Actually I was thinking that I should also talk about quotients of boundary maps and that kind of thing. It could probably be done at the same sort of informal level, and might be useful for people who wanted to relate the above account to what they might have seen in an algebraic topology course.
Do you think that it is worth mentioning explicitly in this discussion that if your closed curve bounds a disk in the domain under discussion, then it can be shrunk down to a point, and so is null-homologous?
Your discussion is quite geometric, and I can see that you are trying to ilustrate how one might go from certain geometric/function-theoretic observations to the realization that the language of homology is an appropriate to employ; hence, the fact that many apparently non-trivial cycles are in fact homologically trivial is not what you want to emphasize, I would guess. So perhaps what I am really asking is whether it would be worth saying more about how to connect your discussion with the usual technical definitions of homology theory, or should one leave well enough alone?
Perhaps this could be explicitly discussed in the article; not in the quick description, but perhaps further down, in the general discussion. One could explain (just as in your comment) that there are various slightly different approaches to avoiding : either by having a positive lower bound on the difference, or by having tend to , staying coprime to .
Presumably at some point there will also be other articles on Diophantine approximation, including Liouville numbers and Roths' theorem, which will be linked to this article. Then one will be able to point out that even if is already known to be irrational (so that in particular is guaranteed to be positive), the question of bounding this quantity from above or below is still of great interest.
Another example that would be nice to include on this page eventually would be Apery's proof of irrationality of . (I suspect that this has been suitably massaged by this stage that one could write down a pretty simple sequence of fractions that does the job.) I don't have the resources to hand at the moment to do this, but if I get a chance at some later point, I might add it. If someone else wants to beat me to it, please do!
Not if you define the class of Borel sets to be the smallest class of sets that contains the open sets and is closed under countable unions and intersections. Then if you want to prove that some fact P is true of all Borel sets it is sufficient to prove that it holds for all open sets, and that a countable union or intersection of sets satisfying P satisfies P.
An analogy: suppose I have some property of invertible matrices and I know that the product of two matrices with that property has the property, and the inverse of a matrix with that property has the property. Now I take a set of matrices with that property. Then I know that every matrix in the group generated by has that property as well. Why? Well, I could build up stage by stage and do an inductive argument. Alternatively, I could say that I know that the set of all matrices with that property forms a subgroup of the group of all invertible matrices, and that subgroup contains . Therefore, it contains the group generated by . The second proof, uses a different notion of "subgroup generated by" and you need induction only to prove that the two notions are equivalent.
In the case of Borel sets, it is neater to use the definition above because it saves you having to mention ordinals every time you want to prove something.
Remark added 16/5/09: I have rewritten the article to reflect the fact that transfinite induction is not needed.
The title could definitely be changed. But transfinite induction is needed: to get from the base class of open sets to all Borel sets requires it, or something similar.
...is another good example of this trick, which I just covered in my class actually. I might put it in here later (and also interlink with "create an epsilon of room".)
The notation "" isn't very clear. My first thought was: "partial derivative with respect to ". After this, I figured you meant the operator on given by the decomposition of (the Jacobian of ), since this is the condition required by the theorem. Still, the notation gives rise the possibility .
The word "instance" leads the reader to expect an example – indeed, from what you say, the simplest non-trivial example.
would be
Do you mean "is"?
to prove that an object satisfies a property ,
Ah – it's not an example after all but a very abstract situation (since the concepts of "object" and "property" are about as general as you can get).
which we know that conmutes with a limit in some sense,
An undergraduate would not be comfortable with this notion of "commutes", and "in some sense" is unnecessarily vague: one should say something more precise, such as "A limit of objects that satisfy also satisfies ."
by finding a sequence of objects such that is invariant on it
The word "invariant" is unnecessarily off-putting. Does this mean just that holds for every ? Or do you mean that is actually a function that is constant on the ?
and such that it converges (in the same limit sense) to .
