Wikipedia:Reference desk/Archives/Mathematics/2011 September 25

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September 25

How to find specific arguments of two functions

How can I find the arguments of two functions, where the value of the first function is a multiple of the value of the second function? To better illustrate what I mean, I will provide an example.

Suppose we are given the following two simple functions

$f_{1}(x)=x^{2}\,\!$

$f_{2}(x)=3x\,\!$

with $x\in \mathbb {N} \,\!$

I want to find those arguments, where the value of $f_{1}$ is a multiple of the value of $f_{2}$. How could I do this? Toshio Yamaguchi (talk) 14:02, 25 September 2011 (UTC)

You introduce $n\in \mathbb {N}$, such that now you have $f_{1}(x)=nf_{2}(x)$ or $x^{2}=3nx$. Thus $x=3n$ for $n\in \mathbb {N}$ (or possibly x=0 if you include 0 in your naturals) 41.234.205.210 (talk) 14:22, 25 September 2011 (UTC)

Differential dx

Hi. In some calc textbooks if you have a separable differential equation of the form ${\frac {dy}{dx}}=F(x,y)$ it recommends multiplying both sides by the differential dx and then separating and integrating. Is this a valid step? Is it ever valid to split up the ${\frac {dy}{dx}}$? Since you're actually just integrating both sides wrt x but using substitution to integrate the lhs wrt y by grabbing the ${\frac {dy}{dx}}$ aren't you? THanks. 41.234.205.210 (talk) 14:12, 25 September 2011 (UTC)

That's exactly right. It seems like you answered your own question.

However, multiplying by $dx$ is a step which just happens to work, even though it's not technically valid. The valid step would be, as you say, to integrate both sides wrt x and use substitution to convert the ${\frac {dy}{dx}}$ side into an integral wrt y. --COVIZAPIBETEFOKY (talk) 14:47, 25 September 2011 (UTC)

In "ordinary," "standard" calculus, as almost universally taught in calculus textbooks, the $\textstyle {\frac {dy}{dx}}$ is just a symbol, not a fraction, although treating it as a fraction often gives correct results (as in the chain rule, integration by substitution, and separable differential equations). This is the reason it's written as a fraction in the first place—it's suggestive notation. However, it is possible to develop a theory of calculus, using non-standard analysis, in which infinitesimals are real things, and then $\textstyle {\frac {dy}{dx}}$ is really a fraction of two infinitesimals, so all of these tricks that pretend it's a fraction can be easily justified because it is a fraction. Non-standard analysis is not commonly taught to calculus students, but if you're interested in it there are books available, such as Elementary Calculus: An Infinitesimal Approach, which is also available online. —Bkell (talk) 16:27, 25 September 2011 (UTC)

The interpretation of dx as an infinitesimal is controversial, but using dx as a formal variable is not. Postulate the rules of differentiation d(x+y)=dx+dy and d(x·y)=x·dy+dx·y, and the algebra of differentiation follows. Bo Jacoby (talk) 20:08, 25 September 2011 (UTC).

What does the mathematical symbol ⊙ represent?

Hi, I recently found the symbol ⊙ (represented in LaTeX as \odot), and I am not familiar with it. Wikipedia does not appear to have an article on the subject, and a Google Search comes up with absolutely nothing. What does it do, and what is its formal name?--82.113.103.164 (talk) 16:40, 25 September 2011 (UTC) — Preceding unsigned comment added by 82.113.103.164 (talk)

Note for those replying: Original post is at WP:HD#What does the mathematical symbol ⊙ represent?, where I posted a link to a Unicode conversion and a document mentioning this symbol. You might be able to better explain for what it is used than the people at the help desk, though. Regards Toshio Yamaguchi (talk) 17:46, 25 September 2011 (UTC)

One of its uses is in representing three dimensional diagrams on a two dimensional page. The $\odot$ symbol represents a vector (or arrow) orthogonal to the page an pointing towards the reader. The symbol $\otimes$, in that setting, represents a vector (or arrow) orthogonal to the page and pointing away from the reader. There will be other uses for $\odot$, just like there are other uses for $\otimes$. — Fly by Night (talk) 18:00, 25 September 2011 (UTC)

It's commonly used for the symmetric tensor product. Sławomir Biały (talk) 18:15, 25 September 2011 (UTC)

See "Circled dot".—Wavelength (talk) 19:08, 25 September 2011 (UTC)

Since nobody said it yet: in an algebraic context (basically anything other than the vector "towards the reader"), there is no standard meaning. It is a convenient symbol to use for operations which are similar to multiplication, but it wouldn't appear without being explained in the text. Staecker (talk) 01:58, 26 September 2011 (UTC)

counting

like so

hello. I've seen this problem countless times and still have not figured out how to do it. Consider a regular polygon, such as a pentagon, with diagonals drawn such that each vertex is connected to each other vertex (or equivalently, that a pentagram is inscribed such that each convex vertex of the pentagram coincides with a vertex of the pentagon). How many triangles are in the interior? The problem is that there are triangles made of many little triangles. How do I do this analytically? There is also the form where a rectangle is divided except not just at the vertices, with a bunch of random lines, I assume it is solved the same way? muchas gracias.

