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# Wikipedia talk:WikiProject Mathematics

 This is a discussion page for WikiProject Mathematics This page is devoted to discussions of issues relating to mathematics articles on Wikipedia. Related discussion pages include: /Conventions /Graphics /PlanetMath Exchange /Proofs /Typography /Wikipedia 1.0 Mathematics Portal MoS Mathematics Reference desk (mathematics) (Please ask general questions of interest here). Please add new topics at the bottom of the page and sign your posts.

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## Mathematician categories

As I was adding some Category:Women mathematicians into non-gendered parts of the tree, I noticed that Category:American mathematicians, and several others, have by-century trees, but that are sparsely populated. What would think about making those century trees (like Category:20th-century American mathematicians diffusing on the head category, and move all mathematicians into those trees, by century, out of the head cat? Something similar was done recently at Category:American novelists (though there is still some controversy about it). What about here? If not, I might suggest otherwise to delete those cats, as having them as non-diffusing is a bit non-standard (by-century cats are usually diffusing elsewhere in the tree). Best --Obi-Wan Kenobi (talk) 23:24, 21 May 2013 (UTC)

Any thoughts? --Obi-Wan Kenobi (talk) 16:33, 3 June 2013 (UTC)

## Yurii Reshetnyak

The new article Yurii Reshetnyak has been proposed for deletion under WP:BLPPROD. Reshetnyak is a very influential figure in geometric analysis and mathematical elasticity. We should have an article about him, even if it's only a stub. Unfortunately most biographical sources appear to be in Russian. Please help to improve the article! Sławomir Biały (talk) 23:02, 2 June 2013 (UTC)

## Disambiguation help needed.

Greetings! Expert mathematical help is needed in determining the correct targets for links to the following disambiguation pages:

1. Facet (mathematics) and Facet (geometry) both redirect to Facet (disambiguation), for a total of 138 links. Done D.Lazard (talk) 16:31, 3 June 2013 (UTC)
3. Sum of squares: 18 links
5. Primitive polynomial: 17 links Done D.Lazard (talk) 09:41, 3 June 2013 (UTC)
6. Simple root: 17 links Done Mark M (talk) 12:55, 3 June 2013 (UTC)

Please also double check to make sure that these terms are indeed ambiguous, and are not susceptible to being presented in a substantive article as a topic an and of themselves. Cheers! bd2412 T 00:18, 3 June 2013 (UTC)

I got to ask: why is Lie bracket a disambig page? Not just a redirect to Lie algebra. (maybe I don't have a good perspective.) -- Taku (talk) 00:54, 3 June 2013 (UTC)
"Lie bracket" by itself is often used to refer to the Lie bracket of vector fields. Although a disambiguation doesn't seem to be necessary. We can do that with a hat note or, even better, mention the algebra of vector fields in the lead of the Lie algebra article, with a link to the relevant notion of Lie bracket. Sławomir Biały (talk) 03:04, 3 June 2013 (UTC)
I agree, per WP:TWODAB; so I redirected Lie bracket to Lie algebra, and put a hatnote. Though the Lie bracket of vector fields is also mentioned later in the article, so I'm not sure if a hatnote is even necessary. Mark M (talk) 08:27, 3 June 2013 (UTC)
Thanks, I appreciate the solution. bd2412 T 12:04, 3 June 2013 (UTC)

For Simple root, there is a strange situation: All articles linking to this dab page are intended for the meaning in group theory (root system), while, IMO, for most people the main topic refers to a root of a polynomial. Both links of the dab page are redirects to sections. Thus I suggest to redirect Simple root to Polynomial#Solving polynomial equations (the target of Simple root (polynomial)) and to add a hatnote to the target section. I believe that it will be more useful for the readers than the symmetric action, which would need less editing work. D.Lazard (talk) 09:41, 3 June 2013 (UTC)

Incoming links can be a tricky indicator, since editors may link to an article merely because it is the existing target, rather than being a primary topic. If "real world" sources on the internet and in print indicate that Polynomial#Solving polynomial equations is the primary target, then this can be redirected per WP:TWODABS. Ideally, a disambiguation page should only exist if there are too many meanings of a term to fit neatly into a hatnote, or if the few meanings that do exist are in equipoise. Cheers! bd2412 T 12:08, 3 June 2013 (UTC)
Okay, I've fixed all of the incoming links to Simple root, and sent all but one to Simple root (root system) (which is a redirect). Only one incoming link was actually intended for Polynomial#Solving polynomial equations, which is where Simple root is now redirected. Mark M (talk) 12:55, 3 June 2013 (UTC)

I have moved the geometric content of Facet (disambiguation) to Facet (geometry), which is now a stub, and redirected Facet (mathematics) to Facet (geometry). There is no reason to tag Facet (geometry) as a dab page, because the different meanings are strongly connected, and a part of the content does not appear elsewhere. D.Lazard (talk) 16:31, 3 June 2013 (UTC)

Fantastic! That leaves only Normed algebra and Sum of squares in the top 500 disambiguation pages. bd2412 T 17:25, 3 June 2013 (UTC)
BTW, I did not find [1] a good move. Unfortunately there are some users who say: if this is a dab page with inbound links, then let us… to disguise a dab page as an article, and disambiguators become happy. I do not think it improves an encyclopedia. Moreover, Bubka42’s edit actually concealed the Normed associative algebra red link which, unlike the current piece of crap, could be a valid encyclopedic topic. May I revert this? Incnis Mrsi (talk) 09:59, 6 June 2013 (UTC)
The over-riding question is whether there is a topic here that is capable of being discussed as an article, or whether there are merely a collection of unrelated terms that happen to share the same name. So far as I can tell from the previous version of the page, there are no title matches at all. It is not as though there is an album titled "Normed Algebra" and bird called the "Normed Algebra" and a person named "Normed Algebra" in the mix. "Normed associative algebra" and "Normed division algebra" are partial title matches, and are therefore not ambiguous to one another. What is left, then, but a single mathematical concept which is merely applied in different ways? bd2412 T 13:24, 6 June 2013 (UTC)
I don't think that your original dab page was truly a disambiguation page either. A normed algebra after all is an algebra with a norm. So the links there were all topics related to this. So I'd suggest reverting the edit if you like, but removing the disambiguation template. Sławomir Biały (talk) 13:51, 6 June 2013 (UTC)
A dab or a stub, but “normed associative algebra” is a more concrete structure which the (ambiguous) wording “normed algebra” usually, but not always, denotes. Recent edits plays this circumstance down. And in any case… fix what occurred at , please. I hoped to fuse two dabs into one, but if there is no support for it, then it would be better to restore its original tiny dab page or do something else but redirecting to a strange stub. Incnis Mrsi (talk) 15:23, 6 June 2013 (UTC)
I note that neither Gelfand–Mazur theorem nor Hurwitz's theorem (composition algebras) mention the term "Normed division algebra", which means that they can not be included on any disambiguation page for the unmentioned term (see WP:DABMENTION). To the extent that the phrase "Normed division algebra" is used with respect to either article, it must be discussed in that article before it can be included on a disambiguation page, because disambiguation pages only disambiguate content that is already in the encyclopedia. Without that restriction, we could not be sure that the disambiguation page itself is strictly reporting notable uses. However, since those are the only two links claimed to be associated with the exact phrase, "Normed division algebra", WP:TWODABS also applies. Why not redirect Normed division algebra to whichever of those two links is more likely the primary topic of the term, with a hatnote pointing to the other link? (I take it no one would look for Banach algebra or Composition algebra under the name "Normed division algebra"). bd2412 T 15:40, 6 June 2013 (UTC)
This is because you are preoccupied with a formal (link-structural, mention rules, etc.) well-being of Wikipedia, not semantic one, and push forward your “WP:TWODABS” on any pretext or even without one. If there is no clear WP:PRIMARYTOPIC, then do not enforce your dreams that there should be a primary topic. Let experts to decide it, please. There will be numerous links to a dab and you will be slightly unhappy, but eventually it would be better for readers, believe me. Incnis Mrsi (talk) 17:01, 6 June 2013 (UTC)
If there are distinct and different topics, then it should be no problem to fix the links themselves, so that they point to the intended article, and not to a disambiguation page. Remember, disambiguation pages are a means to an end, not an end in and of themselves. They exist to clear up confusion where a term can mean different things, and should not exist if this confusion can be cleared up without creating an extra page to do it. bd2412 T 17:05, 6 June 2013 (UTC)

## Long-time requests

Looking at the dates embedded in Wikipedia:Requested articles/Mathematics, I see there are some 40 requests dating back to 2005 and one to 2004. Perhaps we should try to clear the backlog? For reference, they are listed below. Spectral sequence (talk) 17:15, 3 June 2013 (UTC)

IMHO, this list is obsolete and the best is to destroy it. There are many topics of much larger importance which are not, or badly, described in WP and which deserve to have a much higher priority, like Multivariate resultant or D-finite function. D.Lazard (talk) 17:59, 3 June 2013 (UTC)
The traditional approach would be to archive very old entries. Charles Matthews (talk) 15:17, 9 June 2013 (UTC)

## Six operators

The new article titled Six operators definitely needs work! Including links, context, and some explanation of the role and importance of the concept. Michael Hardy (talk) 21:38, 4 June 2013 (UTC)

## L. E. J. Brouwer

Someone moved Luitzen Egbertus Jan Brouwer to Bertus Brouwer. I've always heard him called L. E. J. Brouwer, and I'd never heard Bertus Brouwer before. I've moved it to L. E. J. Brouwer. Michael Hardy (talk) 21:59, 4 June 2013 (UTC)

## Ambox-throwing to sections

Jarble (talk · contribs) (notified) was already well-known to me thanks to his “interesting” work with redirects. Recently he turned attention to mathematics, where can see:

1. Placing {{unreferenced section}} to Real number #Properties section, which unlikely contains anything beyond textbooks on mathematical analysis;
2. Cluttering Quotient ring #Examples with {{unreferenced section}} and {{original research}}(!)

