Talk:Main Page

New $\LaTeX$ macros for your convenience and our internal consistency
There are a number of new $\LaTeX$ macros which have been developed recently as a result of a lot of discussion some time back which never ended up happening at the time.

They can be found on the page Symbols:LaTeX Commands/ProofWiki Specific, also accessible from the sidebar.

A thing or two
I thought I'd dump some ideas here since my activity may wane a bit over the next few months and I might as well say these things while they're fresh. (then I'll have exams late April-mid June, after that I'll resume to current levels with another semester's knowledge) First I wanted to say that I'm planning on maintaining the page User:Caliburn/Job List as well as I can. The intention is for new contributors to be able to come along and find what still needs working on and gaps that people may not have noticed. Also a handy list for myself so I know what still needs to be done. People are free to add to the list or take off stuff they've completed. I am not sure where to put it for maximum visibility, but I'll display it on my userpage and post it here.


 * There are already a number of contributors who have their own way of managing their to-do lists, but a global one may have a use. I say: go for it, but beware that it's just another thing that needs maintenance. There are currently few enough active contributors that this could be extremely useful, especially if they are in the same approximate areas of activity. I can see the points of overlap between measure theory, metric spaces, topology, normed spaces, prob theory, matroid theory etc. etc. where this would be worth applying, in particular. --prime mover (talk) 07:19, 4 January 2022 (UTC)


 * It seems that I am going to tackle some of your topics simply by following Sasane. Namely, I am almost done with distributions. One thing that has been nagging me is the definition of Dirac delta through limits of various functions. This was one of the main reasons why I decided to work on these topics. However, Sasane does not cover this at all. Hence, after I am done with a couple of exercises, I will briefly switch to a different source (not sure which one) to establish how definitions through limits work out in the distributional sense. Then I will work a bit on the Lebesgue integral and resume where I left off (this can be seen by inspecting hyperlinks in Sasane's book page). Most of the work will be less abstract involving normed vector spaces.--Julius (talk) 12:32, 4 January 2022 (UTC)


 * I was looking under Category:Distributions so I didn't realise how much you have got through, fair enough. (I think any category containing results about distributions should be a subcategory of this category) Distributions are a huge topic that have only just started getting worked on, so it is probably worth noting as an area to expand anyway. If you need ideas on sources for more work on distributions, the module at Warwick on distributions recommends Friedlander & Joshi's Introduction to the Theory of Distributions, among some Fourier analysis texts by Duoandikoetxea and also by Grafakos. I haven't looked at these myself yet, learning more about distributions is on my "to-do" list for the summer. Wikipedia's page on distributional limits gives an example of a sequence of functions converging in the distributional sense to $\pi \delta_0$ (not sure if it's the canonical one, if there is one) so that might also be a good jumping off point. Looking forward to seeing how it all turns out. Caliburn (talk) 15:50, 4 January 2022 (UTC)

Second I'm wondering how theorems could be catalogued so that they don't get lost. While having a descriptive name for each theorem makes the most sense, it can make navigation difficult since some pages have to be given unintuitive names, some more so than others. (sometimes leading to mistakenly duplicate pages) I like pages like Properties of Beta Function for this purpose. I'm wondering whether we could branch this out to have easily navigable collections of other results. No specific format to mind yet. My contribution backlog at the moment is massive, so I doubt I will get to this soon, (especially as it's a significant undertaking) I will return to this idea when it's been whittled down a bit. Caliburn (talk) 00:19, 4 January 2022 (UTC)


 * I too am a fan of the "Properties of ..." sort of page. The archetype is Subset Equivalences, which I have found very useful indeed.


 * While duplicate pages are a problem, it's not a serious one. It happens for minor stepping-stone results when multiple source are being used to flesh out an area -- but when we implement categories more incisively, these duplicates are usually caught. The problem mainly happens when a contributor puts every result in a high-level all-inclusive category called, e.g. "Real Analysis" or "Metric Spaces", then it's easier to miss stuff.