Ah, now I start to wonder whether "in some sense" above was referring to the limit rather than the commutation. In any case, better to declare right at the start that we think of as a limit of the .
The power of the method goes in the difficulty of proving the satisfaction of : should be easy to prove for every by some standard techniques, but difficult for with the same techniques.
I'm afraid I completely disagree with you about the status of examples. One example makes this clear, whereas with no examples one might think, "Surely if is the limit of the , then the only way of proving that satisfies will be along the lines you suggest. So what's the trick here?" Of course, the answer is that you actually construct the sequence, and do so in such a way that it consists of simpler objects (something you don't say here).
Note that this instance is somewhat the "reciprocal" of a limiting process (where we replace a sequence by its limit).
I don't really see what you're getting at here.
But of course,
If you're talking to undergraduates, then you shouldn't say "of course" about something like this.
we don't need to be invariant in the sequence: we just need to have a perturbed property for every such that converges to
Without an example, what undergraduate will know what you mean by a sequence of properties converging to another property?
as converges to . Moreover, we don't need the index set to be countable: we can use a net; for example, we can indize with the reals and use the variable .
This is also not a suitable remark for a "friendly" introduction!
Incidentally, I agree that the existing quick description is a little bit sophisticated for an undergraduate. My solution would probably be to start with a simple example, then continue with a more abstract definition of that example, then give the table, and then give the existing examples.
1. Why exactly do you fin it hard to understand? Is it caused by a word? Should it be wholly rephrased? (English is not my native language, so my writing can be convoluted or even plainly wrong!)
2. What does it add to the quick description? I wanted to make the article more "friendly". I wanted to write one paragraph that an undergraduate could fully understand; I think the quick description is too technical for that level, because understanding it implies being fairly used to "trick thinking" already - and then, at this level the trick is rather obvious!
With this in mind (not everyone who comes here should be used enough to "tricks jargon"!), I included the simplest example because it is conceptually near to the "limiting argument" technique, which I think is well-known for undergraduates.
3. Why don't just have an easy example? Because an example, being as good as any other tool, is not an explanation. Actually, to properly show a general technique, you must give a metaexample (a collection of similar examples or an abstract example), and that's what I tried to do: give this example where the approximation is done just by a sequence and the approximating properties are not just converging to the desired property - they are the very same during all the process.
In my humble opinion, examples should be confined to the "examples" section and the general discussion and quick description should give abstract definitions and metaexamples.
4. Does "closed under limits" really sound clearer in English or is it just "more rigorous"? (I ask because I don't really know!). To me, the idea that the "operators" 'limit' and 'property' conmute looked like the easiest picture to grasp, and I just wanted the general idea to be understood. If an english student would understand better "closed under limits", then change it, please!
I find this rather hard to understand. I also find it difficult to see what it adds to the quick description. Perhaps instead one could have an easy example, such as approximating a function in by a trigonometric polynomial by first approximating by a step function, then approximating the step function by a continuous function, and finally approximating the continuous function uniformly by a trigonometric polynomial. The first two steps tell you that you can find a sequence of continuous functions that approximate your function in , and the property of being approximable in is obviously closed under limits. (Incidentally, I think "closed under limits" would be clearer than "commutes with limits in some sense".
I was careful about that in the examples, but as you imply it is annoyingly difficult to find a good formulation for the quick description. I've gone for denominators tending to infinity when the fractions are written in their lowest terms. But in the case of it is slightly easier to use the not-equal-to- formulation. If anyone has strong views about this, feel free to change it.
One probably needs to insist that to avoid the triviality of approximating a rational number by the constant sequence . I'm not sure exactly how this is usually phrased in this context, so I will let someone else make the changes in the text.
It doesn't look to me as though transfinite induction is needed here. All you need is that the Borel sets are the closure of the open sets under countable unions and intersections. So I think the article should have a title such as the following — "To prove a fact about all Borel sets, prove it for open sets and prove that it is preserved by countable unions and intersections" — and should be rewritten accordingly.
