Talk:Main Page

Hosting Fees
Hosting fees are due next month. Looks like it will be about 120 USD or so. I also plan on switching hosts so that I can get shell access. Since the student budget isn't that great, any help with hosting fees would be greatly appreciated! --Joe (talk) 17:50, 21 February 2010 (UTC)

Anonymous edits
Sorry but I'm seriously not a fan on anonymous edits. One has been made which I think is wrong (see its talk page) but as the editor is anonymous it's not so easy to enter into a dialogue. What do the others think? --Prime.mover 21:54, 22 February 2010 (UTC)

I would vote to let the experiment continue at least a little longer, but if it's causing problems, it should be switched back no matter how much I like the idea of openness. On a side note, did the same IP addresses edit to Symmetric Group Center Trivial look ok? --Cynic (talk) 00:09, 23 February 2010 (UTC)

I don't like it, it's a bit terse. If it were me, I would not have replaced a pageful of explanatory pedagogy with a dismissive 2-liner, as space is not of the essence here. I would not welcome this technique as a general trend, it goes against what I would consider this site was "for". But it's not my site, so I can't lay down the law. --Prime.mover 06:25, 23 February 2010 (UTC)

Just had a thought: we could add something to the help page stating something like: "anonymous edits are more likely to be subject to immediate rollback without comment than those made by users who have signed up with a full account." Openness is all well and good, but I like the idea of people being honest enough to put their "name" to stuff they contribute. --Prime.mover 06:28, 23 February 2010 (UTC)

I'm fine with something like that, and I agree that about the 2-liner part. Technically that proof should have been kept, and another section added with a new proof; so we should add a bit telling people to add multiple proofs. --Joe (talk)

The more I look at that 2-line "proof" of Symmetric Group Center Trivial the more I think it's rubbish, so I've reverted, although stuck the edit in, but in comments. --Prime.mover 20:17, 23 February 2010 (UTC)

Nope, from the comment in the talk page he obviously knows what he's talking about, just doesn't believe that mathematical truth is something which deserves to be communicated. So I've reverted his edit, although I rudely split it up into 4 lines and added an ill-mannered link to a definition. I still can't understand it, though. Can you? --Prime.mover 07:01, 26 February 2010 (UTC)

Geometric Distribution
The sharp-eyed among you will have noticed that the definition of the Geometric Distribution has changed from being defined as the distribution that models "the number of successes before the first failure" as opposed to the other way about, which goes against what (for example) Wikipedia has to say on the matter. I've taken the discussion up on that page on Wikipedia. --Prime.mover 22:50, 21 March 2010 (UTC)

Use of "imath" and "jmath" LaTeX tags
I learn today on a Wikipedia discussion that $\imath$ and $\jmath$ are not meant to stand alone as symbols, and in particular not for $\sqrt{-1}$. They were designed so they could be used with other diacritics, e.g. $\hat \imath$ etc. So I have another exercise on the way: to replace all existing $\imath$ and $\jmath$ where used as $\sqrt{-1}$ for the proper letters $i, j\,$. --Prime.mover 19:23, 12 April 2010 (UTC)

Misattribution?
The latest three (admirable) additions to the Wanted Proofs list are attributed to me. Much as I'd like to take the credit for adding these entries, it wasn't me. Must be something to do with how the s/w treats anonymous edits. No matter, just setting the record straight. --Prime.mover 21:50, 30 May 2010 (UTC)

Searches irritatingly case-dependent
I've just wasted several minutes searching for the page Definition:Top-Down by entering "top-down" into the "search" box. Even when I typed in "all:top-down" it still didn't return me any of the pages with this in it.

So I entered the page Definition:Top-Down into the search field and it instantly returned me, via a redirect, to the page Definition:Formal Grammar, which is what I was after all the time.

