User talk:Leigh.Samphier

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Cheers! prime mover (talk) 16:16, 10 October 2016 (EDT)

"Sources" section
Please, if you think the source links are going to be compromised by refactoring you have done, please do *not* just delete the links. Please add an invocation of SourceReview at the top of the Sources section so that anyone who has a copy of that work can go through and fix the links.

Many thanks. --prime mover (talk) 11:35, 1 July 2019 (EDT):
 * I’ll be sure to do this in the future. Although, in this case, the links I removed were links that I inadvertently introduced through a copy/paste operation. So, I thought I was correcting a mistake. The original flow through the source is undisturbed. Of course that doesn’t negate the the need for the SourceReview. —Leigh.Samphier (talk) 17:28, 1 July 2019 (EDT)


 * Without actually checking (I have the book somewhere, not immediately sure where), the actual situation is probably that a subset of the definitions given are in S&S and the statement of their equivalence is mentioned, but not proved. So a true reflection of the contents of the book will be fiddly to establish, and will be a job for a long weekend (of which I do not have many for a while). --prime mover (talk) 17:39, 1 July 2019 (EDT)

Please, I ask again: please add an invocation of SourceReview at the bottom of pages which you refactor. I just happen to be progressing through Steen and Seebach again to correct citation links, and I noticed that work needs doing to correct the prev-next links in Definition:Boundary (Topology). I'm going to need to fix them, which itself is no big deal, but if I had not noticed this, they would not have been done, and integrity would have been compromised. --prime mover (talk) 07:50, 17 May 2020 (EDT)


 * Glad you caught it. My apologies. --Leigh.Samphier (talk) 08:04, 17 May 2020 (EDT)

Adding mathematicians
A quick heads-up: as and when you add new mathematicians to, you may also want to add them to the appropriate pages in the Mathematician:Mathematicians page: one link into the overall chronological list of mathematicians (according to which century (approximately) they were born) and one into the "nation" list (according to exactly where they were born and/or naturalised, again in chronological order of birth date). The latter can get complicated if they were born in a place which has had a lot of political upheaval -- Germany in particular is fiddly because of all the different nation-states pre 1870-ish.

Cheers --prime mover (talk) 08:55, 10 September 2019 (EDT)

overdue proofread exercise
I have been through and performed a proofreading exercise on a large number of your pages, as you have noticed. There are probably some I've missed, but I think it's the main bulk of what you put up in the last 2 years or so.

I will do this again the next time I have a long weekend which I don't know what to do with.

Let me know if there is anything you find I've missed, and/or have questions on what I have done. --prime mover (talk) 09:44, 30 August 2020 (UTC)

Question about Closure in Subspace and Closure of Subset in Subspace
To my eye, these results are exactly the same.

Can you cast your eye over them quickly to see if that is correct?

I intend that they be merged. The proof in Closure of Subset in Subspace is far more neat than (yet appears to say exactly the same thing as) the proof in Closure in Subspace, so I would recommend we keep the former and throw away the latter.

I may have missed something really subtle here, which is why the second opinion.

Note that the corollary has already been moved. --prime mover (talk) 07:39, 12 January 2021 (UTC)


 * The only difference in the theorem statement is that Closure of Subset in Subspace includes $H = \O$ which is not included in Closure in Subspace. The proofs are along the same lines. The proof in Closure of Subset in Subspace is more succinct, and the proof in Closure in Subspace is more explanatory for the novice. But I agree with your recommendation. --Leigh.Samphier (talk) 09:29, 12 January 2021 (UTC)


 * A quick glance confirms that $H = \O$ does not affect the result. If $H = \O$ then $A = \O$ and $\map \cl A = \O$ and so on, and everybody goes home happy.


 * At one time we were being careful to exclude the topological space where $S = \O$ but it was pointed out that this is an unnecessary complication, and that null spaces should indeed properly be considered as valid topological spaces.


 * If you have the headspace, feel free to take a glance at Closure of Subset in Subspace/Corollary 2 and see if you can complete the proof -- I seem to be unable to make headway with it. My ability to figure things out is deteriorating. --prime mover (talk) 10:45, 12 January 2021 (UTC)


 * Proof for Closure of Subset in Subspace/Corollary 2 has been added. --Leigh.Samphier (talk) 05:38, 13 January 2021 (UTC)

Definition:Restricted P-adic Metric
Not too sure about the wisdom of completely removing Definition:Restricted P-adic Metric.

If on the page Integers with Metric Induced by P-adic Valuation we have a line which says "$d$ is known as the restricted $p$-adic metric on $\Z$" then we have a concept, a restricted $p$-adic metric, which merits having a definition page for it.

I wonder why that page was deleted, because I can't see why that would have been done. --prime mover (talk) 20:25, 10 June 2021 (UTC)


 * OOPS!! My Bad. The theorem Integers with Metric Induced by P-adic Valuation should simply state that what is defined is a metric. The source for this is an exercise in and Sutherland does not name the metric as such. While it is induced by the $p$-adic valuation restricted to $\Z$ and induces the same topology on $\Z$ as the $p$-adic metric, it is not the $p$-adic metric restricted to $\Z$.


