Wikipedia:Reference desk/Archives/Mathematics/2011 February 4

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February 4

matrix/vector algebra problem

I am reading a paper, and there is one part that I cannot follow.

Here we have an equation

$\alpha \log(h)=\alpha \epsilon _{i}-G+\log(p_{i})-(1-\alpha )\sum _{j=1}^{n}w_{ij}\log(p_{j})+(1-\alpha )H_{i}$

where $w_{ij}$ are the entries of the $n\times n$ stochastic matrix $W_{n}$, $G$ is a constant, $\alpha$ is a constant between 0 and 1, and $H_{i}\equiv \sum _{j=1}^{n}w_{ij}\log(w_{ij})$

The above equation can be written in vector form, which I assume for $n=2$ will look like:

${\begin{bmatrix}\alpha \log(h)\\\alpha \log(h)\end{bmatrix}}={\begin{bmatrix}\alpha \epsilon _{1}\\\alpha \epsilon _{2}\end{bmatrix}}+{\begin{bmatrix}\log(p_{1})\\\log(p_{2})\end{bmatrix}}+{\begin{bmatrix}-G\\-G\end{bmatrix}}+{\begin{bmatrix}-(1-\alpha )(w_{11}\log(p_{1})+w_{12}\log(p_{2}))\\-(1-\alpha )(w_{21}\log(p_{1})+w_{22}\log(p_{2}))\end{bmatrix}}+{\begin{bmatrix}(1-\alpha )H_{1}\\(1-\alpha )H_{2}\end{bmatrix}}$

Now a row vector $v'_{n}$ is introduced where $v'_{n}={\tfrac {\alpha }{n}}{\begin{bmatrix}1&1\end{bmatrix}}[I-(1-\alpha )W_{n}]^{-1}$

The entries of this row vector sum to one.

Both sides of the equation are then premultiplied by this vector which apparently gives $\log(h)=v'_{n}\epsilon +{\tfrac {1}{n}}\sum _{i=1}^{n}\log(p_{i})-{\frac {G}{\alpha }}+{\frac {1-\alpha }{\alpha }}v'_{n}H$

where $H=[H_{1}\dots H_{n}]'$

Now I cannot figure out how they got this equation from multiplying both sides by $v'_{n}$. Seemingly both sides have also been divided by $\alpha$, so I can understand what happened with the left hand side, and the $\epsilon ,\;G$ and $H$ terms. But I cannot understand what happened to the $\log(p_{i})$ and $\log(p_{j})$ terms, so that nothing else was left except ${\tfrac {1}{n}}\sum _{i=1}^{n}\log(p_{i})$

The paper does not explain this. Does it follow from the information given? —Preceding unsigned comment added by 130.102.78.164 (talk) 04:44, 4 February 2011 (UTC)

Yes, it's clearly designed to. Let P be the (column) vector of log(p_i)'s. Then in the original equation, the terms $\textstyle \log(p_{i})-(1-\alpha )\sum _{j=1}^{n}w_{ij}\log(p_{j})$ give the elements of the vector $P-(1-\alpha )WP=(I-(1-\alpha )W)P$. The left factor of this cancels out with $[I-(1-\alpha )W_{n}]^{-1}$ in the defintion of v', so what is left after you multiply is ${\frac {\alpha }{n}}[1\ldots 1]P$. –Henning Makholm (talk) 09:50, 4 February 2011 (UTC)

Thanks for that, but unfortunately no sooner had I continued reading that I got stuck on the next line...

Again I have this vector $v'_{n}={\tfrac {\alpha }{n}}{\begin{bmatrix}1&1\end{bmatrix}}[I-(1-\alpha )W_{n}]^{-1}$

Then I am told that the equation

$s_{i}=h/n+(1-\alpha )\sum _{j=1}^{n}s_{j}w_{ji}$

implies that

$s'=(h/n){\begin{bmatrix}1&1\end{bmatrix}}[I-(1-\alpha )W]^{-1}=(h/\alpha )v'_{n}$

Now I am trying to get this myself, but not sure how...

I see that the first equation implies something like

$[I-(1-\alpha )W_{n}]s=h/n{\begin{bmatrix}1\\1\end{bmatrix}}$

Then do I premultiply both sides giving

$s=[I-(1-\alpha )W_{n}]^{-1}h/n{\begin{bmatrix}1\\1\end{bmatrix}}$

What is the next step? I forgot all my rules about matrix algebra... —Preceding unsigned comment added by 130.102.158.15 (talk) 03:03, 5 February 2011 (UTC)

Note the position of the indices in $\Sigma _{j}s_{j}w_{ji}$. Since you're now summing over the first index of w, you have a row vector multiplied by W on the right, so your defining equation for s' is

$s'=(h/n)[1\ldots 1]+(1-\alpha )s'W$

and then when you solve for s' you automatically get the $[I-(1-\alpha )W]^{-1}$ on the right side of the [1...1]. Remember that $1-\alpha$ and $h/n$ are scalars and therefore commute with everything. –Henning Makholm (talk) 09:17, 5 February 2011 (UTC)

ah, I was stupid, I didnt notice that s could just be a row vector to start with... thanks again. —Preceding unsigned comment added by 118.208.112.152 (talk) 09:25, 6 February 2011 (UTC)

Love heart

The graph of

$x^{2}+(y-{\sqrt[{3}]{x^{2}}})^{2}=1$

looks like a love heart. Is there a more proper name for this curve, or for a general family to which it belongs? Thanks. (For reference: WolframAlpha.) —Anonymous Dissident^Talk 12:27, 4 February 2011 (UTC)

If you want to say "heart-shaped" and still sound like an intellectual, you can say "cardioid" :) But cardioid has a specific meaning which is not the same as your curve. Tinfoilcat (talk) 12:53, 4 February 2011 (UTC)

See also the MathWorld page.--RDBury (talk) 14:20, 4 February 2011 (UTC)

Finding range of 7+5cos5x divided by 8-6cos6x

Normally to find critical points we use dy/dx=0 then from critical points and the monotonicity of the function we try to find the range, but here, I am unable to find the critical points.