Is there a way of making the "Search" tool case-independent? --Prime.mover 06:46, 22 July 2010 (UTC)

I think there is, since I'm fairly sure Wikipedia search works that way. I believe they may use Google to power their search these days, but I suspect there is an extension that would do it. --Cynic (talk) 00:38, 4 August 2010 (UTC)

Venn Diagrams
Anyone care to demonstrate their skills with a graphics tool to create a 2-circle and a 3-circle Venn diagram? I have been playing around with OpenOffice Draw but am not getting anywhere. What I want is to be able to identity and shade the individual subdivisions of the circles. I haven't worked out how. --Prime.mover 21:00, 24 July 2010 (UTC)

How about these? File:Two Circle Venn Diagram.png and File:Three Circle Venn Diagram.png Feel free to crop and re-upload as desired. --Cynic (talk) 00:38, 4 August 2010 (UTC)

Hmm ... blank ones are no problem at all, I can do those in Geogebra. It's when you want to colour in the individual areas that I haven't been able to find a tool to do. But thanks. --Prime.mover 05:26, 4 August 2010 (UTC)

Solid color? If you're on a PC, just use mspaint. If you're on a mac, try paintbrush (I think it works, but I honestly don't remember its limitations). If you want stripes or some other pattern, GIMP should be able to handle it regardless of platform. --Cynic (talk) 02:50, 5 August 2010 (UTC)

Genius! --Prime.mover 05:37, 5 August 2010 (UTC)

Another problem fixed: script l
Back in December 2009, I raised the question:


 * We're somewhat limited in that LaTeX does not seem to support the "mathscr" font style. So the rendition of the Lebesgue space is somewhat compromised.


 * Anyone know of a workround?

I just happened to notice on the WikiLaTeX Help page there exists:
 * $\ell \,$

which is obtained by:  $\ell$ 

Just that one letter, but that's the one I wanted.

Job done.--Prime.mover 21:30, 7 August 2010 (UTC)


 * UPDATE: MathJax does have mathscr. --prime mover 14:22, 13 February 2011 (CST)

Codomain, range, image
I am going to replace all references to range with equivalent ones for codomain, as it turns out that there is just too much ambiguity with image. Most texts I have consulted equate the range with the image rather than the codomain, so something needs to be done and I'm doing it. This may take some time. Please bear with me. --Prime.mover 14:13, 8 August 2010 (UTC)

P vs NP
Has anyone seen this proof? --Joe (talk) 16:49, 9 August 2010 (UTC)

Two down, five to go ... something of a relief all round that this one has the result it does ... --Prime.mover 18:52, 9 August 2010 (UTC)

Just discovered something about rendition of Definitions
We have (to take an example):
 *  Definition:Open Set  which renders as Definition:Open Set, as you'd expect.

Then we have:
 *  open set  which renders as open set, as we need to do in order to not get the "Definition" part to render and so give the text a chance to flow.

But I just found out you can do:
 *  Definition:Open Set  which renders as Open Set which surprised me. Never thought you could do that.

We can also do it with categories:
 *  Category:Analysis  which renders as Analysis

... and Axioms:
 *  Axiom:Peano's Axioms  which renders as Peano's Axioms

This might save some typing.--prime mover 21:04, 21 August 2010 (UTC)

What they're saying about us
http://mathoverflow.net/questions/3987/proofwiki-anybody-seen-this

"Despite the problems that arrive with effectively naming and categorising proofs for wikis, this site seems to be the clearest and most well maintained resource/reference out there..."

http://www.wykop.pl/link/376524/proofwiki-org-encyklopedia-dowodow-matematycznych/

"Ogromna encyklopedia dowodow matematycznych, po angielsku."

http://www.websvalue.com/www.proofwiki.org

"Estimated worth for proofwiki.org $\$$9,505 USD." --prime mover 05:24, 30 September 2010 (UTC)

For any classificationists ...
What does anyone think about adding MSC2010 classification codes to pages written?

See Mathematics Subject Classification and MSC2010 for background. --prime mover 06:46, 9 October 2010 (UTC)

Mathematician pages
What's the general feeling about the Mathematicians pages? Thought it prudent to split them up into periods (it was getting unwieldy as it was) but the Sorted By Nation page may prove controversial. What I've done is tried to put people into their land of birth, but it may not always be appropriate, particularly as borders change and country names change. --prime mover 06:27, 13 October 2010 (UTC)

Applied Mathematics and Physics
I've bit the bullet and finally got round to making a small start on defining some basic concepts in applied mathematics and physics.

I've tried to write them in as simple language as I can, trying not to get too bogged down in metaphysics and philosophy, without cribbing from Wikipedia too much, but I fear I haven't done a very good job.