 * Best advice while an area is being fleshed out is: if you see a potential duplication, slap a template in there to flag it up and move on with what you're doing, then after all is in place, take a step back to see whether there is a common thread that would need to be refactored out. --prime mover (talk) 07:19, 4 January 2022 (UTC)


 * I think once measure theory is fleshed out I'll have pages like "Limit Theorems for (Riemann/etc.) Integrals" and "Properties of Measures", etc. As an aside - should we archive a few older discussions here? My browser is starting to lag a bit typing this response. Caliburn (talk) 15:50, 4 January 2022 (UTC)


 * Aside -- job done. Good call. --prime mover (talk) 17:16, 4 January 2022 (UTC)

Slow
I don't want to be irritating, but the site has been extremely slow on and off for the past few days. Could this be looked in to? It is making refactoring very sluggish. Caliburn (talk) 18:44, 18 January 2022 (UTC)


 * Haven't noticed it myself. I do hope I'm not the problem. --prime mover (talk) 20:23, 18 January 2022 (UTC)


 * Doesn't seem like it, happens on and off even if I'm the only one editing. Doesn't seem as bad right now. Caliburn (talk) 19:13, 19 January 2022 (UTC)

Weak convergence
Can we get macros for $\rightharpoonup$ (weak convergence) and $\stackrel \ast \rightharpoonup$ (weak-* convergence)? Caliburn (talk) 13:11, 30 January 2022 (UTC)


 * Yes okay, I'll get to it in a bit. --prime mover (talk) 14:03, 30 January 2022 (UTC)


 * ... all good? --prime mover (talk) 18:19, 30 January 2022 (UTC)


 * Yes that's great cheers Caliburn (talk) 18:24, 30 January 2022 (UTC)

Proofs of unsolved conjectures
Without trying to be rude to certain contributors, I'm wondering if codifying disallowing claimed proofs of famous conjectures (Goldabch's, Collatz, RH, etc.), and otherwise obvious crankery, (eg. the Bibhorr nonsense) is a reasonable step. Obviously we shouldn't discard the possibility of original theorems and proofs, (that being one of our selling points over Wikipedia of course - posting interesting-but-not-particularly-useful toy theorems should be encouraged) but I would draw the line before proofs that would very possibly win the poster a Fields Medal (or at least shoot them into international fame) if correct, on the basis that it almost certainly isn't. If by sheer miracle their proof is correct, they can get it published in a peer-reviewed journal, (I know this all might be harder for "outsiders", but Yitang Zhang managed despite being virtually unknown) claim their $\$1$ million dollar prize/Fields Medal/international infamy, and post it here. Thoughts? Caliburn (talk) 22:18, 31 January 2022 (UTC)


 * A well-crafted and coherently-written page to standards (or at least in literate $\LaTeX$) will always be given greater credence than an ungrammatical and careless page written by someone who does not care enough about the target audience to bother to communicate properly.


 * It is a matter of fact that the more unusual and improbable a claim of success at proving an unattainable, the more bletheringly incoherent and incomprehensible the language it is written in. This is not a matter of fluency in what may not be one's native language, this is either the complete inability to string thoughts together, or the writer's belief that they are so cool and awesome that they can write in their own version of street slang and as a result everyone will bow down in worship of the awesome light, brighter than a thousand suns, that emerges from their nether orifice.


 * Hence I give short shrift to crackpots. Life's too short to bother. --prime mover (talk) 23:43, 31 January 2022 (UTC)


 * We are not a peer-reviewed scientific journal. Even if we know a thing or two about maths, we are far from being competent to say anything solid about your favourite unsolved world-class problem. If anyone has proof, please go through the usual peer-review process, get your prize and recognition, and then we can discuss posting your result here. To my understanding, here we are interested in well-established results with anything we can interpolate in between. As for Bibhorr, I could see a historical reason if this was derived thousands of years ago, but to my knowledge, this is a claim to a revolutionary new result, which is not. It is simply an inspiration derived from approximation of standard trig functions.--Julius (talk) 07:46, 1 February 2022 (UTC)


 * This specific exchange was triggered by a recent posting about Collatz. If the post in question is written using standard mathematical symbols in $\LaTeX$ then maybe we'd have been able to have a stab at poking the necessary holes in it like we did Senojesse some months back. But the symbols used were non-printing and non-displaying and hence of zero value. Seriously, I may be getting old, but I have neither time, headspace nor tolerance. As for the request on my user talk page to discuss it over the phone, there's an orifice for such to be inserted. :-/ --prime mover (talk) 09:07, 1 February 2022 (UTC)

Handling TORI as a source
With TORI being offline for approximately 10 years now (according to Wayback Machine https://web.archive.org/web/*/http://tori.ils.uec.ac.jp/TORI/index.php*) I think it is time that we decide what to do with it.

