Wikipedia:Reference desk/Archives/Mathematics/2010 June 29

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June 29

Vector similarity

I can calculate the cosine between two unit vectors in two and three dimensions. Doing some calculations in pencil, I think that doing it in four dimensions is a trivial jump from three dimensions. I would like to know how complex does calculating the cosine between two unit vectors become as dimensions are added. I measure complexity by how many variables are required and how many mathematical operations need to be performed. Is it a linear progression (ie: 10 dimensions is twice the complexity of 5 dimensions)? Is it an exponential progression? Does it actually become easier as you start piling on the eleventh, twelfth, and thirteenth dimension? -- kainaw™ 03:09, 29 June 2010 (UTC)

Usually we say that the angle between vectors A and B is $\arccos \left({{A\cdot B} \over {|A|\cdot |B|}}\right)$.

In that formula, $A\cdot B$ denotes the vector dot product which is straightforward to compute if the vectors are in coordinate form. When A and B are unit vectors, the denominator is 1, of course. 75.57.243.88 (talk) 04:08, 29 June 2010 (UTC)

I didn't realize that I was performing the dot product. So, using 10 dimensions, it would simply be a matter of 10 multiplications and 9 additions. Each dimension adds one multiplication and one addition. So, it is a linear increase in complexity - unless I am missing something. -- kainaw™ 04:13, 29 June 2010 (UTC)

Yes, that's correct.

At the risk of sounding snide (actually, I will definitely sound snide), I'll remind you that I linked to the Cosine similarity article, which includes the known fact that the cosine is equivalent to the dot product, as early as here. It is of course clear to anyone who uses cosine similarity in practice. If you're only now realizing this basic fact, I think you really are biting more than you can chew, and that you shouldn't try to contribute to subject areas you have no grasp on. -- Meni Rosenfeld (talk) 06:35, 29 June 2010 (UTC)

So, what you are saying is that if you don't use vector-based mathematics in health informatics (my area of study), you should avoid learning about it and pretend that it doesn't exist. I was apparently suffering from the illusion that mathematics was a good subject to study whenever work was slow so work could be improved in the future. -- kainaw™ 18:53, 29 June 2010 (UTC)

No, I think Meni was just commenting on the fact that you had failed to learn from the link that he provided six months ago. I'm sure we would all encourage you to learn about vectors, though I would advise starting with the easier articles because some get very technical very quickly. Dbfirs 05:02, 30 June 2010 (UTC)

By all means, you should study it. I was referring particularly to something you mentioned a while back, which as I recall was that you were writing a survey paper about similarity techniques. You shouldn't do that at your current state of knowledge.

Another point is that you should study it in a more structured way. The techniques you ask about assume one is already familiar with the undergrad math. So you should focus on the math first. -- Meni Rosenfeld (talk) 08:30, 30 June 2010 (UTC)

Just to note: A survey paper is a survey of contributions by others. It does not include anything from the author. In my spare time, I have been working on a survey of work by others. If I can't explain something in a survey paper, I figure that I don't understand it well enough, so I keep going back to the topics that I do not understand and reading about them again and again. -- kainaw™ 13:26, 30 June 2010 (UTC)

Invalid proofs

Important proofs which are later seen to be flawed are common throughout the history of mathematics – early proofs of the four colour theorem are prime examples. I'm interested in prominent examples of the following (and what the repercussions were): a result is proved in a seemingly airtight way; the mathematical community accepts the proof; a large amount of mathematical work follows the proof, and new proofs ground themselves in the original result; and then the first proof is exposed as flawed years later, invalidating everything that had followed. Also, I'd like to know whether this kind of thing has happened even in the absence of an initial proof – where mathematicians implicitly assume a result without decisive proof, and then that result is seen to be false and the house of cards collapses. In a documentary I was watching, Andrew Wiles recounted how people were doing new mathematics based on the Taniyama–Shimura conjecture before Wiles used it to prove Fermat's Last Theorem. Although that particular conjecture turned out to be true, it provides an example of what I mean. Thanks in advance. —Anonymous Dissident^Talk 11:03, 29 June 2010 (UTC)