I suggest not having any examples on this page, since numerical analysis is a big subject. Instead, there should be a brief introduction to the various goals of numerical analysis, and links to more specialized articles where the examples would appear. This would bring the article in line with other front pages.
Is this really an example of Occam's razor? It's not 'needless' to hypothesize that air is compressible - after all, it is (slightly) compressible - it just leads to intractable equations!
Two minor comments. First, the parent of this article should perhaps now be changed to "How to solve problems about finite groups". Secondly, this article itself could easily have its title made imperative if one omitted "How to". I think that would be an improvement. Or perhaps better would be "To prove facts about finite groups, use induction on the order". To the objection, "But there are many problems for which that does not work," I would say that I see Tricki titles as a bit like sellers in a busy market: lots of people are shouting out, trying to persuade you to buy their products, and you are not expected to obey them all.
The problem with the add tag was that [/add] was missing entirely at the end. I put it in (in the form }}.[/add]) which seemed to fix things (although you are correct that for reasons of correct punctuation, the full stop should be outside the [/add]).
I think I was brought up with "path integral", but maybe that was more the general definition, which can apply to all functions regardless of whether they are holomorphic, with "contour integral" used more often for holomorphic functions round closed paths. Since that's mainly what I'm talking about here, and since you may well be right that it's more standard, I've made the change you suggest.
The problem with the "add" was that I ended it }}.[/add] instead of }}[/add]. It took me a while to work that out. (Slightly puzzlingly, Matthew Emerton had already fixed the problem, so perhaps that wasn't actually what was causing it.)
Actually I was thinking that I should also talk about quotients of boundary maps and that kind of thing. It could probably be done at the same sort of informal level, and might be useful for people who wanted to relate the above account to what they might have seen in an algebraic topology course.
Do you think that it is worth mentioning explicitly in this discussion that if your closed curve bounds a disk in the domain under discussion, then it can be shrunk down to a point, and so is null-homologous?
Your discussion is quite geometric, and I can see that you are trying to ilustrate how one might go from certain geometric/function-theoretic observations to the realization that the language of homology is an appropriate to employ; hence, the fact that many apparently non-trivial cycles are in fact homologically trivial is not what you want to emphasize, I would guess. So perhaps what I am really asking is whether it would be worth saying more about how to connect your discussion with the usual technical definitions of homology theory, or should one leave well enough alone?
Perhaps this could be explicitly discussed in the article; not in the quick description, but perhaps further down, in the general discussion. One could explain (just as in your comment) that there are various slightly different approaches to avoiding : either by having a positive lower bound on the difference, or by having tend to , staying coprime to .
Presumably at some point there will also be other articles on Diophantine approximation, including Liouville numbers and Roths' theorem, which will be linked to this article. Then one will be able to point out that even if is already known to be irrational (so that in particular is guaranteed to be positive), the question of bounding this quantity from above or below is still of great interest.
Another example that would be nice to include on this page eventually would be Apery's proof of irrationality of . (I suspect that this has been suitably massaged by this stage that one could write down a pretty simple sequence of fractions that does the job.) I don't have the resources to hand at the moment to do this, but if I get a chance at some later point, I might add it. If someone else wants to beat me to it, please do!
Not if you define the class of Borel sets to be the smallest class of sets that contains the open sets and is closed under countable unions and intersections. Then if you want to prove that some fact P is true of all Borel sets it is sufficient to prove that it holds for all open sets, and that a countable union or intersection of sets satisfying P satisfies P.
An analogy: suppose I have some property of invertible matrices and I know that the product of two matrices with that property has the property, and the inverse of a matrix with that property has the property. Now I take a set of matrices with that property. Then I know that every matrix in the group generated by has that property as well. Why? Well, I could build up stage by stage and do an inductive argument. Alternatively, I could say that I know that the set of all matrices with that property forms a subgroup of the group of all invertible matrices, and that subgroup contains . Therefore, it contains the group generated by . The second proof, uses a different notion of "subgroup generated by" and you need induction only to prove that the two notions are equivalent.