Of course, there may be original researches in mathematical articles, but unlikely a reasonable person could claim that a section consisting of disjoint examples, most of which can be found in textbooks, is an WP:original research. Does anybody think that these edits actually helps to improve 10 affected articles? Incnis Mrsi (talk) 06:14, 6 June 2013 (UTC)

These unreferenced sections should be verified by external sources, if possible. I found several lengthy unreferenced sections, and decided to tag them as needing additional references. Adding additional references to these sections would be helpful to readers, since it would enable them to confirm the validity of the statements that are made. Jarble (talk) 06:26, 6 June 2013 (UTC)
How this makes the list of examples an original research? And how do you imagine an intended “externally verifiable” Real number #Properties where most statements are backed by hundreds of textbooks? A dozen of <ref>s on each statement? With page numbers for each instance, or only a chapter name/number? Incnis Mrsi (talk) 06:39, 6 June 2013 (UTC)
I noted a similar phenomenon at Transcendental number#Sketch of a proof that e is transcendental. The section begins We will now follow the strategy of David Hilbert and ends For detailed information concerning the proofs of the transcendence of π and e see the references and external links. In the references, the first item is David Hilbert, "Über die Transcendenz der Zahlen e und $\pi$", Mathematische Annalen 43:216–219 (1893), which is precisely to the point. It seems clear that the section is adequately supported by a reliable source and that any reader would be able to identify the source being relied on. Spectral sequence (talk) 06:42, 6 June 2013 (UTC)
In that case, an inline citation to that particular article should be added to the section, using the <ref> tag. Jarble (talk) 06:55, 6 June 2013 (UTC)
Pointing out what is hopefully obvious: inconsistent citation style and lack of references are two different things. In this case, there was a reference and a citation to that reference contained in the text. If you don't find the citation adequate, then WP:SOFIXIT becomes an issue. Sławomir Biały (talk) 13:20, 6 June 2013 (UTC)

It is not necessary to give citations for each individual statement in an article provided the material is uncontroversial and widely available in general sources. For such kinds of information, general sources are sufficient. WP:SCICITE has more details about this. The "original research" tag pointed out above is obviously ridiculous. Very likely all of these examples will appear in either the textbook referenced by Lang or the textbook by Dummit and Foote. It seems to me that the onus is on the person placing the template to check the references, and I think there is little likelihood that this happened here. I don't really think that these kind of templates in general improve articles. It's more likely that a {{fact}} tag here or there, preferably accompanied by an explanation on the discussion page, will lead to a reference. The placement of large and disruptive templates to articles seems to have more to do with social psychology than with encyclopedia building. Sławomir Biały (talk) 13:44, 6 June 2013 (UTC)

Those tag-tacking edits should be undone. Mct mht (talk) 16:04, 6 June 2013 (UTC)

For a section with uncontroversial knowledge, WP:SCICITE#Uncontroversial knowledge recommends a citation or two for the entire section, often placed in or after the first sentence. So if a section is completely unreferenced, is it within policy guidelines to ask for a citation for the section, but it is also easy to fix this problem. Such a citation can show a reader where to find more information about a section topic; doing so is in my opinion an important part of encyclopedia building. --Mark viking (talk) 10:46, 8 June 2013 (UTC)

Actually it's sufficient to provide a general reference for an entire article. It isn't strictly necessary to have footnotes to the general references in each section. Most (all?) of the articles under discussion include such references already. I do think that citations generally improve the overall quality of the articles, but I disagree very strongly with the application of templates to enforce what is actually just a style issue (especially templates placed without any discussion on the talk page). Sławomir Biały (talk) 12:55, 8 June 2013 (UTC)
Where is the policy that it is sufficient to provide a general reference for an entire article? Again, WP:SCICITE says this is fine for articles of a few sentences, but longer articles should have citations. I agree with you about the drive-by tagging; when I see uncited sections where the material seems fine and I want to know more, I usually either contact the dominant editor, if there is one, or I post a query on the talk page. --Mark viking (talk) 19:21, 8 June 2013 (UTC)
WP:SCICITE specifically says that general references are adequate when most of the content of an article appears in multiple general sources (e.g., textbooks on the subject), regardless of article length. It may be appropriate in some cases to include a citation in the section, the lead of the article, or in a list of references at the bottom. This has all been standard practice in mathematics articles for a long time. Sławomir Biały (talk) 20:39, 8 June 2013 (UTC)

In general examples should not have original research tags stuck on them because they are counted as illustrations and come under WP:PERTINENCE. They may be made up by an editor and don't need citations however if so they must not bring in any new ideas but simply be an obvious demonstration of what has been said around them. Any new idea in them would of course be subject to the verification policy. For instance if in the addition article we say 5+7 is an addition giving the result 12 that does not need a citation showing an example of 5 and 7 being added. Citations can be good if people want to follow up but that is not the purpose of an example. If someone thinks an example is not straightforward from what has been said then a citation can be called for but sticking in block tags is not the way to get clarification of an example. After all one can always say an illustration doesn't seem right but you don't put in tags saying all the images in an article need citations. Dmcq (talk) 13:38, 8 June 2013 (UTC) +WP:IUP we must not copy peoples examples straight as they are copyright even if they involve mathematics. Dmcq (talk) 13:53, 8 June 2013 (UTC)

## Equality, identity and equation

I have recently rewritten the leads of equality, identity and equation. Before my edits, none of these articles contained any understandable definition of their topic. Instead, either they were too technical (introduction of Peano and extensionality axioms in the first line of the lead) or they contained silly assertions like "An identity is a relation which is tautologically true" or "An equation, in a mathematical context, is generally understood to mean a mathematical statement that asserts the equality of two expressions". Editing these leads, I was faced with the problem of lacking of sources providing general definitions of these basic concepts. In fact, they are too difficult to be accurately defined in elementary textbooks. And more advanced studies, such as in mathematical logic, usually deal with only one aspect of these questions.

Therefore I have tried to write down what I believe to be commonly meant by these words. But I may have omitted some important aspect. More important, the lack of convenient sources makes that a consensus is needed on these fundamental questions. Also, I have not edited the bodies of these articles which need also some attention. D.Lazard (talk) 10:15, 8 June 2013 (UTC)

Your edits were partially wrong. An equation does not differ from an equality, it uses the equality binary relation but has a sense different from that of an identity. An (equality-based) identity is a statement of the form ∀⋯: F(⋯) = G(⋯), where F and G are terms, which must be a theorem in certain theory. An equation is a predicate expression of the form F(⋯) = G(⋯) on one or more variables with the equality as the upper (outer) logical relation, and is not a theorem. Incnis Mrsi (talk) 10:44, 8 June 2013 (UTC)
While "equality" may merely mean that two expressions happen to have the same value (extensionally), "identity" is usually taken to mean that they have the same meaning (intentionality). That is, two things are identical if they are necessarily equal, not merely contingently equal.
Equality is characterized by reflexivity and the substitution property of equality. The substitution property implies symmetry and transitivity among other things. However, the article on equality (mathematics) misstates the substitution property as "For any quantities a and b and any expression F(x), if a=b, then F(a)=F(b).". The correct statement would be "For any quantities a and b and any predicate P(x), if a=b and P(a), then P(b).". JRSpriggs (talk) 12:18, 8 June 2013 (UTC)
This explanation is quite enlightening. Sławomir Biały (talk) 12:43, 8 June 2013 (UTC)
Intention and contingency have their own hard to pin down philosophical definitions. From the point of view of mathematical logic, are you saying that identity is like logical equivalence, something that holds in all models, and that equality is more like material equivalence, which is model-dependent? Thanks, --Mark viking (talk) 18:59, 8 June 2013 (UTC)
First of all, I think the desired word is intension, not intention.
I don't really agree with either JR or Incnis here. Two objects are identical just in case they are the same object; necessity or contingency doesn't enter into it, and axiomatic theories still less (Incnis's claims seem to assume a formalist POV). There is an older usage whereby two expressions are said to be "identically equal" if their values are equal for all values of the free variables, whereas just saying they're "equal" may leave open the possibility that you have a particular set of values in mind for the free variables, and the values are equal for that particular assignment. But this is mostly a convenience in exposition, and shouldn't be overinterpreted. --Trovatore (talk) 19:46, 9 June 2013 (UTC)
As a non-expert in these things, I found your version of the leads to be clear and easy to read. Leads are necessarily summaries and approximations of main articles; I don't think one can address all the mathematical and philosophical subtleties of these concepts in a lead, keep a lead short enough for MOS, and keep a lead understandable for non-experts. Equivalence may be a related concept to deal with. --Mark viking (talk) 19:12, 8 June 2013 (UTC)
I have some concerns about them, frankly.
The first point is cosmetic: The changes use boldface repeatedly, which is contrary to Wikipedia style. The article title, and synonyms redirected to it, should be bolded at first occurrence, but not thereafter. Similarly, be careful of WP:OVERLINK.
More substantively, I doubt that some of the proposed distinctions are really standard, specifically the supposed distinction between equation and equality. The distinction I would make is rather that an equation is a syntactic object, something you can write down, whereas equality is a property or relation. --Trovatore (talk) 21:33, 9 June 2013 (UTC)
I agree with Trovatore. I'm not convinced there is a widely held distinction between "equation" and "equality" (in the same way that "inequality" and "inequation" should probably be merged). I understand there are two uses of two similar words, but I'm not convinced mathematicians (or anyone else) actually use these words in the way described. Mark M (talk) 12:34, 10 June 2013 (UTC)
The difference between equation and equality in the leads seems reasonable to me, at least in common usage. We solve for unknown variables in quadratic or differential equations, and the solutions make both sides satisfy an equality or make both sides equal. Also in computer languages, we test variables or objects for equality, which comes down to determining the truth value of a predicate expression whose variables are already determined or bound at the time of testing. --Mark viking (talk) 15:54, 10 June 2013 (UTC)
Several posts are about the difference, if any, between equation and equality. I agree that opposing them, as in equation is too strong. I am thinking to a better formulation. I agree also that, from a formal point of view there is no syntactic difference between the two notions. But this does not means that they have the same semantic. If the notions would have the same semantic, the two words would be synonymous, and one could find sources talking of the equation $(x+1)^2=x^2-x+1$, presenting $x^2+y^2=1$ as the equality of the unit circle or $ax^2+bx+c=0$ as the quadratic equality. As these articles are not intended for specialists of formal logic, not even for experimented mathematicians, it is essential to explain the distinction between these two notions that the mathematicians (not the logicians) make in their common usage . And this explanation must not involve high level notions. I have tried to solve this difficult equation :-), but this may certainly be improved. D.Lazard (talk) 18:20, 10 June 2013 (UTC)
I don't think this use of equality as a count noun is very usual. Equality is generally a property; one ordinarily does not speak of an equality at all, and there is certainly no bar against talking about the "equation describing the unit circle". Is it possible that you are being overly influenced by French usage here? --Trovatore (talk) 18:31, 10 June 2013 (UTC)
I think it may be helpful to some readers to draw a distinction between the two concepts.. but I don't see the point in having two separate articles, equation and equality (mathematics). I think this gives the impression that the concepts are more different than they actually are (and the false impression that it is a really important distinction). It think it would be better to explain terminology in a single article. Mark M (talk) 22:17, 10 June 2013 (UTC)