It's pretty rare for a published "theorem" to turn out to be false, but List of published false theorems mentions a few. 75.57.243.88 (talk) 15:08, 29 June 2010 (UTC)

False proofs get published all the time, but not in peer-reviewed journals. If a proof survives peer-review, then it is extremely unlikely that it's wrong. The mathematical community would not accept the proof without it being in a peer-reviewed journal and wouldn't use the theorem in their own work either. Mathematicians do sometimes assume conjectures to be true (eg. the Riemann hypothesis) and prove other results, but they do so in full knowledge of the fact that the conjecture could end up being false and their result would be invalid. --Tango (talk) 15:33, 29 June 2010 (UTC)

For recent examples of paper published in major peer-reviewed journals (some would say the major math journals: Inventiones and the Annals) see Daniel Biss. I would agree that published, peer-reviewed papers whose main theorems are incorrect are rare. Reviewers often evaluate the general proof strategy and convince themselves that the details could be filled in without too much trouble, so sometimes the exposition or reasoning have minor flaws, or one finds incorrect or redundant intermediate results. Phils 16:33, 29 June 2010 (UTC)

I don't believe any example of your situation "a result is proved...the mathematical community accepts the proof; a large amount of mathematical work follows...then the first proof is exposed as flawed" yet exists. That's not to say it won't ever happen though. The papers making up the Classification of finite simple groups have all been peer reviewed, and a great deal of work assumes it is correct, but the sheer length and complexity (tens of thousands of pages according to our article) is pretty staggering. Our article claims "Jean-Pierre Serre is a notable skeptic" of the proof, but the link given doesn't seem to support that. Tinfoilcat (talk) 18:17, 29 June 2010 (UTC)

Solving for the same exponential variable on both sides of an equation

Those who may know me from the other reference desks will discover that I am a mathematical ignoramus. How do I solve for n in the following equation?

axⁿ − a = byⁿ

Thank you so much. Marco polo (talk) 15:27, 29 June 2010 (UTC)

So a, b, x and y are all known and you want to know n? You can rearrange that equation to $\left({\frac {x}{y}}\right)^{n}={\frac {b}{(1-a)}}$. That can then be solved using logarithms. --Tango (talk) 15:37, 29 June 2010 (UTC)

You seem to have misread the equation as axⁿ − xⁿ = byⁿ.—Emil J. 15:39, 29 June 2010 (UTC)

Oh, yes, I switched the terms when I factorised it... oops! --Tango (talk) 15:53, 29 June 2010 (UTC)

There's probably no elementary way to solve this equation. You can solve it numerically, though. -- Meni Rosenfeld (talk) 15:54, 29 June 2010 (UTC)

It might be possible to make a good approximation if you knew something about the sizes of a,b,x,y Tinfoilcat (talk) 16:07, 29 June 2010 (UTC)

A solution can in lucky cases be computed by the iteration

$n=\lim _{i\rightarrow \infty }f^{i}\left({\frac {\log {\frac {b}{a}}}{\log {\frac {x}{y}}}}\right)$

where

$f(n)={\frac {\log {\frac {b}{a}}-\log(1-{\frac {1}{x^{n}}})}{\log {\frac {x}{y}}}}$

because the original equation is $\scriptstyle n=f(n)$. Bo Jacoby (talk) 19:40, 29 June 2010 (UTC).

This is revealing but somewhat disillusioning to a math dullard such as me. No wonder I struggled with this! I assumed that my weak math skills (and poor memory of algebra class more than 30 years ago) had left me unable to see the solution. I am writing about the global debt crisis and comparing prospects for GDP growth (ax − a, where a represents GDP and x represents the annual rate of growth) with total debt (b) and the rate at which government debt would grow if deficits were eliminated now (or the interest rate on government debt + 1) (y). I am trying to determine the number of years (n) it would take for GDP growth to pay off this debt at different levels of growth. So I have to approximate toward this number by trial and error, then? This destroys my (apparently erroneous) idea that algebra offers an elegant way to solve any equation. Thanks for your responses. Marco polo (talk) 19:54, 29 June 2010 (UTC)

Easier to solve for y, to give the level of growth (y – 1) needed to pay off debt in n years. Qwfp (talk) 20:10, 29 June 2010 (UTC)

It is important to know that x>1, but you did not tell us that. There is no trial and error here. Give me values for a, b, x, y and I will compute n for you by the formula. Bo Jacoby (talk) 21:44, 29 June 2010 (UTC).