In the case of Borel sets, it is neater to use the definition above because it saves you having to mention ordinals every time you want to prove something.
Remark added 16/5/09: I have rewritten the article to reflect the fact that transfinite induction is not needed.
The title could definitely be changed. But transfinite induction is needed: to get from the base class of open sets to all Borel sets requires it, or something similar.
...is another good example of this trick, which I just covered in my class actually. I might put it in here later (and also interlink with "create an epsilon of room".)
The notation "" isn't very clear. My first thought was: "partial derivative with respect to ". After this, I figured you meant the operator on given by the decomposition of (the Jacobian of ), since this is the condition required by the theorem. Still, the notation gives rise the possibility .
What does the notation mean in equation 2?
Instead of using , why not use ?
Any chance of telling us what these principles are? I probably ought to know, but I don't, and maybe others don't either.
Let me try to explain in detail.
The first, simplest instance of this trick
The word "instance" leads the reader to expect an example – indeed, from what you say, the simplest non-trivial example.
would be
Do you mean "is"?
to prove that an object satisfies a property ,
Ah – it's not an example after all but a very abstract situation (since the concepts of "object" and "property" are about as general as you can get).
which we know that conmutes with a limit in some sense,
An undergraduate would not be comfortable with this notion of "commutes", and "in some sense" is unnecessarily vague: one should say something more precise, such as "A limit of objects that satisfy also satisfies ."
by finding a sequence of objects such that is invariant on it
The word "invariant" is unnecessarily off-putting. Does this mean just that holds for every ? Or do you mean that is actually a function that is constant on the ?
and such that it converges (in the same limit sense) to .
Ah, now I start to wonder whether "in some sense" above was referring to the limit rather than the commutation. In any case, better to declare right at the start that we think of as a limit of the .
The power of the method goes in the difficulty of proving the satisfaction of : should be easy to prove for every by some standard techniques, but difficult for with the same techniques.
I'm afraid I completely disagree with you about the status of examples. One example makes this clear, whereas with no examples one might think, "Surely if is the limit of the , then the only way of proving that satisfies will be along the lines you suggest. So what's the trick here?" Of course, the answer is that you actually construct the sequence, and do so in such a way that it consists of simpler objects (something you don't say here).
Note that this instance is somewhat the "reciprocal" of a limiting process (where we replace a sequence by its limit).
I don't really see what you're getting at here.
But of course,
If you're talking to undergraduates, then you shouldn't say "of course" about something like this.
we don't need to be invariant in the sequence: we just need to have a perturbed property for every such that converges to
Without an example, what undergraduate will know what you mean by a sequence of properties converging to another property?
as converges to . Moreover, we don't need the index set to be countable: we can use a net; for example, we can indize with the reals and use the variable .
This is also not a suitable remark for a "friendly" introduction!
Incidentally, I agree that the existing quick description is a little bit sophisticated for an undergraduate. My solution would probably be to start with a simple example, then continue with a more abstract definition of that example, then give the table, and then give the existing examples.
I'll try to answer:
1. Why exactly do you fin it hard to understand? Is it caused by a word? Should it be wholly rephrased? (English is not my native language, so my writing can be convoluted or even plainly wrong!)
2. What does it add to the quick description? I wanted to make the article more "friendly". I wanted to write one paragraph that an undergraduate could fully understand; I think the quick description is too technical for that level, because understanding it implies being fairly used to "trick thinking" already - and then, at this level the trick is rather obvious!
With this in mind (not everyone who comes here should be used enough to "tricks jargon"!), I included the simplest example because it is conceptually near to the "limiting argument" technique, which I think is well-known for undergraduates.