Does anyone care to try and improve on what I've put together? --prime mover 10:26, 7 November 2010 (UTC)

Physics
The question arose a while back as to whether we should be getting into the muddy waters of physics, or keep it as pure mathematics. Truth is, after the last few weeks of postings, I don't think we can keep physics out if we're going to do applied maths at all. So some time in the future I expect to start posting up some pages directly related to physics.

The problem here is knowing exactly what is "axiomatic" and what can be proved from previous stuff - so I plan on glossing over it by referring to such pages as "Physical Law", and if these can be demonstrated from previous results they will be, by having a "Proof" section added. If they can't, they will stay "proofless".

My knowledge of physics is limited to high school (where I graduated summa) and reading textbooks and popular works since, so if anyone is truly more expert than me they can feel free to tighten stuff up - as long as pages don't get overcomplicated.

I won't start immediately, there's a few other pages I want to get done first.--prime mover 06:44, 19 November 2010 (UTC)

I'll keep up with what you do in the physics section and maybe try to add a bit, but as I get up to the end of the term everything is getting crazy, so I don't know that I'll be able to do much before winter break. Since I'm in a relativity class now, I should hopefully be able to go through my notes and get a decent amount of stuff up once I'm through exams, but it might take a while to straighten out. Definitely go ahead and put up whatever you want for now though, and I'll add on what I can. --Alec (talk) 21:00, 19 November 2010 (UTC)

Meter or Metre?
Meter is the standard American spelling, but the International Bureau of Weights and Measures uses metre. I admit as a Brit that "metre" sits better with me, and I appreciate the political difficulties that the US has with France (the origin of the original spelling), but I wonder whether the international spelling "metre" ought to be used rather than "meter"?

Along with Burma and Liberia, the US is one of the last three countries in the world which has not adopted the metric system, so the meter is not even a native measure there - so I wonder whether it would be appropriate to adopt the majority view and spell it "metre"? Thoughts, anyone? --prime mover 06:34, 22 November 2010 (UTC)

I wonder what the site traffic would show. I don't know if we (read: Joe) can see whether more of the page views are coming from US or non-US IP addresses. If it's the US by a landslide, then it might make sense to use meter. If non-US leads or it's fairly close, I'd go with metre since it better fits that audience. And I doubt that Americans who are scientifically minded enough to be looking at online proofs of physics/chemistry/etc theorems are going to be terribly thrown off by international spellings. People will, of course, probably occasionally edit it to the American spelling, but we can change that back easily enough.

Oh, and it's not like the US is going to adopt the metric system any time soon *sigh*. We would never be able to make the switch to thinking in meters, kilos, and celsius. --Alec (talk) 19:18, 22 November 2010 (UTC)

A rival site
ProvenMath - to which I say: the more the merrier. --prime mover 21:59, 30 November 2010 (UTC)

Slow response
Has anyone else noticed that the site has a slow and unreliable response? It's taking several seconds to respond to selecting a link, and frequently comes back with a "not found". Editing becomes hit-and-miss and takes longer than it ought. This has been going on for a couple of days now. I don't think the problem is at my end because other sites are okay.

I'm off line for the day. Holiday prep. --prime mover 11:24, 11 December 2010 (UTC)

Hosting fees and ads
Hosting fees (approx. $\$$120USD), any help would be awesome. As well, what are everyone's opinions on ads ... more specifically Google ads. They are generally pretty unintrusive if placed in nice places. --Joe (talk) 16:02, 8 February 2011 (CST)


 * Google ads are okay if they pay. I've been wondering seriously about whether we can fix it so that this site pays. Now we've got a seriously slick look and feel (or at least we will have, once we have the infrastructure transfer complete), I think we may be worth it. --prime mover 16:48, 8 February 2011 (CST)


 * I'd be perfectly happy to put up some ads. Google's are generally nice and user friendly, as are Project Wonderful's.  --Alec  (talk) 21:59, 9 February 2011 (CST)

Depreciated template
Should we have one? --Joe (talk) 13:46, 13 February 2011 (CST)


 * goodness yes, that was something I was going to do this weekend but I didn't get round to it.


 * Not sure how we'd apply it. It's not the page that needs to be deprecated, it's the constructs in them (i.e. math delimiters, \or, \and, \reals which no longer work, and the old equation template).