I continue to find this situation problematic. I simply do not believe the purported distinction between "an equality" and "an equation" is standard in English-language mathematics (or, indeed, that it is usual to speak of "an equality" at all). D.Lazard, Mark viking, anyone else who makes this distinction, can you back this up? --Trovatore (talk) 18:23, 12 June 2013 (UTC)

I personally am quite comfortable with the phrase "an equality" as a partial synonym for "an equation", and I am more comfortable with "an inequality" than I am with "an inequation". It seems to me that in good encyclopedic fashion the appropriate thing to do here would be to find one or two reliable sources that either do or do not draw distinctions between these terms, and report on what they have to say, rather than trying to formulate the rules ourselves from scratch. --JBL (talk) 21:43, 12 June 2013 (UTC)
I can believe some sources will use the terms in such a way, but I think the main thrust of the articles should follow common scholarly usage, which I think uses equality primarily as a mass noun rather than a count noun. I agree with you about inequation, a very ugly word that is probably attested somewhere but is definitely not standard. --Trovatore (talk) 21:45, 12 June 2013 (UTC)
You are free to dislike "inequation", but, please, remember that in the context of system of equations solving and real algebraic geometry, both inequation and inequality are commonly used with different standard meanings: $A\ne B$ is an inequation, while $A>B$ and $A\ge B$ are inequalities. D.Lazard (talk) 11:40, 13 June 2013 (UTC)
Can you source that? I do not think the term inequation is really standard at all, in any field. But I don't know all fields of mathematics, so I could be wrong. Oh, MathWorld doesn't count, of course, especially in this sort of case, as it frequently seems to borderline make stuff up in regards to terminology, or at least pull it from idiosyncratic sources. --Trovatore (talk) 16:27, 13 June 2013 (UTC)
Not exactly in reply, but syntactically "inequality" much more resembles "identity" than "equality" or "equation". Sławomir Biały (talk) 18:25, 13 June 2013 (UTC)
In common practice in English-based reliable sources like scholarly articles and books, these terms are in no way interchangeable. In Google scholar, "differential equation" nets 1,810,000 results but "differential equality" nets 593 results. In the other direction, "test for equality" nets 31,300 results but "test for equation" nets get 454 results and most of the first page results are bogus. "Equality predicate" gets 4200 results and "equation predicate" gets 24. --Mark viking (talk) 22:52, 12 June 2013 (UTC)
None of that surprises me. As I said, in my usage, "equality" is a property (or predicate), so it's used as a mass noun. I don't have a problem with treating the property of equality separate from the syntactic object of an equation (or the semantic object of the proposition intended by the equation).
What I don't find convincing is talking about an equality, and distinguishing that from "an equation". It's not implausible that such a distinction could be made, and some authors probably do, but I don't believe it's standard. --Trovatore (talk) 23:52, 12 June 2013 (UTC)

## Parity of zero is up for peer review

Wikipedia:Peer review/Parity of zero/archive1

That's right, folks, I'm getting back on this horse. Please feel free to chime in! Cheers, Melchoir (talk) 22:17, 8 June 2013 (UTC)

## Transformed redirect Cartesian tensor into an article

This should be an article. Cartesian tensors are the first approach to tensors and index notation (at least in my experience).

I wrote this from the perspective of someone with a firm grip on basic vector algebra, and some possible exposure to linear algebra and tensor index notation, but not lots, so there is some preamble on notation and terminology.

It is hoped to bridge the gap between basic (Gibbsian, as it were) vector algebra and tensor analysis.

Feel free to complain about the potentially trillions of typos, or anything, on the talk page and we can improve the article.

There is room for more sections and expansion also. M∧Ŝc2ħεИτlk 23:56, 8 June 2013 (UTC)

Could you explain better what does it mean: a tensor in an Euclidean space? There is no such thing as a “tensor which exists in Cartesian coordinates” because a tensor must have a representation in any basis, by its very virtue to be a tensor. Incnis Mrsi (talk) 09:43, 9 June 2013 (UTC)
Fixed, any more specific inaccuracies please take it to the talk page. M∧Ŝc2ħεИτlk 09:56, 9 June 2013 (UTC)

## 3x12 = ?

I have a different opinion about what (or who) was silly in Hexadecimal. Incnis Mrsi (talk) 04:09, 9 June 2013 (UTC)

Perhaps you could explain what you think the purpose of the link is? --JBL (talk) 15:52, 9 June 2013 (UTC)
Perhaps, an inhabitant of Unicodified worlds needs a hint to understand the “humour” invented by ASCII-based tribes in prehistoric times before the advent of HTML? Incnis Mrsi (talk) 16:13, 9 June 2013 (UTC)
This is why the "joke" is followed immediately by an explanation, is it not? --JBL (talk) 17:51, 9 June 2013 (UTC)
<hit with="bone"><monolith/></hit> Sławomir Biały (talk) 19:38, 9 June 2013 (UTC)
Is there a reliable source which supports anything at all in the section Hexadecimal#Common_patterns_and_humor? Spectral sequence (talk) 20:54, 10 June 2013 (UTC)
There are a lot of sources (but of dubious reliability) in the main article on this topic, hexspeak. Apparently the original source for this is Kernighan and Pike The Practice of Programming (1999) but I don't have a copy to check. —David Eppstein (talk) 20:54, 12 June 2013 (UTC)
As far as I can see, hexspeak is a random collection of mildly amusing jokes that can be or have been made using hexadecimal notation, sourced only to primary sources, and with no single reliable secondary source to suggest that the list is complete, or representative, or significant. The section Hexadecimal#Common_patterns_and_humor mentions a few of those jokes and recounts some others, all without any sources at all. It seems completely undue. Spectral sequence (talk) 21:17, 12 June 2013 (UTC)

Fractional-order system is a new article. No other articles link to it. Creating some links to it from other articles, and other work, should get done. Michael Hardy (talk) 18:15, 9 June 2013 (UTC)

Sorry if it is a dumb question, Chaos in fractional order system is another new article. Should they not be merged? Solomon7968 18:24, 9 June 2013 (UTC)
Perhaps. Michael Hardy (talk) 19:05, 9 June 2013 (UTC)
Merged. No significance to the example, so I removed it. It might be required to determine whether the statement makes any sense, even if there is insufficient information in the reference to locate it. — Arthur Rubin (talk) 19:49, 9 June 2013 (UTC)
Proposing a merge into fractional dynamics; same subject, that's almost an orphan, as well, but it's better-connected. — Arthur Rubin (talk) 20:59, 9 June 2013 (UTC)

## Anatolii Alexeevitch Karatsuba

There is a request for further input at Talk:Anatolii Alexeevitch Karatsuba. Spectral sequence (talk) 21:25, 10 June 2013 (UTC)

## New article: Spherical basis

I started it for now. It is incomplete and probably full of typos/inaccuracies, and I intend to finish it soon, but it's been sitting in my userspace for a very long time so for now it's been moved into mainspace allowing others the freedom to edit it, if inclined. M∧Ŝc2ħεИτlk 08:30, 12 June 2013 (UTC)

I also redirected spherical tensor, spherical tensor operator, and tensor operator to spherical basis for now. Do people think they should have their own articles? Spherical tensor and Spherical tensor operator makes sense to redirect since the literature seems to explain spherical basis and spherical tensors and spherical tensor operators together. So tensor operator should have its own article? M∧Ŝc2ħεИτlk 14:16, 12 June 2013 (UTC)
Would some slightly more concrete examples and material, presumably relating the abstract content you develop to the concrete set of spherical harmonics as perhaps the simplest and well-known such basis, be appropriate? (And then perhaps discussing other possible such concretely-realised basis sets?) Or have I missed the point? Jheald (talk) 18:01, 12 June 2013 (UTC)
Yes, as said, more work is needed, including examples and other applications will be added as I learn them in time. M∧Ŝc2ħεИτlk 18:08, 12 June 2013 (UTC)
While tensor operator seems to be used by physicists mostly in the context of tensors under rotation in quantum mechanics, the concept is used in other ways, too. For instance, in image processing, there is the concept of tensor operators for energy tensors, e.g., Energy Tensors: Quadratic, Phase Invariant Image Operators and also in the more general concept of structure tensor. Neither of these necessarily involve 3D rotational invariance/covariance. Tensor operator algebras have also been created for other groups, e.g., Tensor Operator Algebra for Point Groups. I don't think the redirect from tensor operator to spherical basis is a bad move, but if an editor wanted to do so, creating a standalone tensor operator article seems reasonable. --Mark viking (talk) 20:24, 12 June 2013 (UTC)
Yes, I suspected tensor operator would potentially have it's own article for the reasons you mention. It's easy to overwrite a redirect. Thanks, M∧Ŝc2ħεИτlk 20:34, 12 June 2013 (UTC)
In the end I split the tensor operator stuff from spherical basis to overwrite tensor operator, for reasons explained here and at Talk:Spherical basis, Talk:Tensor operator. The redirects are changed as stated on these articles. M∧Ŝc2ħεИτlk 08:56, 13 June 2013 (UTC)

## An instance of erroneous eponymy

I have created this new section, explaining that the so-called Weierstrass substitution was used by Euler long before Weierstrass was born. I don't know how this came to be called the Weierstrass substitution by respectable authors, but Google Books confirms that it is. Maybe the answer to that last question should be included in the article if it can be found. Did Weierstrass even used this substitution? Where, specifically, is it found in Weierstrass' work. If it isn't there, what source could be cited to confirm that it isn't there? Would this warrant renaming the article? How best should the topic of erroneous nomenclature be treated in Wikipedia? Michael Hardy (talk) 13:06, 17 June 2013 (UTC)

