User talk:GFauxPas/Archive1

Convergence and other principles of analysis
Currently I am a teaching assistant for an advanced analysis course, so if you have any questions regarding real (multidimensional or not) analysis, feel free to drop a note on my talk page. When you eventually get there, I might be able to help out on Complex Analysis as well. --Lord_Farin 14:27, 23 October 2011 (CDT)

Awesome, thanks a lot --GFauxPas 14:33, 23 October 2011 (CDT)


 * Thanks for your explanation LF. Just curious, what's the cardinality of $\R^\R$? --GFauxPas 07:18, 17 January 2012 (EST)


 * There are nice rules for cardinalities like $\left|A^B\right| = |A|^{|B|}$, so if we let $c = |\R|, \omega = |\N|$, then $\left|\R^\R\right| = c^c = \left(2^\omega\right)^c = 2^{(\omega\times c)} = 2^c$. So it is 'only' $2^c$. --Lord_Farin 09:38, 17 January 2012 (EST)

Notation
I note from your front page you're setting up some copypasta for yourself. Before you go too far down that route, pls note the following:

1. The raw symbols ≡, · and Δ and so on are never used on ProofWiki. The $\LaTeX$ code is always used: $\equiv, \cdot, \Delta$ (or when appropriate $\triangle$ and its variants).

2. For "defined as" we use $:=$ as this is a specific symbol meaning "is defined as". The $\equiv$ symbol has plenty of other meanings and it is best kept for those.

Hope this is OK. --prime mover 16:29, 24 October 2011 (CDT)

A while back I wrote a proof that all integers are even or odd. The reason I wrote it was just to practice mathematical induction. The theorem itself seems rather unimportant, and so I don't see a reason to make a page for it, but at someone's request I can make it. --GFauxPas 15:56, 4 November 2011 (CDT)
 * I have a feeling that one might already be up, but I don't think it was proved by induction. Can't remember and I don't feel like looking at the moment. --prime mover 17:24, 4 November 2011 (CDT)

Seeing as how we're using an uncommon notation for intervals to avoid ambiguity, what's the defining criterion for whether we use an atypical "better" notation or not? For example, the though the standard notation for function inverse is $f^{-1}$, that notation is the same as that of the multiplicative inverse, very different things. There's $f^\gets$ and $\breve f$ which aren't ambiguous, but they're not common. --GFauxPas 16:41, 8 November 2011 (CST)


 * Good question. There are many notations for intervals, and none of them are very good because it's easy to mistake two numbers separated by a comma for all sorts of other usages. $[a..b]$ is not common but it's completely unambiguous and has a precedent in computer languages, so I'm sort of expecting it to catch on. Getting mathematicians to change their notation, though, is not easy.
 * As for the inverse function notation, "generally speaking" you don't mistake $f^{-1}$ for a multiplicative inverse because the contexts are different. The $f^{\gets}$ notation has been noted on the page for inverse mapping so I suppose we could start using it, if you particularly like it. --prime mover 17:05, 8 November 2011 (CST)


 * I think I oppose to that. The notation for the interval isn't really ambiguous, even if you never saw it before, the meaning is clear. With $f^{\gets}$ I am having the hunch that it will create unnecessary fuss. But that's me, and probably an instantiation of the notation change thing... --Lord_Farin 17:09, 8 November 2011 (CST)

Lord_Farin, just wanted to let you know that I figured out your explanation and PW's approach here Talk:Fundamental_Theorem_of_Calculus/Alternative_Second_Part_Proof, thank you! --GFauxPas 22:19, 5 November 2011 (CDT)


 * Glad I put up sensible stuff. HTH --Lord_Farin 09:01, 6 November 2011 (CST)

Talk page etiquette
I hate having to keep calling you out like this, I sort of hope you will pick up the conventions as you see them being used.

However, it's probably worth mentioning the usual protocol in talk pages, when there's a conversation going on.

You may have noticed that when you start a line with a colon it acts like a tabulator, that is, indents the line by a fixed and noticeable amount. Starting with more colons indents the line further.