I got dy/dx =  150(sin5x)(cos6x) - 200sin5x - 252sin6x - 180(sin6x)(cos5x)  whole divided by  (8-6cos6x)^2
From here, I am unable to find the value of x for which it is 0

Can you please help me with this? — Preceding unsigned comment added by Krishnashyam1994 (talk • contribs) 15:11, 4 February 2011 (UTC)

I don't see why this can't be solved by taking the range of the numerator and denominator since both are cyclic and not synchronized with one another. The numerator has a range of [2,12]. The denominator has a range [2,14]. So, the function's range is [2/14, 12/2] = [0.14, 6]. Doing a quick plot, it does hit both of those extremes, but does not cross them. -- kainaw™ 15:25, 4 February 2011 (UTC)

y=6 is reached at x=0, but the low limit of 2/14 is not actually reached. For that to happen, cos(5x) and cos(6x) must both be -1, so 5x and 6x would both have to be π modulo 2π, so (6-5)x must be a multiple of 2π, which contradicts 5x being π module 2π. Furthermore, since 2π is obviously a period of the function, its range has to be closed, so its lower limit must be strictly larger than 2/14. –Henning Makholm (talk) 19:39, 4 February 2011 (UTC)

For the lower bound, you can only hope for a numerical solution. The actual value is a root of a polynomial of degree 10 (which cannot be resolved by radicals). Newtons method gives a value of 0.15292 for the lower bound. 166.137.140.202 (talk) 22:50, 4 February 2011 (UTC)

See http://www.wolframalpha.com/input/?i=(7%2B5cos5x)%2F(8-6cos6x) Bo Jacoby (talk) 10:40, 5 February 2011 (UTC).

Coin nominal values

Why is it not good to have both 20 cent and 25 cent coins in a single currency? --84.61.176.167 (talk) 15:33, 4 February 2011 (UTC)

This is a classic example of the greedy algorithm when it comes to making change. If you want to give 40 cents in change and you have standard American currency of 1, 5, 10, and 25 cents, the least number of coins will be 1x25, 2x10. If you add a 20 cent coin, the least number of coins will be 2x20 - which requires use of something other than the greedy algorithm. -- kainaw™ 15:36, 4 February 2011 (UTC)

You have to define what you mean by "good". You could have a coin in every denomination from 1 cent to 99 cents, and then you could always make change with a single coin. But that benefit would be outweighed by the problems of having 99 different types of coins to keep track of. At the other extreme you just have a 1 cent coin and it would take up to 99 to make change. There are other factors such as people find it easier to do arithmetic with multiples of 5 and 10. With various conflicting objectives, some of which are cultural, the question doesn't make sense mathematically. You could ask a more specific question like "Given there are to be 4 denominations, what should they be to minimize the average number of coins needed to make change?" Then you might then be able to show there is no solution that uses both a 20 and 25 cent coin.--RDBury (talk) 17:27, 4 February 2011 (UTC)

We need an article on coin nominal value. The simpler solution to the problem of designing currency value systems is to have only the denominations 1, 10, 100, 1000 and so on. Then the representation of a number in terms of coins and banknotes matches the decimal positional notation of the number. Intermediate denominations such as 2, 5, 20, 25, 50, 200, 250, 500 and so on complicates the system but also has a benefit in terms of carrying fewer coins and notes. I agree that this is not a mathematical question. Bo Jacoby (talk) 11:53, 5 February 2011 (UTC).

Trying to be mathematical about it, to get maximum coverage over a wide range of values, it seems that the ideal split is logarithmic ( X A's make a B, X B's make a C, X C's make a D ...). Your 1, 10, 100 system does that, but the jumps of ten are too big. Pay for a 0.99& candy bar with a 10& bill, and you get ten coins in change, nine of them all of the same denomination. Humans are better dealing with smaller numbers; three to four is about the limit we can handle intuitively. A binary system would be ideal for that (denominations 1, 2, 4, 8, 16, ...), but you also want to mesh with our base ten number system. You can't just alter the multiplier on the logarithmic system (10^1/3 is about 2, so we could have 1, 2.15, 4.64, 10, 21.54, 46.41, 100), because now you can't make change easily. A compromise is needed, so if you round the 10^1/3 logarithmic scale to even multiples of ten, you get 1, 2, 5, 10, 20, 50, 100 ... . I'll note that this matches the system used in most modern currencies such as Euro coins and Euro banknotes, as well as Coins of the Australian dollar, Banknotes of the Australian dollar, and even Banknotes of the pound sterling and US Federal Reserve Notes (although the $2 note is rarely used). The presence of the 0.25 coin is somewhat of an aberration to this trend in United States coinage (it's also present in UK coins, but unused in favor of the 0.20 coin), but that's historical, as the US Quarter is it is derived from two pieces of eight, which was the commonly used coinage in the US at one time. The 25p piece of the UK has a similar history, being the post-decimalisation equivalent of the pre-decimilisation crown. -- 174.24.195.38 (talk) 21:50, 5 February 2011 (UTC)

We do have a nice article on Preferred_number. Bo Jacoby (talk) 20:26, 6 February 2011 (UTC).