There are many things that can be done, and many things that cannot be done, and a lot of knowledge required to know the difference. So one should be careful not to jump to conclusions. It is correct that basic algebra cannot solve this equation as well as many other equations. It is wrong that you must resort to trial and error (or at least, depends on your definition of "trial and error").

Suppose that you did have a formula for n. What would you do with it? Punch the numbers for a, b, x, y into a computer and get a result for n. The very same thing - punching numbers and getting a number back - can be done for your equation even without a formula. It does require familiarity with either root-finding algorithms or a CAS. Many here have that and will be glad to give you n if you provide the values of the other variables. -- Meni Rosenfeld (talk) 08:49, 30 June 2010 (UTC)

How does a mathematical ignoramus solve an equation? Well, he dosn't. Either he learns math, or he leaves the job to mathematicians. Bo Jacoby (talk) 13:44, 30 June 2010 (UTC).

Sorry for not being nice. The steps leading to the solution are these.

$ax^{n}-a=by^{n}\,$

$x^{n}-1={\frac {b}{a}}y^{n}$

$x^{n}(1-x^{-n})={\frac {b}{a}}y^{n}$

$\left({\frac {x}{y}}\right)^{n}={\frac {b}{a}}(1-x^{-n})^{-1}$

$n\log {\frac {x}{y}}=\log {\frac {b}{a}}-\log(1-x^{-n})$

$n={\frac {\log {\frac {b}{a}}}{\log {\frac {x}{y}}}}-{\frac {\log(1-x^{-n})}{\log {\frac {x}{y}}}}$

defining

$k={\frac {1}{\log {\frac {x}{y}}}}$

and

$n_{0}=k\log {\frac {b}{a}}$

this is written

$n=n_{0}-k\log(1-x^{-n})\,$

Here the unknown is isolated on the left hand side of the equation, but alas it also occurs in a correction term on the right hand side. So we define

$n_{1}=n_{0}-k\log(1-x^{-n_{0}})$

$n_{2}=n_{0}-k\log(1-x^{-n_{1}})$

$n_{3}=n_{0}-k\log(1-x^{-n_{2}})$

and so on, providing better and better approximation to the exact solution n. This method does not resemble trial and error. Bo Jacoby (talk) 15:35, 30 June 2010 (UTC).

Newton iteration usually converges much faster than this simplistic fixpoint computation method. In the case at hand, it amounts to iterating the function

$f(n)=n-{\frac {ax^{n}-a-by^{n}}{ax^{n}\log x-by^{n}\log y}}$

(which can be simplified in some ways) from a suitable starting value of n (and in absense of any compelling argument otherwise, this starting value can as well be 0).—Emil J. 17:22, 30 June 2010 (UTC)

Please Marco Polo, provide values a, b, x, y for us to compete on speed and simplicity. However, the global debt crisis is easy because the global debt is zero. We are not borrowing money from ETs. Bo Jacoby (talk) 07:23, 1 July 2010 (UTC).

wealth is not a conserved quantity, so why should debt be? —Preceding unsigned comment added by 92.18.186.24 (talk) 10:11, 1 July 2010 (UTC)

Our net global debt is the conserved quantity zero, because we only trade within this globe. Bo Jacoby (talk) 13:36, 1 July 2010 (UTC).