Anyway, in a talk page the convention is to start each reply with one colon more at the start of the line. In that way it can be easily worked out who said what, and when they said it, and what they are replying to. Only when the indentation becomes too much (usually about 7 or 8 colons in) is the conversation reverted to the far left hand side of the page again. Usually this does not happen on ProofWiki because issues are usually solved within two or three postings. --prime mover 12:34, 11 November 2011 (CST)
 * Noted. Sorry to disappoint. I'll cop out and play the "not sleeping enough" card. --GFauxPas 12:41, 11 November 2011 (CST)

Sizing of Images
When you are creating images, there is no need to create different versions depending on what size you want it. All you do in the code is add " px" to tell the browser how many pixels (high? wide? Don't know) to render the image.

In order to keep the images folder tidy (it can't be organized easily) I will update the cone page to use File:ConeVolumeProof.png and size it appropriately, then delete File:ConeVolumeProof2.png.

--prime mover 04:45, 20 November 2011 (CST)

Trig integrals
Good work on all these trigonometric integrals. It's something I've been meaning to get round to doing but haven't done yet. We have the opportunity of providing the best repository of integrals on the internet. --prime mover 12:06, 24 November 2011 (CST)
 * My pleasure. I don't like proofs like the secant proof where the steps come out of nowhere, but it is what it is. Thanks for the compliment. --GFauxPas 12:09, 24 November 2011 (CST)
 * Understood, and I'll stick to x if you like x better, I apologize for wasting your time. As I said, I'm a slow learner at some things. I'm still trying to measure what needs to be said and what doesn't. --GFauxPas 14:06, 24 November 2011 (CST)
 * Okay, here's a general rule: if it's in place using notation you're not happy with, but it's sound, then leave well alone unless there's a good reason not to (e.g. it's incorrect). If you prefer using theta in the proofs you work on, fair enough, but if and when we expand the understanding to take on board complex numbers we might take the opportunity of amending the notation again. Of course, if it has a proofread and/or tidy template (or it's otherwise new by a contributor who has not yet assimilated the house style) then the above does not apply. Of course, if you really don't agree with the presentational style, raise the question in the discussion page. That's always an option. --prime mover 14:58, 24 November 2011 (CST)
 * Thank you for your patience with me, Prime.mover. I don't care what symbols I use, and if it makes you happy you can change any notation I use, I don't care. I just have accustomed myself to using theta for trig, I don't mind using x or whatever if you like it better. The main reason I edited the cosine page there was because the proof didn't address the sign of the sine. Also, can I leave it as a given that differentiation and integration are linear operators? I've been putting it in the proofs, but it's left implied in most of the proofs I see outside of PW. --GFauxPas 15:08, 24 November 2011 (CST)
 * I rarely bother to note the derivative of a minus, because it all follows with simple algebra anyway. If there's a specific need to invoke a complicated linear combination, then perhaps note that, but for a simple constant multiple I would not. Note the corollary to the derivative of the exponential which includes the drv. of $e^{cx}$ - you might want to add something similar as a corollary in the trig functions. In fact, drv of $\sin (ax + b) = a \cos (ax + b)$ is a really useful corollary, so when we get onto complicated substitutions in the messy integrations involving quadratics, you just need to invoke that page and it saves a lot of extra work on the substitution. --prime mover 16:26, 24 November 2011 (CST)

Certainly the integrals of tangent, secant etc. are worth adding, but would you like me to add pages for the integrals of functions like $\sec x \tan x$, $\sec^2 x$? --GFauxPas 13:40, 25 November 2011 (CST)
 * My rule of thumb is: a result is reported in a text book as worthwhile results then they can probably go in. Otherwise, if they're needed in the course of a more complicated proof then we could add them when they were needed. Otherwise I wouldn't bother. --prime mover 17:23, 25 November 2011 (CST)
 * It might be feasible to compute the indefinite integrals for $\cos^n x\sin^m x$ and $m,n\in\Z$. That page would be worthwhile I think as a reference table, and would cover all of these. --Lord_Farin 17:32, 25 November 2011 (CST)
 * That's one that's been on my own list to do in due course - but there's lots of other fiddly stuff I want to get sorted out while I have the particular books in front of me. Feel free to get there first ... --prime mover 17:44, 25 November 2011 (CST)

Grammatical note
Reinstating this section because its still relevant.