My suggestion would be to:
 * Remove the (broken) direct links
 * Include a reference to the Wayback Machine for TORI in general

For me it is up to debate whether or not we want to keep the reference to the licence. We might want to change the wording at least, since it will not be clear any more what part is actually reproduced.

Thoughts welcomed. &mdash; Lord_Farin (talk) 16:03, 16 March 2022 (UTC)


 * Considering how very few pages there are which reference TORI, I wonder whether to just delete the template and references and pretend they never existed. :-)


 * There appear to be a number of primary sources cited in the talk page, which we can plant on the main page after checking them out (these pages date from way before Citation.)


 * If the user who originally posted these pages up returns to us, then he may be invited to link to whatever resources that may contain his work -- although if it was original, such may not exist.


 * If we can explore Wayback and successfully identify the precise source pages from which the pages were generated, then we may be able to amend the template accordingly, but I've not got this anywhere near the top of my Things To Do list.


 * As for the license, I'd say just remove it; the license refers directly to TORI which seems no longer to exist. --prime mover (talk) 16:25, 16 March 2022 (UTC)


 * Good call, I have eliminated the references. As a signpost I have left Template:TORI with a general link to the Wayback Machine. It was not possible to retrieve the content in reliable fashion. &mdash; Lord_Farin (talk) 19:55, 16 March 2022 (UTC)

Differentiating levels of refactoring
As you might have noticed, I have been cherrypicking from Category:Refactoring In Progress recently (i.e. Refactor).

It quickly occurred to me that it contains an overly broad spectrum of tasks, from elementary to highly complex.

To that end, I have introduced Category:Basic Refactoring, Category:Intermediate Refactoring and Category:Advanced Refactoring. It should help to get a better grasp of the level of problem that needs to be addressed.

This brings several questions:


 * 1) Is this useful in general?
 * 2) Is the usage sufficiently explained on the relevant pages or does the wording need to be tweaked?
 * 3) Should we take it further and introduce similar mechanisms for Tidy and the rest? Personally I feel it's a bit early.

Thoughts welcomed. &mdash; Lord_Farin (talk) 09:43, 20 March 2022 (UTC)


 * I like it. In particular I like the categorisation.


 * I wouldn't go with doing the same for Tidy, as this ought to be straightforward to keep under control -- except that last time I did a concerted effort to tidy everything I lost my sanity. --prime mover (talk) 09:47, 20 March 2022 (UTC)

Seth Warner's Modern Algebra exercises
I've been making heavy weather of solving the exercises in sections 11 and 12 of Seth Warner's Modern Algebra, where he derives general results concerning quotient structures and quotient mappings usually introduced in the context of group theory and normal groups. His approach is to investigate structures with the most general properties and derive results which can then be used to further derive the corresponding statements in group theory.

I have documented all of the results in these chapters, but left some of them undone because of lack of ability to concentrate (and I confess also patience).

During the course of this I have been trying to categorise and structure the results in this area so as to make them easier to find, with more or less success.

I'm going to move on from this (to the next chapter, but still in the same area of work), and I would welcome a second pair of eyes to see whether there may be a way to better structure the results in this area so as to make the whole area easier to perceive as the richly patterned structure that it is, rather than as a patchwork of similar-but-not-quite-the-same results.

In particular, the correspondence between groups quotiented both by normal subgroups and congruence relations needs to be better emphasised.

Does anyone want to see whether we can make something really useful here? I would hope that incomplete proofs could be completed using the same approach they have been started with -- the exercises in Warner often say "Use this result to derive (some other result)", where I can see the pattern of what is to be achieved, but have not been able to focus adequately on the fine detail.

I have been asking questions on MathStackExchange a lot recently, and I fear the experts there are losing patience with me. --prime mover (talk) 17:29, 22 March 2022 (UTC)


 * I'll take a shot, but I'll have to ask for a few days of patience till I find a timeslot of sufficient size to really get into it. &mdash; Lord_Farin (talk) 07:36, 23 March 2022 (UTC)


 * Update: done. As you noticed I left remarks and maintenance templates where I felt they were appropriate. Nice job, crisp presentation! &mdash; Lord_Farin (talk) 12:16, 27 March 2022 (UTC)


 * Heh! Sorry, slight misunderstanding: my point was that a number of the exercises have been left incomplete. I was hoping someone was going to do my homework for me add the finishing touches to the incomplete proofs, e.g. $11.12$, $11.16 - 20$, and so on. No worries, I'll return to them later, when I've cleared my head from the current confusion.

