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September 20

composition of distributions and submersions

I don't really understand the distributions article, but I presume the composition of a distribution with a submersion, T(S), would be a distribution rather than a function? Is there anything roughly(or even precisely?) like an inverse to the distribution T, call it T* rather than T^-1, such that T(S(T*)) would be a function?144.35.45.72 (talk) 00:41, 20 September 2017 (UTC)[reply]

Do we have an article on Cancelling out terms (in basic algebra and in otherwise-divergent results)?

By this I mean the simple mathematical/algebraic concept that opposite terms in an equation can be cancelled out, or that problematic terms in group, gauge or other theories can be cancelled out to avoid divergencies.

If a suitable article exists can someone

fix the redlink in the Supersymmetry intro and add a link at Cancel (disambiguation)
set up a couple of #REDIRECTs for it from Cancel out/Cancelling out (or other useful titles)?

(and if not, can a basic stub or something be created too?)

I don't want to create an article myself, yet, in case one already exists.

Thanks FT2 ^{(Talk | email)} 10:39, 20 September 2017 (UTC)[reply]

I guess there should be an article on it, especially as seeing the original book on algebra had a title which translates as 'The Compendious Book on Calculation by Completion and Balancing'. Dmcq (talk) 11:10, 20 September 2017 (UTC)[reply]

There should be, but I don't know if there is (I couldn't find it). If not, can you or anyone who has the requisite knowledge write it, even if just as a stub, and link it as mentioned, because I suspect it's one of those subtle/deceptively non-trivial topics which requires more capability than it seems on the surface might be needed, to cover even crudely. Thank you. FT2 ^{(Talk | email)} 11:52, 20 September 2017 (UTC)[reply]

Created a basic stub at Cancelling out but definitely needs some work done to make it any use. FT2 ^{(Talk | email)} 12:07, 20 September 2017 (UTC)[reply]

@FT2: Nice work! --CiaPan (talk) 12:53, 20 September 2017 (UTC)[reply]

Huh, strange that nothing like that existed, even as sections at other articles, like Algebra or Equation. But I guess this should probably exist on its own, also. --Deacon Vorbis (talk) 12:54, 20 September 2017 (UTC)[reply]

@FT2, Deacon Vorbis, and Dmcq: There exists a Cancellation property article, possibly the two should be linked to each other (but rather not merged). --CiaPan (talk) 13:02, 20 September 2017 (UTC)[reply]

(edit conflict) Well, hold on. We do have Cancellation property. I'm guessing this is what you were looking for, although it's written a bit more abstractly than one for basic algebra should be. (You just beat me to the punch here). --Deacon Vorbis (talk) 13:04, 20 September 2017 (UTC)[reply]

I was on the fence about merging or not merging, but I think I fell on the not merging side, so I went ahead and added an entry to the disambig page at Cancel. --Deacon Vorbis (talk) 13:24, 20 September 2017 (UTC)[reply]

I don't think my capacity for mind wrenching and cruelty goes as far as to point a reader who wants to know about cancelling 2y on both sides of 3 + 2y = 5y, to our article on generalizing invertibility ;-)

Now, we can only cancel factors; we can't cancel terms. That is, you can cancel:

a*c    a
--- = ---
b*c    b

but you can't cancel:

a+c
---
b+c

Georgia guy (talk) 16:11, 20 September 2017 (UTC)[reply]

Can you update the article? Note that worldwide the term "cancelling out" is used for all cancellations, not just for factors in division (if that's a point of confusion anywhere). FT2 ^{(Talk | email)} 16:30, 20 September 2017 (UTC)[reply]

@Georgia guy: You're right. However, some people^[who?] think we sometimes^[when?] can cancel even single digits:

\qquad {\frac {1{\cancel {6}}}{{\cancel {6}}4}}={\frac {1}{4}}

.

CiaPan (talk) 11:02, 21 September 2017 (UTC)[reply]

See this recent paper for more on this. --JBL (talk) 13:03, 21 September 2017 (UTC)[reply]

Requirement that a Measure is over a Sigma Algebra

I'm having trouble understanding why a measure is defined over a sigma algebra, as opposed to just over a set. The Sigma-algebra article mentions subsets of the real line that have no measurable size, but I'm wondering if these are commonly encountered, or just degenerate cases that show what is theoretically possible? OldTimeNESter (talk) 22:13, 20 September 2017 (UTC)[reply]

You are talking with a common sense attitude like, well it'll work in most any cases you come across - just be sensible and anyway if things go wrong you'll find out soon and it won't matter much. That is not how mathematics is done noowadays. It isn't enough to check that something works for the numbers from 1 to 10 and then assume it is always true. What is being talked about needs to be precisely defined and the result has to be true for all numbers, even ones that are too big for a computer to test. As to whether non-measurable are common, well in a theoretical sense they should be overwhelming in number compared to the measurable ones - but they require the use of the axiom of choice to construct, so no if you're not doing mathematics you won't come across them. Dmcq (talk) 22:52, 20 September 2017 (UTC)[reply]