 * The rationale for deleting the page is the it encompassed two definitions which were not equivalent. Both definitions came from exercises from introductory books on topology. One definition was the one defined in Integers with Metric Induced by P-adic Valuation and the other was simply the restriction of the $p$-adic metric to $\Z$. The first definition is not related to the $p$-adic metric and was not named in the source. The second is unnecessary since it was nothing more than the compound of restriction of the $p$-adic metric and so does not need its own definition.


 * If there is a need for a definition for the metric defined in Integers with Metric Induced by P-adic Valuation, I'm happy to create it, it just shouldn't be called restricted $p$-adic metric. I don't know what to call it as I have not seen it discussed anywhere else. The metric in Integers with Metric Induced by P-adic Valuation is the reciprocal of log base $p$ of the $p$-adic metric plus one. Its not unrelated to the $p$-adic metric but it is more than just a restriction. --Leigh.Samphier (talk) 08:41, 11 June 2021 (UTC)


 * Okay no worries. You've thought it through and there are reasons which I get.


 * We do well enough to leave well alone, then, I have very little direct experience in this area anyway. I just encountered it as that exercise in Sutherland, which I have to hand only because it was my set book in my M431 topology course. --prime mover (talk) 10:01, 11 June 2021 (UTC)


 * I have moved the previous discussion Definition talk:Restricted P-adic Metric to Talk:Integers with Metric Induced by P-adic Valuation. I also changed the initial part of the discusion to provide some context to the discussion. --Leigh.Samphier (talk) 01:09, 12 June 2021 (UTC)


 * ... and re-reading that, I notice that it was only 2 months ago that we had that conversation. I remember nothing about it. This "getting old" thing is not funny. --prime mover (talk) 07:16, 12 June 2021 (UTC)

Question: Definition:P-adic Expansion
I am puzzled: why is $\ds \sum_{n \mathop = m}^\infty d_n p^n$ not a power series, but only a series? --prime mover (talk) 12:34, 10 October 2021 (UTC)


 * A power series contains a variable over $\R$ or $\C$ in it and the power series represents a function in $\R$ or $\C$. The $p$ in the Definition:P-adic Expansion is not a variable over $\R$ or $\C$ (or $\Q_p$) and the Definition:P-adic Expansion doesn't represent a function in $\R$ or $\C$ (or $\Q_p$). The Definition:P-adic Expansion is similar to the Definition:Decimal Expansion and Definition:Basis Expansion which are also not power series. --Leigh.Samphier (talk) 13:05, 10 October 2021 (UTC)


 * This is indeed how it is defined on . However, it seems that a power series in $\Q$ has just the right to exist as it does for $\R$ and $\C$. We haven't done so because the concept of a power series in $\Q$ has not occurred in the various sources that have been used so far to develop this notion? Does a rational power series make sense, or are there reasons (like, while its coefficients are rational, its summation may not be) that we specifically do not want to define a rational power series?


 * The point I was trying to make is that the definition of a power series involves a variable over a field. The Definition:P-adic Expansion does not. A power series can be thought of as a polynomial with an infinite number of terms, such as $1 + x + x^2 + x^3 +⋯$. The Definition:P-adic Expansion is not such a beast. The $p$ in the series is a given and not a variable.


 * The power series $1 + x + x^2 + x^3 +⋯$ converges for all $\size x < 1$. But for a given $\size c < 1$, the series $1 + c + c^2 + c^3 +⋯$ converges. This is the difference.


 * I'm no expert on the general power series, but I imagine that it is possible to define a formal power series over any field, or ring. So a power series for the rationals is possible. I also imagine the interesting part of investigating formal power series over a field is when the power series converges on some open set. In which case it is a continuous function on the open set. I wouldn't be surprised if power series on $\Q$ is in fact uninteresting because $\Q$ is incomplete for every norm that you can define on $\Q$. I expect completeness is sufficient if not necessary to make power series interesting.


 * But even if power series on $\Q$ were interesting, the Definition:P-adic Expansion is not a power series as it fails to have the salient feature of a variable. --Leigh.Samphier (talk) 02:32, 11 October 2021 (UTC)


 * There is also a definition of a $p$-adic power series which is analogous to the definitions for real and complex power series and involves a variable over $\Q_p$. --Leigh.Samphier (talk) 13:09, 10 October 2021 (UTC)


 * ... but this is not one of them, right? --prime mover (talk) 13:34, 10 October 2021 (UTC)


 * No, $p$-adic expansion is not a power series over $\Q_p$. Both books that I have on $p$-adic numbers refer to the $p$-adic expansion as a series. They both define a power series over the $p$-adic numbers. Neither book gives the $p$-adic expansion as an example of a power series. --Leigh.Samphier (talk) 02:32, 11 October 2021 (UTC)


 * Thank you, I understand now. --prime mover (talk) 05:27, 11 October 2021 (UTC)