I see you starting lines with a capital letter where it does not need one. Here is an example:


 * The hyperbolic tangent function is defined on the complex numbers as:


 * $\tanh: X \to \C$:


 * $\displaystyle \tanh z := \dfrac {\sinh z}{\cosh z}$


 * Where $\sinh$ is the hyperbolic sine, $\cosh$ is the hyperbolic cosine, and $X = \{ z : z \in \C, \ \cosh z \ne 0 \}$."

The word "where" should not have a capital letter. The above is all (technically) one sentence, like:


 * "The best food is:
 * FISH AND CHIPS
 * where chips are made of deep-fried potato."

See? As "where" is part of the same sentence, it does not start with a capital letter.

I have been changing them consistently where I've seen them, hoping you'll pick it up by following examples, but now I see you changing one in the other direction, I have to mention it.

I understand that Microsoft make things complicated by automatically making the first letter after every new line / return start with a capital, but Microsoft are cracked.


 * Your comment in the edit page about "committed to memory", you might want to amend your subroutines to ensure it's your hard drive not your RAM it gets committed to, as I notice the same is being done on your sandbox page for Riemann Sum. --prime mover 00:19, 25 November 2011 (CST)

email citations
Citing email conversations as source works. I'm going to attempt to give a ruling, as there's no obvious reason why the general technique of citing private correspondence can't be used - within reason.

The idea of a citation is that it allows people to go back to the original source work and see what the original says. I've seen exceptions where books have been written and the citation is "private correspondence" so there's a precedent. But unless there is a unique point mentioned in the page that you are unable to find anywhere in the literature at your disposal then as a last resort you can cite your email conversation. Before that, you want to say to your correspondent: "Where did you get that from?" If he's thought it up himself, obviously you need his permission to use it, but in that case you can cite "personal correspondence" if you are indeed the only person he has shared this with. If he can't remember where he got the information, then google for it, and if you still can't find a citation for that precise piece of information, don't bother citing it at all.

If this piece of information is notable enough, then it may well merit its own page. In the case of the classical probability model (good page, btw, don't worry about notation conventions) I'd be prepared to believe that the email conversation probably (no pun intended) didn't contain anything that can't be found in books.

Citations shouldn't need you to bust a gut. I think what we're doing at the moment is adequate. Your idea of linking to Khanacademy was inspired.

While I'm about it, if you know stuff about [P. Arvanites] (and he consents to this knowledge being in the public domain), feel free to put a page up on ProofWiki and so you'll be able to link to him. --prime mover 14:45, 1 December 2011 (CST)

Gemignani
According to my research, the Gemignani book was first published 1970. I don't know which edition the 1998 one is, but it will either be 2nd or 3rd, or whatever. You might want to restyle the entry you've currently got so as to match how we've done, for example, Books/Roland E. Larson/Calculus. --prime mover 14:10, 7 December 2011 (CST)
 * Okay, how do I fix this? The new edition has the exact same title, no sub-title saying 2nd edition. First copyright is 1970, second copyright in 1998. In the beginning of the book it says:
 * "This Dover edition, first published in 2006, is an unabridged republication of the work originally published in 1970 by Addison-Wesley Publishing Company, Inc., Reading, Massachusetts. The Table of Integrals which appeared on the inside front and back covers in the original edition have been moved to pages 355-358 in the Dover edition."--GFauxPas 14:24, 7 December 2011 (CST)
 * If the one you've got is the unabridged 1970 edition, then make all your references to that 1970 edition. What I've been trying to do is make sure we find the original editions if we can (it's more "absolute", if you get my drift), rather than just the latest one we happen to have when we wrote the page, which is more arbitrary. --prime mover 14:57, 7 December 2011 (CST)
 * I don't have the unabridged 1970 edition, I have the "Dover edition" --GFauxPas 15:47, 7 December 2011 (CST)
 * Yes but the Dover edition (as quoted by you above) says "This Dover edition, first published in 2006, is an unabridged republication of the work originally published in 1970 by Addison-Wesley Publishing Company, Inc., Reading, Massachusetts." That's close enough for any references made to your edition of this work (as long as you don't directly reference page numbers) to be the same as references that would be made to the 1970 work. Looks, walks, quacks like duck, is duck. --prime mover 15:51, 7 December 2011 (CST)

Change to MathWorld citation template
I noticed (based on One-to-One and Strictly Between) that some pages on MathWorld are credited to different authors from Eric Weisstein, and so require that author to be included in the citation.