 * Can we find a way of automatically going through all the pages and producing a category which has (a) old math delimiters and (b) no longer valid LaTeX commands, The equation template is not a problem, we look at the "most linked-to templates" on Special Pages, for example (this also identified some pages where the new interpreter got into a tangle). What else do we indicate as being deprecated?


 * The word is "deprecated" btw, "depreciated" is "decreased in financial value". --prime mover 14:13, 13 February 2011 (CST)

Equation Template development
The template "equation" has been supplanted by a new one "eqn", which is better as follows:


 * 1) "eqn" is shorter and therefore more streamlined than "equation", so takes less typing.
 * 2) You don't need to enter $\LaTeX$ delimiters around the equation you're entering. The "ll", "l", "o", "r" and "rr" environments are already in $\LaTeX$ mode. Note that the "c" and "cc" are not in $\LaTeX$ mode because usually you want plain text in these fields.
 * 3) There's a new "n" environment which you can use to enter a label (numeric or otherwise), which will appear in brackets.
 * 4) All $\LaTeX$ code that you enter in ll, l, o, r, rr appears in "displaystyle" presentation. So no worries about whether your fractions or summations etc. coming out in in-line style or display style.

This lot:

... generates this:

Work is in progress to convert all pages invoking the "equation" template to use the "eqn" one instead.

--prime mover 15:05, 13 February 2011 (CST)

Suggested new template
When you're reading a proof, you think to yourself: "There has to be a better way of doing this, proof, it's horribly inelegant," but you haven't got time or energy or inspiration to provide such a simpler and more elegant proof. I suggest a template   or something like that (maybe with a text comment), and (like our existing Stub and Explain templates) it would log the proof into a Category. I may get round to doing it myself, but not this morning. --prime mover 00:42, 15 February 2011 (CST)


 * Good idea, I like  . If I get time today I'll start it. --Joe (talk) 06:35, 15 February 2011 (CST)

I almost hate to bring this up...
but I just noticed that some of the blackboard bold characters aren't rendering properly in Firefox for me (most commonly \R, \N and \Q seem to just render as plain R, N, and Q in a lot of places). I think this just started recently since I don't remember noticing it before, although it seems to render the \R fine on Help:Editing/No Longer Supported at least.... Screenshot here. Obviously not a particularly serious issue, just figured I'd point it out. --Alec (talk) 00:13, 17 February 2011 (CST)
 * Just logged in on Firefox myself (something I almost never do, I'm a Chrome man) and the Definition:Real Number page is no problem for me. Is it a caching issue? The page it's fine for you on is a new page. --prime mover 00:18, 17 February 2011 (CST)
 * I have the same problem in Firefox, I also noticed it when using \mathfrak for the Def:Category page (hence it's \mathcal) though \mathfrak works fine now. Linus44 15:51, 17 February 2011 (CST)

Another tidying up job
With MediaWiki it was necessary to add some extra dummy characters at the end of each $\LaTeX$ string in order to force it to render as PNG rather than HTML. I believe this is not needed for MathJax, and in fact interferes with the automatic formatting and spacing (in particular, ending with "space backslash space" now makes it force a space at the end of the item. Lots of pages have instances of this in them. It all has to be stripped out. --prime mover 15:06, 17 February 2011 (CST)

Suggestions
I have three suggestions about --Anton 09:12, 26 February 2011 (CST)
 * adding a Counterexample category. A good counterexample can sometimes be more instructive than a proof.
 * merging with Tricki.org. I doesn't make much sense to have this two projects separately.
 * contacting some actively blogging mathematicians (Terence Tao, for instance) for promotion purposes.


 * Immediate reaction:


 * Counterexamples - We already have a "Proof by Counterexample" entry in the "Proof Techniques" category. I've never been keen on creating categories to hold the style of proof (e.g. direct proof, proof by induction, indirect proof, etc. etc.) because you could end up with colossal categories with nothing holding them together but the decision of the prover as to how to set such a proof up.


 * Merge with any other websites: NO.


 * Contact who you like - if they want to contribute they can.

--prime mover 09:26, 26 February 2011 (CST)

Personally,
 * I like the idea of categorizing proof types, though in practice it just turns into a ginormous list. Saying that I don't think it would hurt to have ones for 'Proof by Induction', 'counterexample', etc.. Those would generally be somewhat smaller than "Direct Proof'. Maybe the category page itself could just have a big warning message saying "This page is huge!".