At least until the sun asks to be paid for its services. -- Meni Rosenfeld (talk) 14:32, 1 July 2010 (UTC)

As long as all the people and institutions who think they are owed money agree to forgive the debts of all the people and institutions who have debts, there is no debt crisis. However, here in the real world, we have a problem. As for supplying the values for the variables in my equation, I have values for the United States for every year from 1955 to 2009. I wanted to come up with a formula that I could copy into a row of a spreadsheet to calculate the number of years required for economic growth to pay off the debt starting in each year. I'm not sure that Excel supports the kind of formula that some of you have so helpfully presented above. If Excel does support such a formula, perhaps you could tell me how? I prefer not to respond to Bo Jacoby's remarks, because they reveal a sort of personality that I can maximize my happiness by avoiding. I will say that, in my youth, I made some earnest efforts to learn math, but unfortunately had a series of instructors once I got past algebra and geometry who were unable to communicate in a way that most of the students in my classes could understand. Since then, I'm afraid I've been busy plying my non-mathematical trades and earning a living. I don't think that needs any excuses. Thanks for your attempts to help. Marco polo (talk) 17:42, 1 July 2010 (UTC)

I suggest you look up 'Goal Seek' in the Excel help. I've not used it myself, but I believe it should do be able to do what you want. In fact, there's an example using interest rates. Qwfp (talk) 17:55, 1 July 2010 (UTC)

That's one way. The only problem is that AFAIK it can only be done for one cell at a time.

There is another way, which can work for all rows in one go by practising the dark arts of Excel. First you need to enable iterative circular reference (in Excel 2003 this is done by going to Tools -> Options -> Calculation -> Iteration, for Excel 2007 there is an explanation here.). Then go to the cell where you want the result, and enter Emil's formula for $f(n)$:

$n-{\frac {ax^{n}-a-by^{n}}{ax^{n}\log x-by^{n}\log y}}$

Here's the tricky part: Wherever n appears in the formula, you need to reference the same cell in which you are typing this. Also note that "log" here means the natural logarithm, which in Excel is called LN. The result will magically appear in the cell. Then you can drag the cell to the subsequent rows as usual. -- Meni Rosenfeld (talk) 18:53, 1 July 2010 (UTC)

You may have columns for x, y, a, b, and easily compute columns for k, n0, n1, n2, n3. If n2=n3 then you are done. I am sorry that you do not want to talk to me. I tried to help you. Bo Jacoby (talk) 19:06, 1 July 2010 (UTC).

What are the four values (x, y, a, b) for the United States for one year, say 1955 ? Bo Jacoby (talk) 04:42, 4 July 2010 (UTC).

Folks, thanks to your guidance, I found a way to find the numbers I wanted using Excel. It wasn't exactly the method outlined by Meni, which I'm sure was correct for my original equation. Meni's method helped me to identify errors in my original equation, and I ended up constructing my own iteration in Excel, not by using a single formula, since (forgive me) my math is too weak to come up with the correct formula, but by repeating the same calculation in a vertical series of cells, each using the preceding cell's value as input. I then counted the number of iterations to get the value for each year, which wasn't too onerous, since there are just 54 years in my sample. Thanks again. Marco polo (talk) 15:41, 6 July 2010 (UTC)

Fan Curve

What is the fan curve equation - pressure vs volume flow rate - or how is it derived? In other words how is it related to the specifics of the fan (no of blades, blade angle, blade length etc)? Also how is the efficicency curve of the fan derived (if it can be derived - Im not talking about the friction loss in bearings etc but the hydraulic losses which depend on geometry)? If the answers to the above are not available, can someone tell me how the efficiency varies with fan rpm? Im not talking about ways to calculate/measure the above but get an equation for it. Thanks

Fluid dynamics is tricky business - I suspect that the answers to most of your questions cannot be expressed with a simple formula. However, they should be obtainable from a computerized simulation for any specified system. -- Meni Rosenfeld (talk) 18:31, 29 June 2010 (UTC)

The simplest approximate equation of pressure $\scriptstyle p$ versus volume flow rate $\scriptstyle Q$ is $\scriptstyle {\frac {p}{p_{\max }}}+{\frac {Q}{Q_{\max }}}=1$ where $\scriptstyle p_{\max }$ and $\scriptstyle Q_{\max }$ are proportional to fan rpm. Bo Jacoby (talk) 22:40, 30 June 2010 (UTC).

