It's not clear to me whether this should be a single article with various different ways of using subsequences, or whether each different way deserves its own article. This is a general question about the whole of the Tricki – at what point does one split an article up? Another question: how should the first technique relate to How to use the Bolzano-Weierstrass theorem?

I agree mostly with you in this issue - Imperative and how-to titles are desirable. That's why I created stubs for "How to work without an identity element when you clearly need it", etc.

But my main purpose creating this article was not to explain how to use idempotents (for that, I created the other articles). My main purpose was to help those people that has heard about Peirce decomposition but does not see anything special about it.

The main example I have in mind (I'm new to the field, so there's not that much I can think of!)is when you want to prove a categorical statement about a ring without identity (mostly about R-Mod), and typical "universal" or homological techniques with pretty diagrams do not seem to work; then your "other option" is to switch to ring decompositions like Peirce's (or, in a sort of generalization of this, to exploit some Morita context if you have it).

I've discussed titles quite a bit in this forum post. If you buy my view about imperative titles, then you could consider "Exploit idempotents in a ring." But it depends whether the point of the article is more "You might not have thought of it, but idempotents are actually very useful," or "You have seen or heard that idempotents can be useful, but here is how to produce such arguments for yourself." If the former, then I would be very much in favour of the imperative title. Or one could go for something more specific, such as "Decompose your ring using idempotents." The ideal, in my view is a title that serves as a super-quick summary of the message of the entire article. It's not always possible to achieve that, but often it is.

The above example has a more sophisticated method than the one I tried to describe in the quick description section. In the generalization of the method of example 1, there would be a function and the goal would be to improve our understanding of its smoothness. To that end, we use itself to define another function

such that , and is as smooth as . Now the implicit function theorem says is as smooth as and hence . The argument is continued as many steps as possible. Does anyone know of another application of this argument, or an argument similar to this?

I propose changing the title of this article to ``How to exploit idempotents in a ring''.

I have used this idea many times in my research, but never knew it was called the Peirce decomposition until reading this article. Indeed, it is a standard example in the theory of schemes, and in the representation theory of p-adic groups, which motivate two of the above examples, but I have never seen it referred to by this name in any books on those subjects. (I say this just as a defense in advance of my ignorance of this name; I don't think I will be unique in having an interest in this article, while not knowing the name ``Peirce decomposition".)

If there are no objections within a day or so, I will make this change. I will then add a line to the quick description along the lines of ``this decomposition is sometimes refereed to as the Peirce decomposition of with respect to ''.

Also, the ``More generally'' remark about a complete system of orthogonal idempotents should be elaborated on. (Because if does not have an identity, then is not actually defined as an element of , and so the Peirce decomposition is itself more general than decomposing with respect to orthogonal idempotents. So while the decomposition by a complete system of orthogonal idempotents is very closely related to the Peirce decomposition, it does not always contains the latter as a special case. Indeed, the notion of a complete system of orthogonal idempotents probalby only makes sense if has an identity, at least as far as I can tell.)

Finally, I noticed that the earlier version of this article was to some extent lifted from PlanetMath. Maybe we should be careful doing this in general, just for copyright reasons.

I always thought that a total order could be strict, and that the condition was that , or . Is that a very nonstandard convention?

The relation on the ordinals is more convenient to use than , since one wants many statements to be true for every , for some given . I suppose one could add the word "strict" in brackets before the first occurrence of "total order", but that feels a bit strange, given that you can convert orders of the type into orders of the type and back again. Indeed, because of this I describe them both as total orders and think of them as different ways of describing the same basic underlying object.

