (s)<N$. Then ${\frac {1}{N}}={\frac {c}{b^{m}(b^{k}-1)}}$ for some integer c, thus 1/N has a base b representation with fixed part of length m and a periodic part of length k. (In fact, since m is bounded by log₂r ≤ r, the sum of the lengths of the fixed part and the periodic part are also bounded by N.)—Emil J. 17:17, 6 December 2010 (UTC)

And for a less rigorous, but more elementary and intuitive proof: When you divide by N using long division, once you run out of digits in the dividend, the way the calculation will proceed depends solely on the current remainder. Since there are N possible remainders, after at most N steps you will have a previously seen remainder, at which point the digits since then will repeat themselves. -- Meni Rosenfeld (talk) 18:14, 6 December 2010 (UTC)

(ec) The period of the repeating decimal is always a divisor of N-1. It's exactly equal to N-1 in base k if N is a primitive root mod k. See also Repeating decimal. AndrewWTaylor (talk) 18:18, 6 December 2010 (UTC)

This is not true. For example, 1/4 in base 3 is 0.02020202..., and the period 2 does not divide 4-1=3. Algebraist 18:25, 6 December 2010 (UTC)

(e/c) Umm, no. For example, 1/21 = 0.047619 in decimal has a period of length 6, which is not a divisor of 21 − 1 = 20. What is true, and the above derivation shows it, is that the period length is a divisor of φ(N). It equals N − 1 if and only if N is prime and the base is a primitive root modulo N, not the other way round.—Emil J. 18:27, 6 December 2010 (UTC)

It's true if N is prime.--RDBury (talk) 18:51, 6 December 2010 (UTC)

Oops, apologies for my half-remembered non-facts. AndrewWTaylor (talk) 09:32, 7 December 2010 (UTC)

At every step in long division of 1.0000000... by N, there are N possible remainders:

0, 1, 2, 3, ...., N − 1.

If you ever get 0, you're done. If you never get 0, then there are just N − 1 possible remainders. Once you've gone N − 1 steps without the same remainder recurring, you've used them all up. Therefore the next one must be one you've seen before. If you get a remainder that you've seen before, then you're caught in a loop so it has to repeat. If you get one you've seen before in fewer than N − 1 steps, then you get a short repetend. Michael Hardy (talk) 02:11, 7 December 2010 (UTC)

Thanks all. -- SGBailey (talk) 07:49, 7 December 2010 (UTC)

Numerical integration

It's not obvious to me how numerical integration can be expected to work (i.e. converge towards the actual value of the integral) without assuming something about the function being integrated (piecewise real-analytic, bounded derivative, continuous etc.). What are the necessary assumptions for common techniques like Gaussian quadrature, and is this information tabulated (table of necessary assumptions for different method to work and rate of convergence given these conditions) somewhere in the mathematical literature?--Leon (talk) 18:28, 6 December 2010 (UTC)

Yes these methods tend to have error-estimation theorems that state the conditions precisely (that's why they're called theorems). E.g. for Gaussian quadrature see Gaussian_quadrature#Error_estimates. 67.117.130.143 (talk) 09:27, 7 December 2010 (UTC)

Yes, if the function is differentiable 2n times then you have these estimates. In particular I think the error is $O(\epsilon ^{2n})$. But the OP is interested in worse-behaved functions than that.

I'd be surprised if Riemann integrability is not sufficient for Gaussian quadrature to converge. However, it seems plausible to me that you can find a function differentiable one too few times so that for every $\alpha >0$ the error is $\Omega (\epsilon ^{\alpha })$. -- Meni Rosenfeld (talk) 10:59, 7 December 2010 (UTC)

I don't understand your last statement (I mean, I'm not familiar with the notation, and probably not the concept either). Can you direct me to appropriate articles?--Leon (talk) 18:07, 8 December 2010 (UTC)

If you mean the big-Omega notation, you want Big O notation and in particular this section.

Just in case "differentiable one too few times" was also unclear, it's just my clumsy way of saying a function which is differentiable only $2n-1$ times, where the standard error bounds require it to have a bounded $2n$th derivative.

What I'm thinking about is starting with the Weierstrass function (which is continuously differentiable 0 times) and taking $2n-1$ antiderivatives. I could be wrong about this though. -- Meni Rosenfeld (talk) 18:21, 8 December 2010 (UTC)

Well-definition and definability

If the real numbers are constructed via Dedekind cuts, does this result in each real number being well-defined? From an unrelated question above [2], I learned that most real numbers are not definable. As a mathematical-linguistic curiosity, does this mean there is a real number that is well-defined, but not definable? Thanks, SemanticMantis (talk) 21:32, 6 December 2010 (UTC)

What do you mean by "well-defined" here? Normally it means that a putative definition, for which it is not obvious that it actually defines a unique object, does in fact do so. That is, well-definedness is not actually a property of the object, but of the attempted definition. Obviously, with this meaning an object which cannot be defined cannot be well-defined. Algebraist 21:38, 6 December 2010 (UTC)

I understand well-definition applies to a putative definition, not the object. In this case, I meant putatively defining each real number in terms of a Dedekind cut, as explained in the first link. My intuitive understanding is that this results in well-definition, but I'm not sure, so I asked. From my reading of well-definition, it is possible for a putative definition to unambiguously describe an object (in this case a real number), making it well-defined, but I do not see how this could imply anything about definabilty in the restricted sense the article describes. SemanticMantis (talk) 21:57, 6 December 2010 (UTC)

I think what you mean by well-defined is more like "definite" than "definable". From a realist point of view, every real number is a definite object (that is, it's only one thing), even if it's not definable (you can't write words that pick out that object and exclude all others). --Trovatore (talk) 23:03, 6 December 2010 (UTC)

Thanks, this clears things up a bit. I guess part of my confusion is that the real numbers as a field can be completely defined, even though there are individual reals that cannot be first-order defined in a given finite language. SemanticMantis (talk) 14:54, 7 December 2010 (UTC)

You may ask, why to introduce so a wide class of numbers, if in practice we will use only a very small portion of them. A cheep point of view should be, just introduce a number as soon as one needs it, or at least, just some class of constructible or definable ones, in a suitable sense. After all this is what one does e.g. in field extensions. One reason to buy the whole package of reals, is that that way one has a big scenario once for all, at a lower price than introducing each number as soon as you need it. It's like when you buy your big new computer: you would virtually download the whole internet, even though you know you'll never use most of that stuff. --pma 20:57, 10 December 2010 (UTC)