Internal Group Direct Products
It will be noticed that I have done a bit of work chewing over the definitions and equivalence proofs of Definition:Internal Group Direct Product.

In particular, the definitions are identified by name rather than number, so as (I hope) to make it clearer which one is under discussion at any one time.

The question will be asked: why does it have to be quite as complicated as this?

The answer is: I have a few source works which introduce this concept from different directions:


 * leaves it implicitly understood as a special case of the general abstract algebraic structure: an isomorphism from the cartesian product of two substructures onto the parent structure. He then goes on to demonstrate that if the substructures are subgroups of a group are subgroups, then Definition:Internal Group Direct Product/Definition by Subset Product follows.


 * defines the concept in an introduction on notation as Definition:Internal Group Direct Product/Definition by Subset Product and leaves it at that.


 * takes the general approach of Definition:Internal Group Direct Product/General Definition/Definition by Unique Expression, deriving the $n$-tuplar Internal Direct Product Theorem thence.


 * I also have a vivid memory of my course pamphlets from my degree course which went over the subject finely in detail, which formed the basis of the initial work on on this subject, but I was new to the game and did not do a very good job of it.

My strategy here is to identify those 3 approaches to the definition and generate a $2$-subgroup and $n$-subgroup version of all of these. I appreciate that it may not be the case that all are explicitly documented in the literature, but in order to offer consistency and thoroughness to the approach, I am including them as definitions even if this may not be the case, which is counter to our stated guidelines.

I am now in the process of rationalising all the results which go towards proving the equivalence of all these approaches, with a view towards making them more accessible and internally consistent.

It's a long-overdue job, and at the moment I am getting more frequent periods of an hour or more at a time when I am mentally lucid enough to attack such problems, and I am not wasting them. (Getting old sucks.) --prime mover (talk) 23:34, 24 March 2022 (UTC)


 * Update after brain failure: I have completely lost the thread.


 * Can anyone explain why, from:


 * "Let $\struct {S, \circ}$ be an algebraic structure with $1$ operation.


 * "Let $\struct {A, \circ {\restriction_A} }, \struct {B, \circ {\restriction_B} }$ be closed algebraic substructures of $\struct {S, \circ}$, where $\circ {\restriction_A}$ and $\circ {\restriction_B}$ are the operations induced by the restrictions of $\circ$ to $A$ and $B$ respectively.


 * "Let the mapping $\phi: A \times B \to S$, where $A \times B$ denotes the Cartesian product of $A$ and $B$, defined as:


 * $\forall \tuple {a, b} \in A \times B: \map \phi {a, b} = a \circ b$


 * be an isomorphism."


 * ... it can be deduced that:
 * "$\forall \tuple {a, b}, \tuple {c, d} \in A \times B: \tuple {a, b} \circ \tuple {c, d} = \tuple {a \circ c, b \circ d}$"




 * It seems to be taken for granted in all the books I've managed to find on this subject (there aren't many) -- but I don't see why. --prime mover (talk) 23:23, 25 March 2022 (UTC)


 * Actually I think I may have worked it out.


 * The external direct product is defined as $\tuple {a, b} \circ \tuple {c, d} = \tuple {a \circ c, b \circ d}$ but the internal direct product is not the same thing at all, it is in fact defined as $\tuple {a, b} \circ \tuple {c, d} = \tuple {a circ b, c \circ d}$ and they are not the same.


 * I'll try and make some sense of this tomorrow. --prime mover (talk) 23:32, 25 March 2022 (UTC)

Codifying a category structure
I'm wondering if we should codify categories, which is a bit fragmented as things stand. My idea is that categories such as Category:Measure Theory, Category:Galois Theory, Category:Functional Analysis should on the first level contain the fewest results possible, limiting only to major theorems that are too wide-reaching to fit in a specific subcategory. The next level down, contained in the topic category, should then consist of the fundamental objects of that field of study, for example Category:Measure Theory should contain stuff like Category:Measures, Category:Sigma-Algebras, and so on. These categories should then contain the bulk of the results, (bar Corollaries, I think there is established convention that these should be categorised as theorems) but not subpages thereof. Pictorially we have for example:

Category:Proofs by Topic
 * Category:Measure Theory
 * Category:Complex Measures
 * Category:Variation of Complex Measure (have not created this yet)
 * Category:Variation of Complex Measure is Finite Measure
 * Variation of Complex Measure is Finite Measure

Do people agree on this? To clarify, my picture is that Characterization of Left Null Space should be contained only in Category:Null Spaces, and Category:Linear Algebra should primarily contain other categories rather than the 100 or so pages it does now.