 * As for merging with another website, that's a tricky subject. Personally I would like to see ProofWiki remain its own entity. Saying that, having information on proof techniques would be nice and I would be interested to look at some sort of content merge (Very hesitant to write that sentence, not really sure how I feel about it. Depends on the rest of the community and the other website involved, ...).

--Joe (talk) 10:02, 26 February 2011 (CST)
 * The more exposure we get the better. If anyone is in contact (or would like to contact) anyone who would like to promote ProofWiki, then by all means do so. The more people we have viewing and editing the better!

... if another site has a particularly fine dissertation on a subject which we are unwilling to go into the levels of detail on, we can always post up links. We already do this on the mathematician biographies sections (as that's not where we major). If it comes to linking to other sites' proofs, then no because we (well, me in particular) want this site to be the best and greatest repository of actual proofs as such. --prime mover 10:32, 26 February 2011 (CST)


 * Re merger, I think merging the sites would annoy and alienate too many users from both sites, a 'content merge' would be better but hard to implement. It'd be hard to weigh up the benefits of doing so globally, and would probably be most practical on the scale of freely trading individual pages. But even then they'd need considerable editing to change from one style to the other, and this seems like not much less work than writing an article from scratch.


 * My view is that it makes sense to have more than one site for much the same reasons as it makes sense to have one than one book on the same subject.


 * Re Terry Tao, he has `pioneered', with some success, some massively collaborative research projects which suggests some interest in this kind of thing. Certainly worth a try. FWIW I think attention to this or similar sites from the academic community will be minimal for some time; I don't think many working mathematicians really believe such a thing will be useful in a way that books aren't, and will continue to dismiss the content based on reliability.


 * Of course, I disagree, but I'd be surprised if the support of a fields medallist swayed many people, certainly no-one cares what I think. It'll happen, just not until the Wikipedia generation permeate academia. --Linus44 10:37, 26 February 2011 (CST)

Fair enough. I want to clarify some points.
 * I don't know how to treat such statements like "pointwise convergence doesn't imply convergence of integrals". Such examples are definitely useful for understanding some theorems (e.g. Dominated Convergence).
 * If by "how to treat" you mean: how to incorporate as a page on this site, what you do is set up a page with that as the title (appropriately capitalised - see some examples and go and do thou likewise, or you'll find somebody changes it to the "house style". Then in the body of the proof you specify the proof you're wishing to add, along with (and this is the vital bit which is the life and soul of this site) links to the pages which specify the meanings of each and every one of the terms you are using in that proof. Then write the proof, in this case, by probably specifying a counterexample to show that there's an example of a pointwise convergence that doesn't imply convergence of integrals, remembering to cite by link any other results used in the process of writing this one. Job done.--prime mover 14:34, 26 February 2011 (CST)


 * By merging I obviously meant incorporating content.
 * If by this you mean "copy and paste" then I would guard against that. We want to offer something here that doesn't exist anywhere else. If we just copy and paste, that limits our uniqueness. "Oh forget that proofwiki site, it's just rehashes of wikipedia / tricki / planetmath / etc." would be bad. "Hey check out proofwiki, there's stuff on there that's been published nowhere else - and get this - they tell jokes on it too, man!" would be good. The latter is the direction I have been more or less bullying it into for the last 3 years or so.--prime mover 14:34, 26 February 2011 (CST)


 * I found this site because I wanted it to exist. And usually if you think to yourself "I would be nice to have X" there is someone in America who's already making X. This site needs promotion. Many people (especially students) read Tao's blog. Maybe, just maybe, they are exactly the kind of people who would be also happy to find this site.
 * If Tao thinks this site is worth citing, then we'd be delighted. As for me, I don't (yet) feel comfortable that we have something here that is quite worth bragging about yet. People who stumble upon it and say, "Just what I'm looking for, now I've got somewhere to download $n$ years of accumulated wisdom" are rare (as for myself I had about 1300 pages of LaTeX without a publisher). Every so often someone turns up who says: hey there's nothing here on (pick whatever subject) and fills it. That's where the energy really ought to be going. I confess that mine is all in maintenance of existing pages and tidying up the really fundamental stuff. And there's always so much more to do, and there's always more people coming along who disagree with what we're doing who want to change its direction. Those directions may not be bad as such, but every fundamental change along those lines takes away the energy that should be flowing into making it grow. Read back through the discussion archives to see people who have joined for a week, made big pronouncements on what ought to be done, then vanished without a trace. Sorry to sound cynical and jaded, but there are few people who are actually working on growing the site. Are you? So far you are - please keep up with the work on functional analysis. It's an important gap to be filled. --prime mover 14:34, 26 February 2011 (CST)

--Anton 13:10, 26 February 2011 (CST)

End of a Template
The Equation template has been completely replaced by the Eqn template (a mere 1100-ish pages)..