I have never heard of this name for the tangent half-angle substitution and I would never find this article by myself. I have created the redirect Tangent half-angle substitution, and I support to move Weierstrass_substitution to this new page. Also it could be worth to merge this page with Tangent half-angle formula. D.Lazard (talk) 13:58, 17 June 2013 (UTC)
Is there an independent reliable source stating that this substitution was used by Euler? So far, all we have is a reference to Euler E342.V.261 where a substitution equivalent to this appears for one specific case. Michael Hardy may be correct about Euler, and in deed I personally think he is, but we go by what the sources say. It seems to be at best original research by way of synthesis to suggest from this reference alone that Euler was aware of the substitution for general rational functions. It is also unwarranted to assert that the assignment to Weiesrstrass is "erroneous". There may be a good historical reason for it -- we just do not know, because there are no sources cited to support the assertion either way. Whereof one cannot speak, thereof one must be silent. Spectral sequence (talk) 17:03, 17 June 2013 (UTC)
Euler knew that
\begin{align} \text{if }y & = \tan\frac\theta2 \\[8pt] \text{then }\frac{2y}{1+y^2} & = \sin\theta \\[8pt] \text{and }\frac{1-y^2}{1+y^2} & = \cos\theta \\[8pt] \text{and }\frac{2\,dy}{1+y^2} & = d\theta. \end{align}
Certainly he was aware of that substitution. Whether he knew it could be applied to all or to only some rational functions of sin θ and cos θ, I can't say for sure. Whether Weierstrass knew that, or ever wrote anything about this substitution, I don't know. What we can say is erroneously attributed to Weierstrass is the first use of this substitution. Michael Hardy (talk) 22:51, 17 June 2013 (UTC)
. . . . and I should add: Secondary sources are needed to establish notability, and we certainly have those secondary sources for this substitution. But primary sources are what is needed to establish that a particular author used this substituion. Michael Hardy (talk) 22:52, 17 June 2013 (UTC)
"I don't know". That's exactly my point. Wikipedia reports what reliable sources say, not our personal guesses. Is there a reliable source that says that the name "Weierstrass" is erroneous? The primary source indicates that Euler was aware of one particular use of this substitution. Does any source state that Weierstrass was the first person to use this substitution, which would indeed be an error if so? Perhaps it was named after Weierstrass because he rediscovered it, generalised it, popularised it, used it heavily, or whatever -- I don't know and it seems that you don't either. Saying that the attribution to Weiestrass is "erroneous" appears to be synthesis at best. Spectral sequence (talk) 06:15, 18 June 2013 (UTC)
Additional: I see that you have removed the word "erroneous" anyway so this part of the discussion is perhaps moot. In any event it seems to belong at Talk:Weierstrass substitution, not here. Spectral sequence (talk) 06:23, 18 June 2013 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── The general points raised do seem worth continuing to discuss here. I prefer the term "unhistorical" to "erroneous". It seems to me that we should use the terminology most commonly used in the current literature, whether or not that happens to be historically accurate. That is what readers expect and will think it odd not to find. We are here to follow, not lead, the scientific literature. We are not here as advocates or to right great wrongs. Spectral sequence (talk) 06:32, 18 June 2013 (UTC)

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# Ebook Ο Ηγεόας

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# Is the derivative of 2x^2 = 4x or 8x?

Call me crazy, but I do recall the power rule of integration viz: f(x) = x^n, f(x)' = n*x^n-1. Therefore, it seems as though 2x^2 would have a derivative of 4x. Fine. So why have I encountered someone else claiming that it's 8x? WHAT?! Who's right?

Mark44
Mentor
Call me crazy, but I do recall the power rule of integration viz: f(x) = x^n, f(x)' = n*x^n-1.
Do you really mean ##n * x^n - 1##? That's what you wrote. As inline text, use parentheses -- n*x^(n - 1)

5P@N said:
Therefore, it seems as though 2x^2 would have a derivative of 4x.
Yes. ##d/dx(2x^2) = 4x##.
5P@N said:
Fine. So why have I encountered someone else claiming that it's 8x? WHAT?! Who's right?
Maybe they're working a different problem.

S.G. Janssens
So why have I encountered someone else claiming that it's 8x?
I can think of many reasons, but none of them involve mathematics.

I haven't yet applied Latex (I'm going to read the article after I finish reading this other long article on u-substitution - I'm really trying here), but what I mean is the super basic power rule of derivatives: f(x) = xn, f(x)' = nxn-1

S.G. Janssens
You are right, the other person must have a brain worm. Also, I would write ##f'(x)## instead of ##f(x)'##.

EDIT: Could you prove that the other has a brain worm?

Last edited:
And just for the record: I meant DERIVATION, not integration.

Mark44
Mentor
And just for the record: I meant DERIVATION, not integration.
Contrary to much popular opinion, the opposite of integration is differentiation, not derivation. You can derive the quadratic formula using the completion of squares technique, but you differentiate ##2x^2## to get the derivative, 4x. Yes, English is weird...

fresh_42
Mentor
Contrary to much popular opinion, the opposite of integration is differentiation, not derivation. You can derive the quadratic formula using the completion of squares technique, but you differentiate ##2x^2## to get the derivative, 4x. Yes, English is weird...
Thanks for clarifying. I sometimes get confused because the result of differentiation is a derivative (obeying the product rule) and the result of integration is an anti-derivative. Is that correct or am I still confusing terms?

Mark44
Mentor
Thanks for clarifying. I sometimes get confused because the result of differentiation is a derivative (obeying the product rule) and the result of integration is an anti-derivative. Is that correct or am I still confusing terms?
No, you have it right.

Erland


text:
Estimating detector systematics at T2K's far detector: A neutrino oscillation adventure

Seria:
Seminarium doktoranckie
Prelegent i afiliacja:
Yashwanth Prabhu (Szkoła Doktorska NCBJ)
Data:
czw., 2023-01-19 09:15 do 10:15
Miejsce:
room 207, Pasteura 7 / https://www.gotomeet.me/NCBJmeetings/phd-seminar / https://events.ncbj.gov.pl/e/PhDSeminar2223
Streszczenie:

T2K is a long-baseline neutrino experiment that measures neutrino oscillations. Oscillation of the muon (anti)neutrino beam to electron (anti)neutrinos is sensitive to the leptonic CP violating phase. In this talk, I will discuss the  addition of a new electron neutrino signal sample at T2K's far detector,  which can improve T2K's sensitivity to the CP-violating phase. I will also talk about how the systematic errors at T2K's far detector are estimated using Markov Chain Monte Carlo techniques, along with some outlook to future studies.

Załączniki:
ZałącznikWielkość
12.86 KB
98.82 KB


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## Algorithm for finding “mean center” of unweighted graph

I have a large sparse unweighted undirected graph (20M vertices, 60M edges) and would like to find what I’m calling the “mean center” (the vertex w/ shortest mean distance to all other vertices. Does this already have a name?). I know of an $$O(N^2)$$ solution: Starting at every vertex: BFS to calculate the mean distance to all other vertices, choose the minimum. But $$N^2$$ is too slow for a graph this size, is there any way to calculate the “mean center” in faster than $$N^2$$ time? Are there any good approximation algorithms? Or more efficient algorithms which find some similar “center” of a graph?

## Maximum flow on a tripartite graph

I have to solve an assignment problem between $$\{1,\dots, N\}$$ agents and $$\{1,\dots, M\}$$ objects, which comes to maximize : $$$$\sum_{ij}\beta_{ij}x_{ij}$$$$ where $$x_{ij}$$ equals one if object $$j$$ is assigned to $$i$$, $$0$$ otherwise. The benefit of such an assignment is the non-negative value $$\beta_{ij}$$.

This problem usually comes wih some constraints : $$$$\sum_i x_{ij} = 1~,~~\sum_j x_{ij} = 1$$$$ stating that an object can only be assigned once and that an agent can only assign one object.

As far as I understand, this problem can also be viewed as a maximum flow on a bipartite graph and can be efficiently solved by an auction algorithm (see Bertsekas et al.)

I have to implement additional constraints labeled $$k=1,\dots, K$$ which are all of the same form : the $$k^{th}$$ one imposes that on a subset of object $$\mathcal{J}_k$$, only a limited number of them, say $$P_k$$, can be assigned in the same time : $$$$\sum_{j\in \mathcal{J}_k}\sum_i x_{ij} \le P_k$$$$

My question is that I find quite difficult to solve former problem with these new constraints on a bipartite graph and I wonder if a tripartite graph would be more convenient.

Here is below how I would do, I would like to call upon more experienced people in this field to assess whether this approach is well suited or if there is a simpler one.

I put the former problem with the additional constraints in the form of a tripartite graph : the first column is still that of agents and the third one that of objects, except that agents and objects are no more directly linked.

Between first and third column, I insert the column of the $$\{1,\dots,K\}$$ constraints. Let $$y_{ik}\in\{0,1\}$$ be the link between agent $$i$$ and constraint $$k$$. The additional constraint $$k$$ now reads : $$$$0\le\sum_i y_{ik} \le P_k$$$$ The flow of agents onto constraint $$k$$ is limited.

Furthermore, let $$z_{kj}\in\{0,1\}$$ be the link between constraint $$k$$ and object $$j$$.

That is to note, an object $$j$$ can be linked to several constraints, let $$\mathcal{K}_j$$ be the subset of constraints in which object $$j$$ is involved, i.e. $$j\in \mathcal{J}_k$$.

The flow from constraint $$k$$ to objects must equal that of agents to this constraint :

$$$$\sum_i y_{ik} = \sum_j z_{kj}$$$$ which means that an agent $$i$$ is connected to constraint $$k$$ because there is some object $$j$$ linked to it, and conversely.

Then, coming back to the bipartite graph, the link $$x_{ij}$$ between agent $$i$$ and object $$j$$ is now : $$$$x_{ij} = \sum_{k\in\mathcal{K}_j} y_{ik} + 1 – |\mathcal{K}_j|$$$$ This equals one only if the agent $$i$$ is connected to all constraints $$k$$ in which object $$j$$ is involved. The inconvenient is that $$x_{ij}$$ can be negative. Another defintion could be : $$$$x_{ij} = \min_{k\in\mathcal{K}_j} \{y_{ik}\}$$$$ but I would like to remain as most linear as possible.

Because $$P_k< |\mathcal{J}_k|$$ there will always remain at least $$|\mathcal{J}_k| – P_k$$ unassigned objects due to constraint $$k$$.

Let’s define a virtual agent $$s$$ to which we attach all objects (at zero benefit) not linked with any constraint $$k$$, $$\forall k~~ z_{kj} = 0$$. The link is denoted $$x_{sj}\in\{0,1\}$$ because it is a direct link between an agent and objects.

Each object $$j$$ must fulfill : $$$$\sum_{k\in\mathcal{K}_j} z_{kj} + |\mathcal{K}_j| x_{sj} = |\mathcal{K}_j|$$$$ Then, either object $$j$$ is connected to all its constraints, $$\forall k\in\mathcal{K}_j~,~z_{kj} = 1$$, or it is connected to the virtual node, $$x_{sj}=1$$.