Caliburn (talk) 17:57, 3 May 2022 (UTC)


 * This is exactly the structure that I have more recently been trying to evolve into. Note that the bottom line of what you put has recently been formalised by adding another use case for SubjectCategory where you have a result parameter which is the page title of the result being documented. --prime mover (talk) 18:26, 3 May 2022 (UTC)


 * Agreed, this makes sense. We might allow ourselves reasonable time to tackle this by an appropriate maintenance template that can be used on categories needing to be subcategorised. &mdash; Lord_Farin (talk) 18:44, 3 May 2022 (UTC)

Progress on generalized functions and Dirac Delta
Dear all, I think I am done with covering the basics of generalized functions, distributions, Dirac delta etc. Obviously there is much more to cover, but I have accomplished my two little goals - absorption of the last chapter of Sasane and some clarification of Dirac delta properties, sequences and connection to the sketchy presentation in standard applied literature. The majority of the work is based on Sasane, Arfken and Kanwal. I tried to stick to $1D$ to understand basic arguments before we jump into arbitrary dimensions since this will require invention of a few arguments. Also, in my work I chose to denote the distribution as $\map {T_f} \phi$, while many books choose $\innerprod f \phi$. I wonder if this is legible and how this is presented in other people's studies. I may still add additional material, but I won't steamroll it anymore. So, if someone feels that something is really worth adding, please be specific with your wishes. Otherwise I am coming back to continuous and linear mappings on normed vector spaces.---Julius (talk) 21:50, 4 May 2022 (UTC)


 * What you've done seems pretty solid. The opportunity is now there for anyone to build on it who wants to and is able to. I think if anyone has questions from here on in, it will take a while for them to evolve. Lots of study will be needed, and that will take time, at least from my perspective. Good job . --prime mover (talk) 05:07, 5 May 2022 (UTC)

Progressive Mapping -- the cat jumps
I have taken the radical decision to amend the terminology concerning "inflationary mappings" and "progressing mappings".

While I am retaining the name "inflationary mapping" when in the context of the general ordered set (although this may change), I am moving the class-theoretical definition of "Inflationary Mapping/Subset" completely over to the pave Definition:Progressing Mapping.

Rationale:

The concept of "progressing mapping" is used extensively in class theory in the development of the principle of induction to a much more general footing than on the natural numbers. The work which I am using to do this job is Smullyan and Fitting, which is the only one I have to hand which defines the concept. I have found no mention of "inflationary mapping" in any of the works I've seen (which is not saying much, I haven't studied all that many), and no other contributor has actually provided any other sources on the subject.

I note that the concept of an "inflationary mapping" appears to be used in Measure Theory, but again, no sources are cited and I have come across none myself.

Because the concepts are basically the same, I am going to continue to keep them nested one within the other as a structure, but weasel-word my way around it by saying that the two things have different names in the different mathematical contexts: "In the development of class theoretical approaches to development of the natural numbers, an inflationary mapping is better known as a progressing mapping" or some such.

Forefront in this development is because the concept of a "slowly progressing mapping" appears not to be mirrored in the context in which "inflationary mappings" are raised, and of course, in the context of the latter, on a close-packed domain, such a concept cannot be realised.

Please feel free to offer your thoughts. --prime mover (talk) 07:44, 5 May 2022 (UTC)


 * The reason inflationary (also: "extensive") mappings do not get a lot of love separately is because they are mostly interesting in combination with other properties. It is therefore not a concept which is carefully named and explored. Personally I find "inflationary" and "progressing" clearer names than "extensive" which bears too much reference to the set-theoretic case (and even there is not very clear). I have no real preference for either name except "inflationary" being already established and therefore requiring less work. But sourced names trump easy names, so I support your endeavour for "progressive" in covering S&F. &mdash; Lord_Farin (talk) 08:00, 5 May 2022 (UTC)


 * Thanks L_F -- this work will progress (no pun intended) progressively. --prime mover (talk) 08:07, 5 May 2022 (UTC)