I have fixed the template (which is now "MathWorld" not "Mathworld", that's just me tidying up) so as to be able to include the author (which, if not given, defaults to the "Weisstein, Eric W." format as per normal).

What you need to do is add "author=author-name" and "authorpage=author-pagename" where "author-name" is the displayname of the author and "author-pagename" is the name of the html file on MathWorld (not including the full path, not including the extension).

An example:

which gives:
 *  



If the page is given as written by "Weisstein, Eric W." then you should not add the "author" and "authorpage" tags.

I have included this info in the usage section of the Template:MathWorld page itself, but I'm bringing it to your attention because I know you've been active in using it.

Chx. --prime mover 02:55, 31 December 2011 (CST)

Limit of a Sequence
Hello PW friends, what is the name of this theorem? I wanted to see the proof but I haven't had luck finding it.

If $\displaystyle \lim_{x \to +\infty} \left \langle {x_n} \right \rangle$ exists and $f: \R \to \R$ agrees with $\left \langle {x_n} \right \rangle$ for all $x$ in the domain of $\left \langle {x_n} \right \rangle$ then $\displaystyle \lim_{x \to +\infty} f(x)$ exists and the limits are the same. Thanks --GFauxPas 14:22, 11 January 2012 (EST)

fixed variables
LF, you said "Fixed variables cannot be quantified over". Can you please elaborate on that a bit? What's wrong with a statement like: $\exists 2 \in \N: 1 + 1 = 2$? --GFauxPas 06:35, 23 January 2012 (EST)
 * Nothing (at least, not for me; I am used to axiomatic set theory). But you wrote something like: 'Let $M, N \in \R$. Then $\forall M \in \R$...' which is of the category I meant. You produce a de facto constant (though arbitrary), and then quantify over it; this is void. --Lord_Farin 07:18, 23 January 2012 (EST)
 * Sorry, that was my fault: I moved the M and N into the categorical statement without removing their declaration above. --prime mover 08:40, 23 January 2012 (EST)

don't kid yourself
they are not cool. :-) - primemover 1/22/2012

intuition
Are people all right with my "intuition" sections, like I put here and here? --GFauxPas 06:54, 23 January 2012 (EST)
 * I am. I think some theorems benefit from a sort of colloquial phrasing of the result; this can prevent annoying misconceptions and confusion. --Lord_Farin 07:14, 23 January 2012 (EST)
 * I think so, as long as they are grammatical enough and don't use too much colloquial language. Not that colloquialism is bad in itself, but it can grate a little, particularly to those whose culture is not the same as the one from which those colloquialisms originate. For example, I can't stand "... and we're done" because I have a problem with using "done" to mean "finished". Possibly just my problem, but then others may also share this. --prime mover 08:44, 23 January 2012 (EST)

Tarski Geometry
I have written the Axiom:Upper Dimensional Axiom page. Make sure to disregard my previous comment about empty models. Every treatise on model theory requires specifically that a model be non-empty. In any case, refrain from quantifying over a variable that has already been quantified over. --Lord_Farin 04:10, 24 January 2012 (EST) Furthermore, I have rephrased the axiom pages to actually present the material more clearly by using some more words. --Lord_Farin 04:22, 24 January 2012 (EST)
 * Aah okay, I just couldn't think of a way to present "for every x, if x exists...", so I was seduced into quantifying bound variables. I will make sure to make pages for the rest of the axioms more like your improved versions. --GFauxPas 06:49, 24 January 2012 (EST)
 * Remind me, when I'm back on Planet ProofWiki on a more permanent basis, to put a template together to define the citation for these works. For a start it needs to include the authors. --prime mover 17:47, 24 January 2012 (EST)

I recall you had questions about Equidistance Independent of Betweenness, but can't find them anymore. Could you post them again (here)? --Lord_Farin 13:14, 26 January 2012 (EST)
 * I deleted it because I figured it out. Or more accurately, I almost figured it out, and I want to see if I can finish figuring it out on my own. My question was why Padoa's method works. Is it something like this?:

Assume that it is possible to define $\equiv$.

