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January 6

Tetrahedral angles

If I take a regular tetrahedron and draw segments from each of the vertices to the center, what's the angle between two of those segments? --75.60.13.19 (talk) 00:54, 6 January 2011 (UTC)[reply]

If you look at the Tetrahedron article, i.e. click here, then you'll find out everything you need to know, any many things you don't. — Fly by Night (talk) 01:33, 6 January 2011 (UTC)[reply]

By symmetry, the average of the three vectors is zero; hence the sum of the three is zero; hence the sum of their x-coordinates is zero if one of them points in the direction of the x-axis. Again by symmetry, the x-coordinates of the other three are equal to each other. Since the one pointing in the axis diretion is 1, the others must each be −1/3. Hence the angle is arccos (−1/3). Michael Hardy (talk) 02:49, 11 January 2011 (UTC)[reply]

Learning to read and write proofs

What books would you recommend for learning the general techniques of mathematical proofs? 74.14.111.188 (talk) 07:17, 6 January 2011 (UTC)[reply]

I suppose it really depends on the level of mathematics you're concerned with, but one course I found very helpful in the early years of my undergraduate studies used the textbook "A Transition to Higher Mathematics," by Smith, Eggen, and St. Andre. I found it be very illuminating at the time, particularly in regards to learning methods of proof. Nm420 (talk) —Preceding undated comment added 15:41, 6 January 2011 (UTC).[reply]

How to read and do proofs by Daniel Solow is another one to look at. There are probably others in the same vein.--RDBury (talk) 00:54, 7 January 2011 (UTC)[reply]

I think there's one by Daniel Velleman. And some others...... Michael Hardy (talk) 02:46, 11 January 2011 (UTC)[reply]

Websites first postings

I need to find out when the sites listed below first established a website/page on the internet. Thanks

Taylor & Francis Group: an informa business - http://www.taylorandfrancisgroup.com/

- www.tandf.co.uk

Association for Childhood Education International - http://acei.org/

- www.acei.org/cehp.htm

National Council of Teachers of Mathematics - http://www.nctm.org/

- http://my.nctm.org

National Association for the Education of Young Children - http://www.naeyc.org/yc/

- www.journal.naeyc.org —Preceding unsigned comment added by 24.210.25.124 (talk) 13:30, 6 January 2011 (UTC)[reply]

Question reformatted for legibility. The Wayback machine should be able to help - e.g. it suggests that www.tandf.co.uk first appeared in early 1997. AndrewWTaylor (talk) 20:50, 6 January 2011 (UTC)[reply]

Remainder term

If the Taylor series becomes arbitrarily close to the original function for all analytic functions (in other words, all function that we are normally interested in), what is the purpose of the remainder function? 24.92.70.160 (talk) 21:34, 6 January 2011 (UTC)[reply]

Analytic functions define a very small subset in the space of all function. Asking a function to have continuous derivatives of all orders and then asking for a series to converge is a very, very big ask. High school functions like 1/x aren't analytic; it fails to be continuous at x = 0. Take a look at the article on flat functions. The exponential

\scriptstyle f(x)\,=\,e^{-1/x^{2}}

is an example of a flat function. It is well defined at x = 0 because both the positive and negative limits x → 0 give ƒ(x) → 0. In fact, it is a smooth function because each of its derivatives exist and are continuous for all x (particularly at x = 0). But you will find that each and every derivative vanishes at x = 0. So the function is always contained in the remainder function; no matter how far along the Taylor series you go. As for non-smooth functions, only the first few derivatives may be continuous, so we can only define the Taylor series up to a certain, finite order. Then the remainder term takes up the slack. — Fly by Night (talk) 22:03, 6 January 2011 (UTC)[reply]

Basically its purpose is to provide you with language for reasoning about whether or not the function you're considering at any given time happens to be one of the (in practice) rarely occurring exceptions to analyticity. –Henning Makholm (talk) 23:15, 6 January 2011 (UTC)[reply]

Numerical analysts often use a finite series as an approximation for a function; having an upper bound on the remainder term allows them to draw conclusions about how accurate their final results will be. Knowing that the infinite series converges exactly is not good enough if you can't compute an infinite series. Eric. 82.139.80.114 (talk) 01:54, 7 January 2011 (UTC)[reply]

Just to clarify a few points: analytic functions are a small subset theoretically (measure zero in L^2 I think...), but a huge subset of functions commonly used (i.e. in practice outside pure math research). Another thing to consider is the domain. Everyone seems to be assuming the whole real line, but 1/x and exp(-1/x^2) are both analytic on R\{0}. SemanticMantis (talk) 02:06, 7 January 2011 (UTC)[reply]

As a tangent here, are there smooth functions that are not analytic almost everywhere? The smooth function article claims that they exist (even nowhere analytic ones), but does not give details. Can they be constructed without the axiom of choice? –Henning Makholm (talk) 03:14, 7 January 2011 (UTC)[reply]

Here for instance is an explicit example. Algebraist 03:18, 7 January 2011 (UTC)[reply]

And it turns out we have an article on it. Algebraist 03:25, 7 January 2011 (UTC)[reply]

Oops, didn't notice you had already added a link to the smooth function article. It wasn't my intention to have two of them. –Henning Makholm (talk) 07:36, 7 January 2011 (UTC)[reply]

Another example. Algebraist 03:30, 7 January 2011 (UTC)[reply]

Interesting, thanks. Those appear to be impeccably constructible. –Henning Makholm (talk) 03:42, 7 January 2011 (UTC)[reply]

LaTeX

Resolved

Can someone suggest a good LaTeX writing program for Windows. I mean a program where I type the code and it compiles it, turns it into .dvi, .ps or .pdf. I use Kile on Linux but it needs some fiddling with to run on Windows and I don't know how. A nice user friendly interface would be perfect, with some symbol buttons that substitute the LaTeX code when you click them, etc. — Fly by Night (talk) 22:25, 6 January 2011 (UTC)[reply]

there aren't any. just install miktex like everyone else. 87.91.6.33 (talk) 22:42, 6 January 2011 (UTC)[reply]

There's a commercial program called PCTeX that some people like. Personally I don't like it; if I recall correctly (but this was years ago) it has its own style and/or class files, and it's a bit of a pain to produce TeX source that other people can use without the program. But maybe they've fixed that for all I know. --Trovatore (talk) 22:48, 6 January 2011 (UTC)[reply]

I use TeXnicCenter; some of my friends use LEd. But there are plenty of editors (free or not) listed here. The Menu for Inserting Symbols column may be of interest to you. Invrnc (talk) 22:52, 6 January 2011 (UTC)[reply]

I use Lyx, quite good and free editor and writer. It can give output in .div, .ps and .pdf formats. Anyway, the list given above indicates many options. I haven't used it yet, but have heard good comments of Scientific WorkPlace. Pallida Mors 00:53, 7 January 2011 (UTC)[reply]

Thanks for all the suggestions. I went for TeXnixCenter in the end. — Fly by Night (talk) 12:54, 7 January 2011 (UTC)[reply]