It is clear that if the flow on all constraints reach its maximum, there will remain at least : $$$$\sum_k (|\mathcal{J}_k| -P_k)$$$$ objects unassigned, thus : $$$$\sum_j x_{sj} \ge \sum_k (|\mathcal{J}_k| -P_k)$$$$

Then, each time the links between an object and its constraints are broken, the object becomes unassigned, so that : $$$$\sum_j x_{sj} – \sum_k (|\mathcal{J}_k| -P_k) = \sum_k y_{sk}$$$$

where $$y_{sk}$$ is the link between the virtual agent and the constraint $$k$$, whose capacity is $$\{0,\dots,P_k\}$$ and checks : $$$$\sum_i y_{ik} + y_{sk} = P_k~,~~0\le y_{sk}\le P_k$$$$

Coming back to the assignment problem, the maximization becomes : $$$$\begin{split} \sum_{ij}\beta_{ij}x_{ij} =& \sum_{ij}\beta_{ij}\biggl(\sum_{k\in\mathcal{K}_j} y_{ik} + 1 – |\mathcal{K}_j|\biggr)\ =& \sum_{ik}\biggl(\sum_{j\in\mathcal{J}_k}\beta_{ij}\biggr) y_{ik} + \sum_j (1 -|\mathcal{K}_j|)\sum_i \beta_{ij} \end{split}$$$$ so that the maximization now takes place between agents and constraints. I would like then to write the dual formulation of this problem in order to solve it with an auction algorithm : agents have to buy all constraints to which an object is related in order to be assigned this one.

## Coloring a graph with odd number of vertices with $k$ (which is close to $\Delta$) colors in linear time

We have an undirected simple connected graph with odd number of vertices. We also know the number $$k$$ which is actually the closest odd number greater than or equal to $$\Delta$$. (So if $$\Delta$$ is even, $$k = \Delta +1$$ else $$k=\Delta$$.) i.e $$k$$ is the least odd number that is greater than or equal to degrees of all vertices.

We want to find a linear time algorithm that colors the graph with $$k$$ colors.

I am very new to graph coloring algorithms. I know that a greedy algorithm is actually linear time and can color the graph with $$\Delta +1$$. But I can’t guarantee I can color it with $$k= \Delta$$ when $$\Delta$$ is odd. Also, we probably should use all the information the question gives us. i.e using odd number of vertices somehow, but greedy algorithm don’t use this extra information.

How can I solve this?

## How to detect self loop in graph using greedy algorithm if given list of number of degrees

if you are given list of n integers that represents the degree of a graph. How to detect if there self loop in the graph using greedy algorithm.

## Artificial ant colony algorithm for graph

let’s assume we have ant at node $$1$$ and she has $$\{2,3,4\}$$ vertices. How do I compute which one she choose? I mean if the formula $$p_{ij}(k)$$ is the probability of $$k$$-th ant at node $$i$$ choose $$j$$ node, for example, do I take $$j=2$$, compute the probability and if it satisfies $$random(0,1), the ant choose $$j=2$$, if not move to next vertex $$j=3$$ until the inequality is satisfied? The next question is, can you recommend me the sources, where I can find conversion from this algorithm graph version to optimizing function version. Thank you.

## Clique-problem for planar graph

I have to show, that the clique problem is in P for planar graphs. I found a post, that anwers this here. However I don’t get the conclusion “This follows already from Kuratowski’s theorem: a clique is at most of size 4.”

I’ve never heard of that theorem before and the exercise indicates to proof it with the fact, that the class of planar graphs is closed under sub graphs. Thank you for your help!

## Algorithm for breaking a graph into sub-graphs based on an attribute

I am a biologist doing some computational work. I encountered a problem and need some help since I don’t have much algorithmic background.

The problem:
Assume we have a non-directed, weighted graph, in which every node has some categorical attribute, e.g. color. I want to break the graph into sub-graphs in which there is at most one node of each color. Decisions should be made by trying to preserve the strongest connections (edges with highest weights). For example, let’s look at the graph below (I didn’t include weights as this would be hard to see):

In this case, I would like to break the graph into two sub-graphs (assuming that this is what the weights indicate):
Two notes (I don’t know if they matter though):

• The maximal number of colors is known in advance
• After the graph was broken, I don’t really care about the topology anymore – I just want to know which nodes ended up together.

My questions:
My first question is: does this look familiar to you? Has this problem been solved before? Maybe it’s a variant of something every CS student learns? If so, can you please refer me to relevant information/literature?
If not, Then I’d like some help on how to approach this. Can you give some ideas on how to start developing and algorithm that can solve this type of problem?
Any other advice would be useful. Thank you!

## Literature request: Generating all vertex subsets of a graph

I made the same post here in Mathematics Exchange but maybe here is a better place.

I am working in an algorithm which finds a unique maximal independent set of vertices which generates all other vertex subsets. This is done using two measures of ‘importance’ for vertices. I assume this might have some applications outside mathematics, especially in computer science, but I am not sure where I can read about such applications. Does anyone have any good literature suggestions?

## Graph problem with expectation

Consider an undirected graph G = (V, E) representing the social network of friendship/trust between employees. We partition the set of node triples into four sets T0, T1, T2, and T3. A node triple v1, v2, v3 belongs to T0 iff no edge exists between the nodes v1, v2, and v3, – T1 iff exactly one of the edges (v1, v2), (v2, v3), and (v3; v1) exists,- T2 iff exactly two of the edges (v1, v2), (v2, v3), and (v3, v1) exist,- T3 iff all of the edges (v1, v2), (v2, v3), and (v3; v1) exist. |T3| denotes the number of connected triples in the graph that is the quantity we need to estimate. if i apply the following algorithm:

• Sample an edge e = (a, b) uniformly chosen from E

• Choose a node v uniformly from V \ {a, b}

• if (a, v) ∈ E and (b, v) ∈ E then x = 1, else x = 0

How can I show that |T1| + 2|T2| + 3|T3| = |E|(|V | − 2)?

## Find a longest path with k vertices in a directed graph

Assume a directed acyclic graph G=(V,E), the goal is to find a path (with k vertices) so that the sum of edge weights is the maximum.


text:
# Quality of coding life with a terminal multiplexing startup script

Saving minutes every time I (re-)boot or remote login to a machine, using terminator, screen and ssh.

My terminator setup

For my various jobs (academic and business) I have spend a lot of time behind a computer doing code-related things. I have been in the situation many times (actually, I should say all the time, before I started doing what I will describe below) where I had many windows open; specifically a bunch of terminals, editor windows (and browser of course). Like, way too many. Different terminals had different purposes, and were grouped and named so I could find them back easily; I would be working on multiple projects simultaneously in different workspaces; a couple of terminals would be remotely logged in to other machines. So every time I had to reboot (or lost connection) I would have to go back and open all the windows, and start all the processes again, and rename all the terminals, and put them all back in the same place (because I had gotten used to everything being in a certain place).

This could take a lot of time. My guess is up to 15 minutes every time, although I never timed it. It did take waaaay too long for me before I finally automated this whole thing, but I did in the end. Perhaps some of my findings can be useful to other people who have a mild anxiety attack every time they have to reboot.

I use terminator for my terminal multiplexing nowadays. Just

apt-get install terminator

and the important thing is that you can configure various layouts, and make it so that each terminal window executes a predefined command when they start.

For the layout, start terminator, and begin creating and resizing the windows to your fancy. It’s very easy. Then save this layout by right-clicking any terminator window > Preferences > Layouts > Add, then give it a name and hit save.

Open the config file in an editor (should be in ~/.config/terminator/config) and then then manually set all of the following for each terminal

directory = /home/me/projects/someproject
# which directory to start in, obviously

profile = default
# the visual profile: colors, fonts, etc. You can create these from the terminator settings.

title = jupyter notebook  # the name of this terminal window

# here is the important bit
command = '''bash --rcfile <(cat ${HOME}/.bashrc; echo 'export PROMPT_COMMAND="source activate my_env; jupyter notebook; unset PROMPT_COMMAND"') -i''' This layout has a name (you set it before in the preferences window), but you can edit the layout name in the config file as well. If you edit the layout named ‘default’, this will be the layout that is loaded by default when you start terminator without any options (what’s in a name, huh?). Let’s call it mylayout for now. So after (re)booting my system, I just do: terminator --layout=mylayout and then, for example, automatically I will get my jupyter notebook server in the correct virtualenv, pycharm, a window running htop, a continuous rsync process for pushing code to a remote, and whatever else have you. My terminator after start-up I’ll explain a little about the command, because it is a bit fiddly. First of all, terminator can only execute a single command, so we have to hack everything we want to happen into one command: bash. Props to this stackoverflow answer. The explanation below is mostly a copy-paste from that SO answer, but I’ll add some notes of my own. Here is the command again: command = '''bash --rcfile <(cat${HOME}/.bashrc; echo 'export PROMPT_COMMAND="source activate my_env; jupyter notebook; unset PROMPT_COMMAND"') -i'''
• We execute bash in interactive (-i) mode. So the bash session stays open at the end.
• We execute commands from a custom command file (--rcfile) instead of .bashrc.
• This file is created with the contents of .bashrc plus one more command.
• This extra command exports PROMPT_COMMAND with a value of “whatever we want to execute”.
• The PROMPT_COMMAND is a bash built-in, and gets executed at every new prompt line.
• Therefore, the PROMPT_COMMAND must be unset just after it is executed the first time to avoid multiple executions after each interaction with the shell.
• Within the PROMPT_COMMAND I can mostly chain together all the commands that I want. So in this example, I activate my virtual environment, and start my jupyter notebook server inside it.