3. Why don't just have an easy example? Because an example, being as good as any other tool, is not an explanation. Actually, to properly show a general technique, you must give a metaexample (a collection of similar examples or an abstract example), and that's what I tried to do: give this example where the approximation is done just by a sequence and the approximating properties are not just converging to the desired property - they are the very same during all the process.
In my humble opinion, examples should be confined to the "examples" section and the general discussion and quick description should give abstract definitions and metaexamples.
4. Does "closed under limits" really sound clearer in English or is it just "more rigorous"? (I ask because I don't really know!). To me, the idea that the "operators" 'limit' and 'property' conmute looked like the easiest picture to grasp, and I just wanted the general idea to be understood. If an english student would understand better "closed under limits", then change it, please!
I find this rather hard to understand. I also find it difficult to see what it adds to the quick description. Perhaps instead one could have an easy example, such as approximating a function in by a trigonometric polynomial by first approximating by a step function, then approximating the step function by a continuous function, and finally approximating the continuous function uniformly by a trigonometric polynomial. The first two steps tell you that you can find a sequence of continuous functions that approximate your function in , and the property of being approximable in is obviously closed under limits. (Incidentally, I think "closed under limits" would be clearer than "commutes with limits in some sense".
I was careful about that in the examples, but as you imply it is annoyingly difficult to find a good formulation for the quick description. I've gone for denominators tending to infinity when the fractions are written in their lowest terms. But in the case of it is slightly easier to use the not-equal-to- formulation. If anyone has strong views about this, feel free to change it.
One probably needs to insist that to avoid the triviality of approximating a rational number by the constant sequence . I'm not sure exactly how this is usually phrased in this context, so I will let someone else make the changes in the text.
It doesn't look to me as though transfinite induction is needed here. All you need is that the Borel sets are the closure of the open sets under countable unions and intersections. So I think the article should have a title such as the following — "To prove a fact about all Borel sets, prove it for open sets and prove that it is preserved by countable unions and intersections" — and should be rewritten accordingly.
Example 1 could use some motivation: where does this sort of statement come up and why would one want to prove it?
Some examples for multidimensional integration would be nice to see here.
I suggest not having any examples on this page, since numerical analysis is a big subject. Instead, there should be a brief introduction to the various goals of numerical analysis, and links to more specialized articles where the examples would appear. This would bring the article in line with other front pages.
Is this really an example of Occam's razor? It's not 'needless' to hypothesize that air is compressible - after all, it is (slightly) compressible - it just leads to intractable equations!
I updated the link to the How to prove facts about finite groups by induction on the order page to take into account the new name of that page, and moved it so as to be under the How to solve problems about finite groups link.
I changed the title following the second of your suggestions.
... of the diagonalisation article, so I've interlinked and parented accordingly.
Two minor comments. First, the parent of this article should perhaps now be changed to "How to solve problems about finite groups". Secondly, this article itself could easily have its title made imperative if one omitted "How to". I think that would be an improvement. Or perhaps better would be "To prove facts about finite groups, use induction on the order". To the objection, "But there are many problems for which that does not work," I would say that I see Tricki titles as a bit like sellers in a busy market: lots of people are shouting out, trying to persuade you to buy their products, and you are not expected to obey them all.
There is some overlap between this and convergent subsequences and diagonalization. Perhaps some interlinking or merging would be appropriate.
The problem with the add tag was that [/add] was missing entirely at the end. I put it in (in the form }}.[/add]) which seemed to fix things (although you are correct that for reasons of correct punctuation, the full stop should be outside the [/add]).
I think I was brought up with "path integral", but maybe that was more the general definition, which can apply to all functions regardless of whether they are holomorphic, with "contour integral" used more often for holomorphic functions round closed paths. Since that's mainly what I'm talking about here, and since you may well be right that it's more standard, I've made the change you suggest.
The problem with the "add" was that I ended it }}.[/add] instead of }}[/add]. It took me a while to work that out. (Slightly puzzlingly, Matthew Emerton had already fixed the problem, so perhaps that wasn't actually what was causing it.)