What's the policy - delete it? That would make sure nobody used it, and the Eqn is appropriately documented. Yay, nay? --prime mover 16:11, 4 March 2011 (CST)

My vote, would be to leave it there, but maybe change what's transcluded to say something like: "Warning: This template is no longer in use." Deleting the template doesn't delete it fromt he database, so it doesn't hurt to just leave the page in tact.

By the way, good work on switching everything over! --Joe (talk) 16:16, 4 March 2011 (CST)


 * Okay, that's a plan, I'll do that.--prime mover 16:28, 4 March 2011 (CST)

Futile rework
It's tempting, when coming to this site for the first time, to head straight for the pages containing (usually elementary) material one is already familiar with, and to read them and say: "Aha! This is wrong!" when it uses language or a direction different from that which one has been taught.

Under these circumstances, it is probably better not to replace what has gone before with your own understanding, lucid and enlightened though it may possibly be. There may well be a good reason for an exposition to have been crafted the way it has.

If you have a different way of looking at a subject, then it is usually probably better to add your view to the page, rather than replace what's there.

If what you wish to add is radically different from what's there, you may wish to place a note on the talk page of the page in question, or even directly to the author of the page, explaining the matter.

This approach is far less likely to seriously alienate the people who may have spent a lot of time and effort crafting the page in quesion in the particular form you encountered it. In extreme cases, a moment's courtesy along these lines may well make the difference to whether you are barred from the site or not. --prime mover 15:05, 9 March 2011 (CST)

MathJax crediting
Should we include this file on our front page somewhere?
 * [[File:Mathjax-badge-square.png]]

... they like sites to give them the credit:

Technically it needs to be like this:

<a href="http://www.mathjax.org/"> <img title="Powered by MathJax" src="http://www.mathjax.org/badge-square.png" border="0" alt="Powered by MathJax" /> </a>

or:

<a href="http://www.mathjax.org/"> <img title="Powered by MathJax" src="http://www.mathjax.org/badge.gif" border="0" alt="Powered by MathJax" /> </a>

--prime mover 16:32, 19 March 2011 (CDT)

Seems like it would make sense add it to the MediaWiki and GNU FDL badges in the bottom right corner (re-sized appropriately, of course). --Alec (talk) 15:49, 20 March 2011 (CDT)


 * Looks like a job for Superjoe. --prime mover 17:17, 20 March 2011 (CDT)

Problems with LaTeX
Worth starting a separate page documenting various shortcomings of LaTeX and the various workrounds? Could also be used to indicate where certain constructs don't work and need to be repalced with others (yes I know we already have such a page on Editing Help but we could transclude it).

For starters: noticed how the $P$ symbol needs to have a small gap added to its right?

The thought being that MathJax could then be notified of this. I would have done it myself but I'm not sure whether the terms of installation to use in ProofWiki insist that its use be registered, in which case we should communicate with them under that aegis. --prime mover 02:28, 21 March 2011 (CDT)

Proof 3000
Proof 3000 coming up. Anyone want to claim it, before I nob it with something trivial? --prime mover 18:05, 22 March 2011 (CDT)

create account
Moments ago I created an account here at ProofWiki. I hope you don't mind me pointing out something that's a teensy bit confusing during that process. At one point, it asked me:


 * To help protect against automated account creation, please solve the simple sum below and enter the answer in the box (more info):
 * Failed to parse (PNG conversion failed; check for correct installation of latex, dvips, gs, and convert): 91 + 5 =

I'm guessing it really meant to ask something more like:


 * To help protect against automated account creation, please solve the simple sum below and enter the answer in the box (more info):
 * 91 + 5 =

Is there some way to fix this? --DavidCary 19:23, 28 March 2011 (CDT)


 * How's your PHP? --prime mover 00:09, 29 March 2011 (CDT)