Another example; starting some application but having it run in the background, so we can use the terminal windows for something else afterwards:

command = '''bash --rcfile <(cat ${HOME}/.bashrc; echo 'export PROMPT_COMMAND="unset PROMPT_COMMAND; pycharm &"') -i''' To run in the background, you usually add the & at the end of the command, but apparently this doesn’t work if you want to chain another command after it using a semicolon. Luckily, we can switch the commands around so the & is at the end, and the unsetting of the PROMPT_COMMAND doesn’t interfere with the execution of itself. Another very related issue is doing things on remote machines over ssh. We have the same problems as above, having to re-setup the ssh connections every time we reboot our own system. But there are two more liabilities when working over ssh (seasoned ssh users will know the solutions to this already, of course): if you lose connection to a process on a remote machine, the process will exit, and if the remote machine reboots, you obviously also have to restart all the stuff that you were running there. My most convoluted terminator startup command solves all three of these at the same time. For connection loss / client machine reboot, we are going to use good old screen. The essential primer is this: type screen, hit enter, and you just started a new terminal session, but it is sort of like a background process. You are currently connected to it, and it is practically indistinguishable from when you normally work in a terminal. However, if you close the window, or press CTRL-A CTRL-D, this sessions get detached. This means it is still running, it’s just no longer attached to a user I/O. To reconnect simply do screen -r. Screen can do a LOT more that I won’t go into right now. For convenience, instead of writing out the entire command into the PROMPT_COMMAND, I make an alias out of it. So # in my terminator config file command = '''bash --rcfile <(cat${HOME}/.bashrc; echo 'export PROMPT_COMMAND="unset PROMPT_COMMAND; remote_tensorboard &"') -i'''

# and in my ~/.bash_aliases:
remote_tensorboard () {

SCRIPT='bash --rcfile <(cat ${HOME}/.bashrc; echo "source activate tf_env; cd project/output; tensorboard --logdir=runs")' ssh -t myremote "screen -RR -q -S tensorboard bash -c '${SCRIPT}'"

}

This command is going to:

• ssh into my remote machine (myremote is the ssh alias defined in my ~/.ssh/config, defining username, ssh-key location, port forwarding, etc)
• execute a command on this remote machine (the -t option)
• this screen will get a name (-S tensorboard in this example) and will not show the standard screen welcome screen (using -q)
• if there is already a screen session with the same name, we will attach to that screen session instead without running the bash command (-RR)
• run a ‘single’ bash command inside this screen session using bash -c (the -c tells bash to execute a command from a string)
• if inside this bash command we want something complex, we use the same --rcfile trick as before. In this case, we start tensorboard from a virtual environment and from a specific directory.

Honest disclaimer: I always have to muck about with the quotation marks and string escapes and such in bash, so there might be a better way to compose all this, but at least it works!

Having set all this stuff up has made me inappropriately happy, and has probably saved me between 5 and 15 minutes (depending on the project) every time when I reboot/reconnect, not to mention the annoyances!

Of course I’m on a linux system here. I expect macOS probably has something very similar. I’m using terminator and screen, but tmux I think has combined functionality, and byobu as well I believe. Haven’t tried them yet, but they look nice and I might try them in the future.


text:
Abstract

We introduce a new BSP tree construction method for set of segments in the plane. Our algorithm is able to create BSP tree of linear size in the time $O(n \log^3 n)$ for set of segments with low directional density (i.e. it holds for arbitrary segment $s_i$ from such set, that a line created as extension of this segment doesn't intersect too many other segments from the set in a near neighbourhood of $s_i$) and a directional constant $\delta$ belonging to this set.


text:
## University Calculus: Early Transcendentals (3rd Edition)

$$\lim_{x\to0}\frac{x\csc2x}{\cos5x}=\frac{1}{2}$$
$$A=\lim_{x\to0}\frac{x\csc2x}{\cos5x}$$ $$A=\lim_{x\to0}\frac{x\times\frac{1}{\sin2x}}{\cos5x}=\lim_{x\to0}\frac{x}{\sin2x\cos5x}$$ $$A=\lim_{x\to0}\frac{x}{\sin2x}\lim_{x\to0}\frac{1}{\cos5x}=\lim_{x\to0}\frac{x}{\sin2x}\times\frac{1}{\cos0}=\lim_{x\to0}\frac{x}{\sin2x}\times\frac{1}{1}$$ $$A=\lim_{x\to0}\frac{x}{\sin2x}$$ $$A=\lim_{x\to0}\Big(\frac{\sin2x}{x}\Big)^{-1}=\Big(\lim_{x\to0}\frac{\sin2x}{x}\Big)^{-1}=X^{-1}$$ Considering $X$: $$X=\lim_{x\to0}\frac{\sin2x}{x}$$ To be able to apply Theorem 7, we need $2x$ in the denominator, instead of $x$. So we would multiply both numerator and denominator by $2$: $$X=\lim_{x\to0}\frac{2\sin 2x}{2\times x}=2\lim_{x\to0}\frac{\sin 2x}{2x}$$ Apply Theorem 7 with $\theta=2x$ here: $$X=2\times1=2$$ Therefore, $$A=X^{-1}=2^{-1}=\frac{1}{2}$$


text:
# On automorphisms of infinite abelian 2-groups

Let $G$ to be a infinite abelian 2-group. I want to know information in structure of group automorphism of $G$. For example $\alpha$ such that $\alpha(x)=x^{-1}$ for all $x\in G$ is an automorphism of $G$. Do we can find other automorphisms of $G$?

-
In general? Depends on your set theory. There are set theories in which there exist infinite abelian groups of exponent $2$ with no non-trivial automorphisms (see this). The automorphism you give would be the identity in that case. –  Arturo Magidin Jan 28 '12 at 22:46


text:
## From Linear Regression to Linear Classification

Preliminaries An Introduction to Linear Regression A Simple Linear Regression Bayesian theorem Feature extraction Recall Linear Regression The goal of a regression problem is to find out a function or hypothesis that given an input $$\mathbf{x}$$, it can make a prediction $$\hat{y}$$ to estimate the target. Both the target $$y$$ and prediction $$\hat{y}$$ here are continuous. They have the properties of numbers1: Consider 3 inputs $$\mathbf{x}_1$$, $$\mathbf{x}_2$$ and $$\mathbf{x}_3$$ and their coresponding targets are $$y_1=0$$, $$y_2=1$$ and $$y_3=2$$....

February 17, 2020 · (Last Modification: April 28, 2022) · Anthony Tan

## Neuron Model and Network Architecture

Preliminaries linear classifier An Introduction to Neural Networks Theory and Notation1 We are not able to build any artificial cells up to now. It seems impossible to build a neuron network through biological materials manually, either. To investigate the ability of neurons we have built mathematical models of the neuron. These models have been assigned a number of neuron-like properties. However, there must be a balance between the number of properties contained by the mathematical models and the current computational abilities of the machines....

December 10, 2019 · (Last Modification: April 29, 2022) · Anthony Tan


text:
# Conformal equivalence of resistance

Link to the question in the physics portal.

I'm currently working on a system that uses a logarithmic and a Schwarz-Christoffel transformation to calculate the resistance of a specific area. With resistance I mean the resistance that would apply to that area if it were between two equipotential irregular plates (which are defined by their vertices). In my case it is the magnetic reluctance but it is analogous to electrical resistance.

Right click on the image and choose the appropriate option to view a larger image or click here

I have multiple areas of the canonical domain that have to be translated to a resistance value in the physical domain(with the intermediate step of a logarithm). Is the resistance (defined analogous to electrical resitance) that can be calculated in the homogeneous field of the canonical domain directly related to the resitance in the phyisical domain? The geometry of a resistor in the canonical domain is a simple rectangle formed from four prevertices. In the physical domain it is far more complicated (see figure).

The measure of resistance can be formulated with the help of an "equivalent length". Could it be that this $\Lambda$ is not affected by the conformal tranformations? \begin{align} &R=\rho\dfrac{1}{\Lambda}\\ &\text{cylinder gives: } &\Lambda_\text{cyl}=\frac{2\pi \ell}{\log\frac{r_2}{r_1}}\\ &\text{cuboid gives: } &\Lambda_\text{cub}=\dfrac{A}{\ell} \end{align}

I have done some calculations by discretising the physical domain and comparing it to the resistance generated in the canonical domain. The results were comparable, maybe because of accident, maybe because of a strong mathematical implication, I can't say.

Question in short: Can the prevertices that define a rectangle in the canonical domain define the resistance value of the mapped rectangle in the phyisical domain. (With the intermidate step of a logarithm)

I use the SC-Toolbox for Matlab created by Tobin A. Driscoll.

Lets say that we have a rectangle which has been rotated and translated (Möbius transformations if you want) so that its sides are perpendiculary oriented to the real and imaginary axis, one of its vertices is (0,0) and that it is in the 1st quadrant. We have to calculate its resistance.

Lets say the opposite vertex of (0,0) is $z=x+iy=|z|e^{i\varphi}$, the imaginary part of $z$ represents the length of the path, while the real part represents the projected area(the area is only visible in 3D, we are looking at it in 2D). We write $R'$ because of the 2D representation.

The resistance of this rectangle will therefore be: \begin{align} R'=\rho \frac{\mathfrak{Im}(z)}{\mathfrak{Re}(z)} \end{align} Which can be written as: \begin{align} R'&=\rho \frac{|z|\sin\varphi}{|z|\cos\varphi}=\rho \frac{\sin\varphi}{\cos\varphi}\\ R'&=\rho\tan\varphi \end{align}

The thing that the resistance is only dependent on the angle $\varphi$ and the definition of conformal transformations as angle preserving, provoked me to ask this question. I found a lot about space being conformally invariant, but in this case it is a function that is discussed.

I'm not sure how to proceed. I can prove that the transformations(maps) are conformal, but I don't know what to do about a function in the canonical space.

• You should give a mathematical definition of resistance (at least, a link to one) since otherwise your question does not make sense. – Moishe Kohan Apr 23 '14 at 17:33
• I edited it, I hope it is clearer now what I was talking about. Is it now comprehensible, or should I add further explanations? – WalyKu Apr 23 '14 at 17:54
• I still do not see a mathematical definition. Wikipedia also does not provide one. However, in view of en.wikipedia.org/wiki/Extremal_length (the very end of the article), I think, resistance defined via two disjoint continua $C_1, C_2$ in the plane, is just the extremal length of the family of paths which connect $C_1$ to $C_2$. If so, it is well-known to be conformally invariant, see the same link. – Moishe Kohan Apr 23 '14 at 18:17
• I read it up. I'm no mathematician but I'm doing my best. As far as I comprehend it there should actually be two families of paths. One represents the lengths(green lines in the figure), and the other one represents the projected area(red lines)[the system is a 2D representation of a 3D elongated object]. The aproach to the calculation is mathematically sounder using this. I used to map the canonical domain point by point back to the phyisical domain and calculate the euclidean distances between them to get the resistance. It doesn't give a direct relation to the Resistance(max comment length) – WalyKu Apr 24 '14 at 13:37
• My suggestion is that you ask your advisor for a reading material on mathematical definition of resistance: Without such definition you will never be able to compute it. Also, without such definition, your question at MSE cannot be answered (one cannot prove/disprove conformal invariance of a function which is not defined). You can also ask for help at physics.stackexchange, maybe they can direct you towards a reference. – Moishe Kohan Apr 24 '14 at 17:30

Given what you wrote and what this wikipedia page says (references can be found there as well), here is, I think, the mathematical interpretation (and answer to) of your question. However, in order to understand it you need to know more math than you do (you need some basic complex analysis, general topology and differential geometry). In short, yes, resistance is conformally invariant.