But as both of these models have the same primitive terms, i.e. precisely the ones we are asserting stuff about, it might be that we have to make $\equiv$ a primitive term? --GFauxPas 13:27, 26 January 2012 (EST)

The theorem states that you cannot define $\equiv$ from first-order logic with $\mathsf B$. So there is no formula $\phi$, containing only first-order logic and the ternary relation $\mathsf B$, with free variables $abcd$ such that $\forall abcd: ab \equiv cd \iff \phi(abcd)$ is a theorem (i.e., true in all models).

This is because fixing an interpretation for $\mathsf B$ in a model apparently does not fix $\equiv$. Therefore, $\equiv$ cannot be defined as such a formula (or by the standard rules of model theory, we would have already a unique interpretation in terms of $\phi$. But we have two interpretations). Apparently it is *not* proved that no formula in mentioned language (FO-Logic + $\mathsf B$) can satisfy $\forall abcd: ab \equiv cd \iff \phi(abcd)$ in *some* model. Just that no $\phi$ can suffice for all models at once.

Gah, I am confusing myself by thinking that there might be two formulae $\phi, \psi$ as above that both satisfy all axioms for all models... In any case, there is no *unique* $\phi$ that does the trick for all models. Just thinking out loud now. --Lord_Farin 13:47, 26 January 2012 (EST)

Vector Arrows
Book: Linear Algebra, 3rd edition, by John B Fraleigh and Raymond A. Beauregard.

Context: $\R^n$ considered as a euclidean space.

O = $\mathbf{0}$.

Visual representation in 3-space: Where the x,y,z axes intersect.

Here's the juicy part, I'll paraphrase some parts.

We are accustomed to visualizing an ordered pair or triple as a point in the plane or in space and denoting it geometrically by a dot...Physicists have found another very useful geometric interpretation in their consideration of forces acting on a body...[stuff about magnitude, direction]...It is natural to represent a force by an arrow...such an arrow is a force vector.

Using a rectangular coordinate system in the plane, note that if we consider a force vector to start from the origin (0,0), then the vector is completely determined by the coordinates of the point at the tip of the arrow. Thus we can consider each $x \in \R^2$ to represent a vector in the plane as well as a point in the plane. When we wish to regard an ordered pair as a vector, we will use [x,y] instead of (x,y).

''Mathematically, there is no distinction between (1,2) and [1,2]. The different notations merely indicate different views of the same element of $\R^2$.'' Each n-tuple can be viewed as both a point and as a vector.

So there you go, it's purely a matter of perspective. He's saying that they're both legitimate ways to view an n-tuple, but ultimately there's no mathematical difference, just different connotations. --GFauxPas 13:52, 26 January 2012 (EST)


 * Brilliant. Way to go. I note the amendments to the Vector page. --prime mover 16:18, 26 January 2012 (EST)


 * For the formal part of it, I quote: ...if we consider a force vector to start from the origin (0,0), then.... Also, in many (particularly mechanical) situations, the starting point of a vector is very significant for its effect on a system (for example, forces on a rigid axis of a wheel do practically nothing; forces on some surface point of the wheel generally make the wheel turn). Therefore, this assumption is quite questionable, especially when thinking about the starting point of a vector like $\mathbf u -\mathbf v$ where $\mathbf u,\mathbf v$ are vectors... I consider this case not closed yet. --Lord_Farin 17:27, 26 January 2012 (EST)


 * At one point during my mid-teens mathematics education, the concept "position vector (of a point)" was encountered, whose meaning was "the vector from the origin to that point", so one can call $\mathbf 0$ the "position vector of the origin" if that helps. --prime mover 17:40, 26 January 2012 (EST)


 * I was under the impression that vectors are not defined by their location, i.e., the vector issuing from $(0,0)$ and ending at $(1,0)$ is the exact same vector as the one starting from $(5,5)$ and ending at $(6,5)$. Certainly if we define a vector as "magnitude and direction" we don't see "location" there. --GFauxPas 17:44, 26 January 2012 (EST)

It is precisely that approach that I am questioning, on mentioned physical grounds. --Lord_Farin 17:49, 26 January 2012 (EST)
 * Well, in Khan Academy, Khan is very sure about that, and this is what my Linear Algebra professor, er, professes. I'll let you know if I find a book that says otherwise. Oh, if you want a physics source, check out http://www.learner.org/resources/series42.html video 5, which also takes this approach, though it may be dated. --GFauxPas 17:56, 26 January 2012 (EST)

LF I looked through my books to see if I can find any clues for you. I'm paraphrasing. They're only dealing with $\R^n$.