Usually when I've seen a problem like this, it's asking how many triangles are in a specific picture, and the best way I know of to solve that problem is just to count them one by one. It's laborious, and it takes some thought to make sure you aren't missing any or counting any twice, but it gets the job done eventually. If the question is phrased so that you have a regular n-gon, say, and you're asked how many triangles are in the picture in terms of n, then that becomes a much more difficult problem, I think, and I don't know how to solve that. —Bkell (talk) 18:30, 25 September 2011 (UTC)

Classify the triangles according to shape and size. The acute triangles are big or medium or small. The obtuse triangles are big or small. Now count each class. Acute: 5 big and 10 medium and 5 small. Obtuse: 10 big and 5 small. Total 35 triangles. Bo Jacoby (talk) 19:58, 25 September 2011 (UTC).

This reminds me of the question asked here: Talk:Diagonal#Challenging_Question:_2_for_a_triangle.2C_5_for_a_square.2C_12_for_a_pentagon..., where some answers are given. I can't see a way to find a formula for the regular n-gon in terms of n. It seems to be a matter of examining intersections and subtracting lost regions Dbfirs 09:17, 26 September 2011 (UTC)

How many for a hexagon? Then we could see if OEIS had it. Grandiose (me, talk, contribs) 10:32, 26 September 2011 (UTC)

http://oeis.org/A006600. I searched OEIS using the prefix you gave and found nothing. Then I started writing a program to do the computation. It didn't handle correctly the case that 3 segments meet at a single point so it gave a result of 12 for a square. I drew a square to find the correct answer, which is 8, not 6. Using this new knowledge, finding the sequence in OEIS is easy ([1]). -- Meni Rosenfeld (talk) 14:30, 26 September 2011 (UTC)

And the number of regions in the regular n-gon is http://oeis.org/A007678.

What an amazing collection of sequences! It doesn't have the answers for shapes divided by random lines, of course, because both the regions and the number of triangles depend on the exact configuration of the lines. The maximum number of regions for a given number of intersecting lines is just the well-known pancake cutting problem with answer: 0.5(n² + n + 2) (independent of the bounding polygon). Dbfirs 22:27, 26 September 2011 (UTC)

A problem with Bkell's answer is that it works separately in individual cases, but if n is the number of vertices, it doesn't say in what way the answer depends on n. For example, how fast does it grow as n grows?

Another question is: What if it's slightly irregular? Look at the case of the hexagon, with three concurrent lines meeting at the center. As soon as they're not quite concurrent, some more triangles appear. Michael Hardy (talk) 17:27, 27 September 2011 (UTC)

Game Theory Q

Hi, this is explicitly a homework question, but I just need a push in the right direction. There's a game where you can choose between a nickel and dime, if the dime is chosen the game ends, if you choose the nickel you are presented with the same choice.

"Consider a variant of the game in which if the dime is taken the game stops, but if the nickel is taken then the game is repeated with probability p. Assume that p < 1, so the game will eventually stop. What is the optimal strategy?"

I'm just confused about how to go about solving such a problem. Any advice?209.6.54.248 (talk) 20:12, 25 September 2011 (UTC)

Think about it this way: if you choose the nickel and are lucky enough to have the game go on, then the game has the same value at turn 1 as it did on turn 0. You need to convert that fact into an equation, and then solve it. Looie496 (talk) 20:56, 25 September 2011 (UTC)

What am I missing? I see that the game stops. I see that the game can continue. I see the probability of continuing (or stopping). I don't see how to win. So, I don't understand how an optimal strategy could be developed. -- kainaw™ 21:00, 25 September 2011 (UTC)

The goal is to maximize the expected value of your earnings.--Antendren (talk) 21:52, 25 September 2011 (UTC)

So, the strategy appears to be very simple. Always take the nickel unless you know that for some strange reason there will certainly not be another turn. Then, take the dime and end the game by your decision. -- kainaw™ 18:04, 26 September 2011 (UTC)

Only if $p\geq 1/2$. -- Meni Rosenfeld (talk) 18:51, 26 September 2011 (UTC)