I am thinking that the use of polar coordinates could be another entry in methods for simplifying integrals (or in general methods for estimating integrals). Of course the example in this article is maybe an elementary one but one could think of more involved examples in singular integrals (e.g. method of rotations) where essentially this principle is the basic trick. I understand that 'polar coordinates' is probably quite restrictive. We should probably have an entry along the lines 'try to change your coordinate system'. I think it is a standard trick in estimating oscillatory integrals as well. Stein does this in order to prove the multi-dimensional version of the van der Corput lemma with a bump function, and I think also Michael Christ has done that in a couple of papers in order to prove sub-level set estimates (but I can't say i have all the details in my head right now). That is, adopts the coordinate system to the direction along which the phase function has a derivative that stays bounded away from zero and uses a one dimensional estimate along this direction. I guess there are plenty of other cases that I don't know of. On the other hand maybe this should be part of an 'exploit symmetries and invariance' article if one thinks for example the way Stein proves the dimension free bounds for the Euclidean ball-maximal function. I am getting a bit confused concerning which is the natural or 'right' place for each article.

yannis

Amusingly, I planned an article ages ago that was very similar in spirit to this one. I've just looked for a dead link but can't find one. Anyhow, it's great that you've started this one, which I hope may eventually fit in nicely as a special case of a more general principle, which is what my intended article was going to be about. The general principle is that whenever you need to choose something and it feels difficult to do so because you need to guess too much about how the rest of the proof will go, you have the option of not choosing it and instead taking some abstract object instead, running the rest of the proof, and seeing what properties you need your unspecified object to have. At that point, you have a new problem: is there an object with those properties? Often this problem is much clearer and easier than the problem you had before when you didn't know how the proof would go.

Numerical examples in analysis are an obvious source of examples. I use the general idea a lot, but when it came to writing the article I suddenly found I couldn't think of any. Perhaps I'll write a stub with a general discussion – roughly equal to the above paragraph – but no examples! And then it could be a parent for this article.

In this article, "decoupling" sounds as an interesting general strategy. We could add a parent article about this more general metatechnique (I don't have the knowledge to elaborate it). Why is it important? In which cases ought we try to get a decoupling? Etc

Another "decoupling" technique that comes to mind now is that of "separating variables" in ODEs and PDEs.

I think there should exist some examples involving trigonometric (or inverse trigonometric) functions, where we forget about the actual evaluations of the functions but we use their main properties, like periodicity, a sine is a cosine with a phase change, bounds between and , , , etc.

More generally, I think we almost always do this abstraction: for us, mathematical objects are mostly what their main properties say they are; for example, we just associate every function with their properties, not with their sets of values (e.g., exponentials are the functions that verify the rules of exponentiation, are always positive, eigenfunctions for the derivative operator, etc). And when we define a family of "well-known" functions (for example, "elemental" functions and operations to represent an undefined integral), we are just restricting ourselves to functions whose properties we know well. I think this main idea could be the subject of another, even more abstract article about "Forget the actual object and stand by its properties" (not just for functions).

(Actually, I also want to write an article some day titled "Just check the definition", because sometimes it shows more fruitful to just stick to the definition of the object in question rather than trying characterizations and well-known properties of it. If you read this and feel like writing that article, feel free to do so!).

Would the following example be too trivial?

One sees that any open set is the union of disjoint intervals using a greedy algorithm. For an element in let , be the collection of all open intervals lying in and containing . Let and The intervals in are open, connected and they are not disjoint of each other because they all contain . These imply that is also a member of and by its definition it is the maximal element of .

We now enumerate the rationals in and iterate over them as follows. Suppose , ,..., are the intervals chosen upon iterating over the first rationals. For the rational we check if intersects with , ,...,. If there is a nonempty intersection with a then by the maximality of and in this case we complete the step and proceed to the next rational. If is disjoint from , ,..., then we let .

Let be the largest number the index reaches in the above iteration (it may be ). Because the rationals are dense in it follows that .

I think the identification of random variables and measurable functions is standard. For example, Varadhan's Probability Theory book, page 7:

``An important notion is that of a random variable or a measurable function.''

then the next definition:

``Definition 1.5. A random variable or a measurable function is a map ...''

Another example is the book Probabilities and Potential by [[w:Paul-Andr