I have heard the pressure varies with the square of rpm and the velocity proportionally, then is he above equation right? Further I am looking for an equation if possible relating the above to the geometry of fan. Or to some extent given my ideal pressure and volume flow, how do I decide on my fan - in terms of no of blades and geometry? Thanks

Changing sign on the rpm does not change the square of rpm, so pressure does not vary with the square of rpm. You may have heard that the power $\scriptstyle P=p\cdot Q$ varies with the square of rpm. (I changed the notation to match that of the article: power (physics)). The above equation is not right, but it is the best you have got so far: a useful approximation, interpolating the facts that if $\scriptstyle p=0$ then $\scriptstyle Q=Q_{\max }$, and if $\scriptstyle Q=0$ then $\scriptstyle p=p_{\max }$. The maximum volume flow $\scriptstyle Q_{\max }$ varies with the section area $\scriptstyle A$ of the fan, but not with the number of blades $\scriptstyle n$, assuming that the fan is running sufficiently fast such that the air can be considered a heavy fluid. The maximum pressure $\scriptstyle p_{\max }$ varies not with $\scriptstyle A$, (assuming constant mean linear velocity of the fan blades) but with $\scriptstyle n$ . The geometry of the blades is complicated, as Meni told you; see Propeller (aircraft). Bo Jacoby (talk) 10:54, 3 July 2010 (UTC).

Forgive my ignorance but is the above equation at a specific rpm or changing with rpm...? for instance looking at a fan curve I find it reasonable to believe the above equation, that for a specific rpm, at zero static pressure volume flow is max and the converse. So r u defining Pmax and Qmax as maximums at a specific fan rpm or Pmax as the maximum pressure which can be delivered (at highest rpm). Also I read that power varies with cube of rpm, pressure with square and volume directly. Further can anyone tell me how efficiency varies with rpm - say if given an efficiency curve at a specific rpm how do I transpose it to a different rpm? My query is more efficiency oriented as I want to design/choose a fan for operation at a specific point of pressure and volume flow, hence ensure that at this point efficiency is as high as possible. So I wanted the derivation of the fan curve to find out (a)what is the maximum efficiency attainable (b)is this max efficiency attained at my desired point of pressure and volume flow. Thanks

1. $\scriptstyle p_{\max }$ and $\scriptstyle Q_{\max }$ are proportional to fan rpm. At constant rpm, $\scriptstyle p_{\max }$ and $\scriptstyle Q_{\max }$ are constants.
2. Power varies with the square (not the cube) of rpm, maximum pressure varies directly with rpm (not with the square), maximum volume flow varies directly with rpm.
3. The power $\scriptstyle P=pQ$ is zero when either $\scriptstyle p=p_{\max }$ or $\scriptstyle Q=Q_{\max }$. The maximum power theorem says that $\scriptstyle P_{\max }={\frac {1}{4}}p_{\max }Q_{\max }$ is obtained when $\scriptstyle {\frac {p}{p_{\max }}}={\frac {Q}{Q_{\max }}}={\frac {1}{2}}$
4. The energy conversion efficiency depend not only on rpm but also on $\scriptstyle p$ and $\scriptstyle Q$.
5. The ratio $\scriptstyle {\frac {p_{\max }}{Q_{\max }}}$ is the mechanical impedance of the fan. It is independent on rpm.

Bo Jacoby (talk) 20:49, 4 July 2010 (UTC).

Thanks Jacoby, that is a very clear explanation. If Pmax/Qmax is independent of rpm then certainly they must vary directly with rpm. However Im not sure whether this is true, for instance look at http://en.wikipedia.org/wiki/Affinity_laws. And isnt there some basic equation regarding centrifugal fans which goes: Power=mu^2(1-vcotβ/u), where u = impeller tip speed (proportional to rpm), v = normal component of flow exiting impeller (proportional to rpm), β = blade angle, m = mass flow rate (proportional to rpm) which would indicate pressure variance with square and power with cube of rpm? I am looking for such an equation which is possibly more elaborate, taking into account losses etc.