We consider open connected bounded subsets $\Omega$ of the complex plane ${\mathbb C}$. Fix in the boundary of $\Omega$ in two disjoint subsets $A$ and $B$. I will assume that the sets $A$ and $B$ are closed and connected. The triple $Q=(\Omega, A, B)$ is a condenser. (The physical meaning here is that the domain $\Omega$ is filled with a material that has constant conductivity, which is set to be 1. The unit electric charge is applied along the continuum $B$.)

Example. $\Omega$ is the open rectangle $$R_{a,b}=\{(x,y): 0<x<a, 0<y<b\}.$$ Set $$A=\{(0, y): 0\le y\le b\}, B= \{(a, y): 0\le y\le b\}$$ (the vertical sides of the rectangle).

Define the capacity of a condenser
$$c(Q)=\inf \{E(u)| \ u:\bar{\Omega}\to {\mathbb R}, u|_A=0, u|_B=1\}$$
where we assume that functions $u$ are continuously differentiable in $\Omega$ and are continuous on its closure $\bar\Omega$. The quantity $E(u)$ is the energy: $$E(u)=\iint_{\Omega} |du|^2 dxdy,$$ where $du$ is the differential of $u$. It is known that infimizing sequences $u_n$ for $c(Q)$ converge (uniformly on compacts in $\Omega$) to harmonic functions $h$ on $\Omega$ so that $$E(h)=c(Q)= \lim_{n\to \infty} E(u_n).$$ However, the harmonic functions $h$ (in general) do not extend continuously to the closure $\bar\Omega$. Lastly, define the resistence of the condenser as $$r(Q)= \frac{1}{c(Q)}.$$ This quantity is also know as the extremal length of the family of paths $\gamma: (0,1)\to \Omega$, such that $\gamma(t)$ accumulates to $A$ as $t\to 0$ and accumulates to $B$ as $t\to 1$.

Example. Suppose that $\Omega=R_{a,b}$, a rectangle with the sets $A$ $B$ chosen to be its vertical sides as above. Then $$r(\Omega, A, B)= \frac{a}{b}.$$ The extreme function in this case is $h(x,y)=x/a$.

It is now well-known that for a general condenser $Q$, the capacity $c(Q)$ is a conformal invariant of $Q$, in the following sense. Suppose that $f: \bar\Omega\to \bar\Omega'$ is a homeomorphism which restricts to a conformal map $f: \Omega\to \Omega'$; define the new capacitor $$Q'=(\Omega', f(A), f(B)).$$ Then $c(Q)=c(Q')$. The reason for this is the following. One can define energy of functions with respect not only to the Euclidean metric $g_0=dx^2 + dy^2$ but to an arbitrary Riemannian metric $g$ on $\Omega$: $$E(u,g)= \iint_{\Omega} |d u|_g^2 dA_g,$$ where $dA_g$ is the area form of $g$ and $|du|_g$ is the norm of the differential of $u$ with respect to the Riemannian metric $g$. Then for a function $v: \Omega'\to {\mathbb R}$ and $u=v\circ f$, we clearly have $$E(u, g)= E(v),$$ where $g=f^*(dx^2 + dy^2)$. Since $f$ is conformal in $\Omega$, the metric $g$ is conformal, i.e. $$g= \lambda(z) g_0,$$ where $\lambda$ is a positive continuous function on $\Omega$. Lastly, one observes that $$E(u, g)= E(u)$$ for every function $u$ on $\Omega$, since the $$dA_g= \lambda^2 dxdy$$ while $$|du|^2_g= \lambda^{-2} |du|^2.$$ The two scalar functions $\lambda^2, \lambda^{-2}$ cancel each other and integrals are the same since you are integrating equal forms.

Since capacity is conformally invariant, so is reciprocal $r(Q)$.

• Thank you for the answer, I value your effort a lot. I must go trough it in detail. – WalyKu Apr 25 '14 at 20:11
• I think I got it roughly. I still need to get a few details settled though. This is indeed a very interesting field. Thanks! – WalyKu Apr 28 '14 at 12:38


text:
## Basic idea

After the nice_violin() function, here’s how to make nice scatter plots easily!

Let’s first load the demo data. This data set comes with base R (meaning you have it too and can directly type this command into your R console).

data("mtcars")
head(mtcars)
##                    mpg cyl disp  hp drat    wt  qsec vs am gear carb
## Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
## Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
## Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
## Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
## Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
## Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1

### Getting started

Load the rempsyc package:

library(rempsyc)

Note: If you haven’t installed this package yet, you will need to install it via the following command: devtools::install_github("rempsyc/rempsyc"). Package devtools is necessary for this command. If you haven’t installed devtools yet, install it via install.packages("devtools").

nice_scatter(data = mtcars,
predictor = "wt",
response = "mpg")

### Save a high-resolution image file to specified directory
ggsave('nice_scatterplothere.tiff', width = 7, height = 7, unit = 'in', dpi = 300,
path = "D:/R treasures/")
# Change the path to where you would like to save it.
# If you copy-paste your path name, remember to use "R" slashes ('/' rather than '\').
# Also remember to specify the .tiff extension of the file.

Pro tip: Change .tiff for .pdf or .eps for scalable vector graphics for high-resolution submissions to scientific journals!

## Customization

### Change x- and y- axis labels

nice_scatter(data = mtcars,
predictor = "wt",
response = "mpg",
ytitle = "Miles/(US) gallon",
xtitle = "Weight (1000 lbs)")

### Have points “jittered”

Meaning randomly moved around a bit to prevent overplotting (when two or more points overlap, thus hiding information).

nice_scatter(data = mtcars,
predictor = "wt",
response = "mpg",
has.jitter = TRUE)

### Change the transparency of the points

nice_scatter(data = mtcars,
predictor = "wt",
response = "mpg",
alpha = 1) # default is 0.7

### Remove points

nice_scatter(data = mtcars,
predictor = "wt",
response = "mpg",
has.points = FALSE,
has.jitter = FALSE)

nice_scatter(data = mtcars,
predictor = "wt",
response = "mpg",
has.confband = TRUE)

### Set x- and y- scales manually

nice_scatter(data = mtcars,
predictor = "wt",
response = "mpg",
xmin = 1,
xmax = 6,
xby = 1,
ymin = 10,
ymax = 35,
yby = 5)

### Change plot colour

nice_scatter(data = mtcars,
predictor = "wt",
response = "mpg",
colours = "blueviolet")

### Add correlation coefficient to plot and p-value

nice_scatter(data = mtcars,
predictor = "wt",
response = "mpg",
has.r = TRUE,
has.p = TRUE)

### Change location of correlation coefficient or p-value

nice_scatter(data = mtcars,
predictor = "wt",
response = "mpg",
has.r = TRUE,
r.x = 4,
r.y = 25,
has.p = TRUE,
p.x = 5,
p.y = 20)

### Plot by group

nice_scatter(data = mtcars,
predictor = "wt",
response = "mpg",
group = "cyl")

### Use full range on the slope/confidence band

nice_scatter(data = mtcars,
predictor = "wt",
response = "mpg",
group = "cyl",
has.fullrange = TRUE)

nice_scatter(data = mtcars,
predictor = "wt",
response = "mpg",
group = "cyl",
has.legend = TRUE)

### Change order of labels on the legend

nice_scatter(data = mtcars,
predictor = "wt",
response = "mpg",
group = "cyl",
has.legend = TRUE,
groups.order = c(8,4,6)) # These are the levels of 'mtcars$cyl', so we place lvl 8 first, then lvl 4, etc. ### Change legend labels nice_scatter(data = mtcars, predictor = "wt", response = "mpg", group = "cyl", has.legend = TRUE, groups.labels = c("Weak","Average","Powerful")) # Warning: This applies after changing order of level **Warning**: This only changes labels and applies after changing order of level! Always use groups.order first if you also need to use groups.labels! This is to make sure to have the right labels for the right groups! ### Add a title to legend nice_scatter(data = mtcars, predictor = "wt", response = "mpg", group = "cyl", has.legend = TRUE, legend.title = "Cylinders") ### Plot by group + manually specify colours nice_scatter(data = mtcars, predictor = "wt", response = "mpg", group = "cyl", colours = c("burlywood", "darkgoldenrod", "chocolate")) ### Plot by group + use different line types for each group nice_scatter(data = mtcars, predictor = "wt", response = "mpg", group = "cyl", has.linetype = TRUE) ### Plot by group + use different point shapes for each group nice_scatter(data = mtcars, predictor = "wt", response = "mpg", group = "cyl", has.shape = TRUE) ### Plot by group, point shapes, line types, legend + no colours (black and white) nice_scatter(data = mtcars, predictor = "wt", response = "mpg", group = "cyl", has.legend = TRUE, legend.title = "Cylinders", has.linetype = TRUE, has.shape = TRUE, colours = rep("black", 3)) ### Putting it all together If you’d like to see all available options at once (a bit long): nice_scatter(data = mtcars, predictor = "wt", response = "mpg", ytitle = "Miles/(US) gallon", xtitle = "Weight (1000 lbs)", has.points = FALSE, has.jitter = TRUE, alpha = 1, has.confband = TRUE, has.fullrange = FALSE, group = "cyl", has.linetype = TRUE, has.shape = TRUE, xmin = 1, xmax = 6, xby = 1, ymin = 10, ymax = 35, yby = 5, has.r = TRUE, has.p = TRUE, r.x = 5.5, r.y = 25, colours = c("burlywood", "darkgoldenrod", "chocolate"), has.legend = TRUE, legend.title = "Cylinders", groups.labels = c("Weak", "Average", "Powerful")) ## Special situation: Add group average There’s no straightforward way to add group average, so here’s a hack to do it. We first have to create a second data set with another “group” that will be used as the average. ### Black line + coloured lines new.Data <- mtcars # This simply copies the 'mtcars' dataset new.Data$cyl <- "Average" # That would be your "Group" variable normally
# And this operation fills all cells of that column with the word "Average" to identify our new 'group'
XData <- rbind(mtcars,new.Data) # This adds the new "Average" group rows to the original data rows

Then we need to create a FIRST layer of just the slopes. We add transparency to the group lines except the group average to emphasize the group average (with the new argument groups.alpha).