Jewett: Physics for Scientists and Engineers
An example of a vector quantity is displacement...The direction of the arrowhead represents the direction of the displacement, and the length of the arrow represents the magnitude of the displacement.

For many purposes, two vectors $\mathbf{a}$ and $\mathbf{b}$ may be defined to be equal if they have the same magnitude and point in the same direction:


 * $\mathbf{a} = \mathbf{b} \iff \left({||\mathbf{a}|| = ||\mathbf{b}|| \land \text{the vector arrows run along parallel lines}}\right)$

This property allows us to move a vector to a position parallel to itself in a diagram without affecting the vector.

Larson
Larson bolds terms that he's defining for the first time. He's using $\R^2$ for the moment, he addresses $\R^n$ later.

A directed line segment is used to represent a vector quantity. The directed line segment $\overrightarrow{PQ}$ has initial point $P$ terminal point $Q$. Directed line segments that have the same length and direction are equivalent.

(He's defining a new term here. Note he doesn't say equals. - GFP)

The set of all directed line segments that are equivalent to a directed line segment $\overrightarrow{PQ}$ is a vector in the plane and is denoted $\mathbf{v} = \overrightarrow{PQ}$...be sure you see that a vector in the plane can be represented by many different line segments--all pointing in the same direction and all the same length.

The component form of $\mathbf{v}$ is given by:


 * $\mathbf{v} = \langle{v_1,v_2}\rangle$, $v_n \in \R$

where it's implied that the initial point is the origin. If both the initial point and the terminal point lie at the origin, then $v$ is called the zero vector and is denoted by $\mathbf{0} = \langle{0,0}\rangle$.

Two vectors $\mathbf{u} = \langle{u_1,u_2}\rangle$ and $\mathbf{v} = \langle{v_1,v_2}\rangle$ are equal iff $u_1 = v_1 \land u_2 = v_2$.

FWIW, note he's defining equal at the end here, he put it in bold. Also, Equality of Ordered Pairs.

I'm not convinced he has the same approach as Khan and Fraleigh. --GFauxPas 07:35, 27 January 2012 (EST)


 * I hallow the distinction between 'directed line segment' and 'vector' (where the latter is an equivalence class of the former). If this can be implemented, I am satisfied. It is good that we have found at least one reference that calls full rigour to arms on this subject. --Lord_Farin 08:12, 27 January 2012 (EST)


 * On closer investigation, Khan treats vectors differently in the Linear Algebra videos than in the physics videos. I think the video I linked to on the definition:vector page is formal enough for your expectations. He distinguishes the arrow representation of a vector from the vector itself. A vector in $\R^n$ is and only is an ordered n-tuple of n elements of $\R^1$. I guess this supports your view of physics as math without rigor. I'll have to watch his linear algebra videos much more carefully, then I'll try to fix the page.

On a side note, is it more common to say n-toople or n-tuh-pl? --GFauxPas 08:56, 27 January 2012 (EST)


 * I always say n-toople, but that is because this is also how it is pronounced in Dutch. Dealing mostly with Dutch students around, I consider this common practice. I actually enjoyed watching the Khan academy video (even though I already knew all the material covered). Especially when he cares to warn about identifying a vector with the arrow interpretation; note that this distinction is consistent with proposed difference between 'directed line segment' and 'vector'. --Lord_Farin 10:14, 27 January 2012 (EST)
 * I'm glad you like it. I find it's worth watching Khan's videos even if you know the material, as it gives me idea to explain it to people who don't know. cf. my intuition sections. Anyway, how about this. I'll do an entry on euclidean n-space, I'll consolidate the stuff about arrows and mention that this is the primary interpretation used in physics even though it's not 100% correct, and you do other vector spaces? --GFauxPas 10:18, 27 January 2012 (EST)