I know about the importance of precise definitions. I was referring to (and I should have been more clear about this) the practice of a lot of math textbooks to state that a theorem applies to "well-behaved" functions, without explicitly defining what "well-behaved" means. In my experience, the authors do this to avoid confusing students when there are pathological cases where the theorem doesn't hold that are not of practical importance. I wanted to know if the cases where a subset isn't measurable are similarly pathological (which is probably what I should have just said: it was late, and I'd had too much caffeine). OldTimeNESter (talk) 10:11, 21 September 2017 (UTC)[reply]

The article Non-measurable set my be of use here. In fact, most subsets of the real line which you will encounter are Lebesgue measurable, but assuming all subsets are measurable leads to difficulties and it's probably better to have an abstraction such as Sigma Algebra for the theory. This is analogous to studying vector spaces (another abstraction) in general rather than assuming all vectors are in Rⁿ. --RDBury (talk) 11:45, 21 September 2017 (UTC)[reply]

Well-behaved is pretty meaningless without some context. There is an article Pathological (mathematics) which gives some meanings but if it is used it should refer to a previous definition for the context. However in physics they might mean twice differentiable for instance without defining it, and in a social science they might mean it is monotonic with no nasty dip or hump in the middle - neither of which implies the other. Dmcq (talk) 12:10, 21 September 2017 (UTC)[reply]

The context is actually probability theory: even what I assume would be graduate-level texts use the descriptor "well-behaved" rather loosely, in my experience. That said, I found some good Youtube videos that explain the Banach-Tarski_paradox, and that helped things to "click" for me. Thanks for taking the time to respond. OldTimeNESter (talk) 22:14, 22 September 2017 (UTC)[reply]

This may help: if it's a set you can easily picture, it's measurable. (I mean sure, set theorists, topologists, and lots of people with PhDs can easily picture non-measurable sets, for some value of "easily", but this is generally a skill developed through lots of formal training and practice.) Even fairly exotic things are measurable, like the Cantor set (measure zero), or the Fat Cantor sets (positive measure). Those are particularly good to think about, because they are both nowhere dense, and have positive measure. If you are just starting out with measurability and sigma algebras, it's good to spend plenty of time making sure you firmly grasp measure zero. This will help you with things like almost never and almost surely, without having to get in to as much detail as necessary for formally defining positive measure. SemanticMantis (talk) 15:22, 23 September 2017 (UTC)[reply]

We can get more precise than that. Actually in several different ways, which is what makes the discussion complicated.

Roughly speaking, any set you can define, unless you're using very high-powered techniques, is going to be Lebesgue measurable. Even if you are using very high-powered techniques, it's still at least consistent with ZFC that the set is Lebesgue measurable (that is, you won't be able to prove it's not).

Getting a little more specific, all Borel sets of reals are Lebesgue measurable. So all open sets, all closed sets, all countable unions of closed sets, all countable intersections of those, and so on, for quite a strong notion of and so on. And if you accept large cardinals, then we can push on past that, through the so-called projective hierarchy.

In practice, this is going to cover any particular set that you're going to run across in any ordinary application of probability theory. The only way you're going to encounter non-measurable sets is by applying the axiom of choice to show that such a set exists, but not telling you specifically which one it is. --Trovatore (talk) 20:00, 23 September 2017 (UTC)[reply]

Zzzzz...

The Emperor's New Clothes Count Iblis (talk) 21:36, 23 September 2017 (UTC)[reply]

That's a completely irrelevant and useless comment, Iblis. --Trovatore (talk) 03:49, 24 September 2017 (UTC)[reply]

Cut all invisible parts of analysis away and you'll arrive here.

Count Iblis (talk) 07:33, 24 September 2017 (UTC)[reply]

Having lost thereby the best parts of mathematics. Look, you're totally free to take the ultrafinitist viewpoint if it pleases you for some reason. But then please just ignore questions about the parts of math it cuts out. Your comments contribute nothing to the discussion. --Trovatore (talk) 07:43, 24 September 2017 (UTC)[reply]

Nothing is lost, because you can only ever manipulate a finite number of symbols using a finite number of rules, no matter how large the sets are that some symbols are supposed to refer to. So, all of math is ultimately just discrete math. This is relevant to this discussion as it demystifies the point you made above about not being able to capture the unmeasurable sets. Count Iblis (talk) 22:19, 25 September 2017 (UTC)[reply]

You're confusing the objects with the language that discusses the objects. That's a basic error.

Your comments are utterly useless in this discussion. If you don't believe in the objects, then that's fine; it means you think the whole discussion is meaningless. You can think that. But it adds nothing to a discussion that is asking about them. --Trovatore (talk) 22:27, 25 September 2017 (UTC)[reply]

Formalism (philosophy of mathematics). Count Iblis (talk) 01:23, 26 September 2017 (UTC)[reply]

Even formalists don't generally commit your error. In theory at least, they treat mathematical statements as mere formal strings without interpretation. But they do not confuse the objects of an interpretation with their names; they simply dispense (again, in theory) with an interpretation at all. In order to make practical progress, they speak, and presumably think, as Platonists, but holding a mental reservation as to the "real meaning" of their assertions. --Trovatore (talk) 02:32, 26 September 2017 (UTC)[reply]

A good example that the Law of the instrument applies even to the best mathematicians I think. Dmcq (talk) 22:58, 25 September 2017 (UTC)[reply]