(p <- nice_scatter(data = XData,
predictor = "wt",
response = "mpg",
has.points = FALSE,
has.legend = TRUE,
group = "cyl",
colours = c("black", "#00BA38", "#619CFF", "#F8766D"), # We add colours manually because we want average to be black to stand out
groups.order = c("Average","4","6","8"), # We do this to have average on top since it's the most important
groups.alpha = c(1,0.5,0.5,0.5))) # This adds 50% transparency to all lines except the first one (Average) which is 100%

Finally we are ready to add a SECOND layer of just the points on top of our previous layer. We use standard ggplot syntax for this.

library(ggplot2)
p + geom_point(data = mtcars,
size = 2,
alpha = 0.5,
shape = 16, # We use shape 16 because the default shape 19 sometimes causes problems when exporting to PDF
mapping = aes(x = wt,
y = mpg,
colour = factor(cyl),
fill = factor(cyl)))

### Black line, coloured dots

If you’d like instead to still show the group points but only the black average line, you can do the following as first layer:

(p <- nice_scatter(data = mtcars,
predictor = "wt",
response = "mpg",
has.points = FALSE,
has.legend = TRUE, # This argument is important else the next legend won't appear on the second layer!
colours = "black"))

Then to add the points as second layer we do the same as before:

p + geom_point(data = mtcars,
size = 2,
alpha = 0.5,
shape = 16,
mapping = aes(x = wt,
y = mpg,
colour = factor(cyl)))

### Thanks for checking in

Make sure to check out this page again if you use the code after a time or if you encounter errors, as I periodically update or improve the code. Feel free to contact me for comments, questions, or requests to improve this function at https://github.com/rempsyc/rempsyc/issues. See all tutorials here: https://remi-theriault.com/tutorials.


text:
## anonymous one year ago A reaction uses up 20 grams of reactant at a rate of 0.50 grams/s. How much time will it take for all of the reactant to be used up? 0.50 s 20 s 40 s 80 s

1. Rushwr

What do u think the answer is ?

2. Rushwr

Looking at the rate's unit. you can simply see that the rate here was calculated by dividing grams by seconds. $Rate = \frac{ grams(reacted amount ) }{ seconds (time taken \to react) }$ so the answer would be 40 s


text:
## § Crash course on DCPO: formalizing lambda calculus

In lambda calculus, we often see functions of the form $\lambda x \rightarrow x(x)$. We would like a way to associate a "natural" mathematical object to such a function. The most obvious choice for lambda calculus is to try to create a set $V$ of values which contains its own function space: $(V \rightarrow V) \subseteq V$. This seems to ask for a set whose cardinality is such that $|V|^|V| = |V|$, which is only possible if $|V| = 1$, ie $V$ is the trivial set $\{ * \}$. However, we know that lambda calculus has at least two types of functions: functions that terminate and those that don't terminate. Hence the trivial set is not a valid solution. However, there is a way out. The crucial insight is one that I shall explain by analogy:
• We can see that the cardinality of $\mathbb R$ is different from the cardinality of the space of functions over it, $\mathbb R \rightarrow \mathbb R$.
• However, "the set of all functions" isn't really something mathematicians consider. One would most likely consider "the set of all continuous functions" $\mathbb R \rightarrow \mathbb R$.
• Now note that a function that is continuous over the reals is determined by its values at the rationals . So, rather than giving me a continus function $f: \mathbb R \rightarrow \mathbb R$, you can give me a continuous function $f': \mathbb Q \rightarrow \mathbb R$ which I can Cauchy-complete, to get a function $\texttt{completion}(f') : \mathbb R \rightarrow \mathbb R = f$.
• Now, cardinality considerations tell us that:
$|\mathbb R^\mathbb Q| = (2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0 \cdot \aleph_0} = 2^\aleph_0 = |R|$
• We've won! We have a space $\mathbb R$ whose space of continuous functions $\mathbb R \rightarrow \mathbb R$ is isomorphic to $\mathbb R$.
• We bravely posit: all functions computed by lambda-calculus are continuous! Very well. This leaves us two questions to answer to answer: (1) over what space? (2) with what topology? The answers are (1) a space of partial orders (2) with the Scott topology

#### § Difference between DCPO theory and Domain theory

• A DCPO (directed-complete partial order) is an algebraic structure that can be satisfied by some partial orders. This definition ports 'continuity' to partial orders.
• A domain is an algebraic structure of even greater generality than a DCPO. This attempts to capture the fundamental notion of 'finitely approximable'.

#### § Posets, (least) upper bounds

• A partial order $(P, \leq)$ is a set equipped with a reflexive, transitive, relation $\leq$ such that $x \leq y$ and $y \leq x$ implies $x = y$.
• A subset $D \subseteq P$ is said to have an upper bound $u_D \in P$ iff for all $d \in D$, it is true that $d \leq u_D$.
• An upper bound is essentially a cone over the subset $D$ in the category $P$.
• A subset $D \subseteq P$ has a least upper bound $l_D$ iff (1) $l_D$ is an upper bound of $D$, and (2) for every upper bound $u_D$, it is true that $l_D \leq u_D$
• Least upper bounds are unique, since they are essentially limits of the set $D$ when $D$ is taken as a thin category

#### § Directed Sets

• A subset $D \subseteq P$ is said to be directed iff for every finite subset $S \subseteq D$, $S$ has a upper bound $d_S$ in $D$. That is, the set $D$ is closed under the upper bound of all of its subsets.
• Topologically, we can think of it as being closure , since the upper bound is sort of a "limit point" of the subset, and we are saying that $D$ has all of its limit points.
• Categorically, this means that $D \subseteq P$ taken as a subcategory of $P$ has cones over all finite subsets. (See that we do not ask for limits/least upper bounds over all finite subsets, only cones/upper bounds).
• We can think of the condition as saying that the information in $D$ is internally consistent: for any two facts, there is a third fact that "covers" them both.
• slogan: internal consistency begets an external infinite approximation.
• QUESTION: why not ask for countable limits as well?
• QUESTION: why upper bounds in $D$? why not external upper bounds in $P$?
• QUESTION: why only upper bounds in $D$? why not least upper bounds in $D$?
• ANSWER: I think the point is that as long as we ask for upper bounds, we can pushforward via a function $f$, since the image of a directed set will be directed.

#### § Directed Complete Partial Order ( DCPO)

• A poset is said to be directed complete iff every directed set $D \subseteq P$ has a least upper bound in $P$.
• Compare to chain complete, CCPO, where every chain has a LUB.
• QUESTION: why not postulate that the least upper bound must be in $D$?
• In a DCPO, for a directed set $D$, denote the upper bound by $\cup D$.

#### § Monotonicity and Continuity

• A function $f: P \to Q$ between posets is said to be monotone iff $p \leq p'$ implies that $f(p) \leq f(p')$.
• A function $f: P \to Q$ is continuous, iff for every directed set $D \subseteq P$, it is true that $f(\cup D) = \cup f(D)$.
• This has the subtle claim that the image of a directed set is directed.

#### § Monotone map

• A function from $P$ to $Q$ is said to be monotone if $p \leq p' \implies f(p) \leq f(p')$.
• Composition of monotone functions is monotone.
• The image of a chain wrt a monotone function is a chain.
• A monotone function need not preserve least upper bounds . Consider:
$f: 2^{\mathbb N} \rightarrow 2^{\mathbb N} f(S) \equiv \begin{cases} S & \text{S} is finite \\ S U \{ 0 \} &\text{S is infinite} \end{cases}$
This does not preserve least-upper-bounds. Consider the sequence of elements:
$A_1 = \{ 1\}, A_2 = \{1, 2\}, A_3 = \{1, 2, 3\}, \dots, A_n = \{1, 2, 3, \dots, n \}$
The union of all $A_i$ is $\mathbb N$. Each of these sets is finite. Hence $f(\{1 \}) = \{1 \}$, $f(\{1, 2 \}) = \{1, 2\}$ and so on. Therefore:
$f(\sqcup A_i) = f(\mathbb N) = \mathbb N \cup \{ 0 \}\\ \sqcup f(A_i) = \sqcup A_i = \mathbb N$

#### § Continuous function

• A function is continous if it is monotone and preserves all LUBs. This is only sensible as a definition on ccpos, because the equation defining it is: lub . f = f . lub, where lub: chain(P) \rightarrow P. However, for lubto always exist, we need P to be a CCPO. So, the definition of continuous only works for CCPOs.
• The composition of continuous functions of chain-complete partially ordered sets is continuous.

#### § Fixpoints of continuous functions

The least fixed point of a continous function $f: D \rightarrow D$ is:
$\texttt{FIX}(f) \equiv \texttt{lub}(\{ f^n(\bot) : n \geq 0 \})$

#### § $\leq$ as implication

We can think of $b \leq a$ as $b \implies a$. That is, $b$ has more information than $a$, and hence implies $a$.

#### § References

• Semantics with Applications: Hanne Riis Nielson, Flemming Nielson.


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# Is there a reason it is so rare we can solve differential equations?

Speaking about ALL differential equations, it is extremely rare to find analytical solutions. Further, simple differential equations made of basic functions usually tend to have ludicrously complicated solutions or be unsolvable. Is there some deeper reasoning behind why it is so rare to find solutions? Or is it just that every time we can solve differential equations, it is just an algebraic coincidence?

I reviewed the existence and uniqueness theorems for differential equations and did not find any insight. Nonetheless, perhaps the answer can be found among these?

A huge thanks to anyone willing to help!

Update: I believe I have come up with an answer to this odd problem. It is the bottom voted one just because I posted it about a month after I started thinking about this question and all you’re inputs, but I have taken all the responses on this page into consideration. Thanks everyone!

Let’s consider the following, very simple, differential equation: $$f′(x)=g(x)f'(x) = g(x)$$, where $$g(x)g(x)$$ is some given function. The solution is, of course, $$f(x)=∫g(x)dxf(x) = \int g(x) dx$$, so for this specific equation the question you’re asking reduces to the question of “which simple functions have simple antiderivatives”. Some famous examples (such as $$g(x)=e−x2g(x) = e^{-x^2}$$) show that even simple-looking expressions can have antiderivatives that can’t be expressed in such a simple-looking way.