Wikipedia:Reference desk/Archives/Mathematics/2009 September 11

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September 11

Matrices and equations

I know that one may use matrices to solve linear equations of the form

${\begin{cases}a_{1}x_{1}+a_{2}x_{2}=c_{1}\\a_{3}x_{1}+a_{4}x_{2}=c_{2}\\\end{cases}}$

Where the c and a are constants. There are a number of methods, but the one I know best is Cramer's rule, which makes the solving of x₁ and x₂ a relatively simple matter:

$x_{1}={\frac {\begin{vmatrix}c_{1}&a_{2}\\c_{2}&a_{4}\end{vmatrix}}{\begin{vmatrix}a_{1}&a_{2}\\a_{3}&a_{4}\end{vmatrix}}}={{c_{1}a_{4}-a_{2}c_{2}} \over a_{1}a_{4}-a_{2}a_{3}}$

$x_{2}={\frac {\begin{vmatrix}a_{1}&c_{1}\\a_{3}&c_{2}\end{vmatrix}}{\begin{vmatrix}a_{1}&a_{2}\\a_{3}&a_{4}\end{vmatrix}}}={{a_{1}c_{2}-c_{1}a_{3}} \over a_{1}a_{4}-a_{2}a_{3}}$

This is all well and good, but Cramer's rule hits a stumbling block as soon as we have an x₁ or x₂ with a degree ≠1. My question is this: can matrices be used to solve systems of equations that are not linear, such as those containing polynomials and more complex expressions? For example, could a matrix be used to solve this?

${\begin{cases}3x^{2}-2y+5z=356\\4x+y-10z^{\frac {1}{2}}=145\\6x+11y+15z=512\\\end{cases}}$

Thanks for the help. 58.168.51.217 (talk) 07:41, 11 September 2009 (UTC)

To a great extent, no. Solving polynomial systems involves much, much more mathematics, both in the theory and algorithms. Special systems, of course, can be reduced to linearity, e.g. (changing your example a little) you can solve this

${\begin{cases}3x^{2}-2y+5z^{\frac {1}{2}}=356\\4x^{2}+y-10z^{\frac {1}{2}}=145\\6x^{2}+11y+15z^{\frac {1}{2}}=512\\\end{cases}}$

first as a linear system in x² ,y , z^1/2. --84.221.209.213 (talk) 09:14, 11 September 2009 (UTC)

Can you point me toward that mathematics? And I think I'm seeing what you've done here. What we find for z (for example) through Cramer's is really the square root of the z we're looking for. Is that correct? 58.168.51.217 (talk) 10:54, 11 September 2009 (UTC)

Correct. But are you more interested in explicit computations, algorithms, approximation of solutions, or more in the qualitative aspect: what is the shape and the structure of the set of solutions, in particular, existence results and so on. The two aspect are linked, of course. Another point is, what kind of functions a have you in your mind: polynomials, analitic functions, differentiable functions, continuous. Each has a whole theoried behind: commutative algebra algebraic geometry differential geometry nonlinear functional analysis degree theory fixed point theory...

Maybe the most simple interesting result, which is satisfactory both theoretically and for the practical point of view, is the contraction principle. --131.114.73.84 (talk) 12:09, 11 September 2009 (UTC)

If you don't put in additional restrictions then even if all the coefficients are integers and there is only one equation there is no general solution. This is Hilbert's tenth problem. Dmcq (talk) 12:25, 11 September 2009 (UTC)

...yes: talking about integers solutions. That's another main issue: does the OP want integer, real or complex solutions? That changes completely theory and methods.--131.114.73.84 (talk) 13:23, 11 September 2009 (UTC)

There is also the issue that even single-variable polynomial equations may have no easy solution, e.g. x⁵−x−1 = 0 has no solution in radicals. — Carl (CBM · talk) 13:31, 11 September 2009 (UTC)

As an applied mathematician, I would point you to Newton's Method. If you just want numerical values for x, y, z that solve the nonlinear system, that is you best bet. If you want to know if you can express the solution algebraically (i.e. something like x = sqrt(5)+log(4)), then you'll have to ask the pure mathematicians...

BTW: Cramer's rule is a pretty ineffcient way to solve a linear system of equations. Gaussian Elimination is much better. Cramers rule requires you to find n+1 nxn determinants - the most efficient way to calculate a determinant is ... Gaussian Elimination! So better to use that straightaway.195.128.250.134 (talk) 23:25, 11 September 2009 (UTC)

Let the n equations in n unknowns be p_i(x_j) = 0, i = 1..n, j = 1..n where the p-functions are polynomials. Then you may eliminate variables in the same way as if the equations are linear, ending up with n polynomial equations each in 1 unknown in the form p_i(x_i) = 0, i = 1,..,n. (See Buchberger's algorithm for technical details in the general case. In simple cases just eliminate variables). These polynomial equations are solved numerically by a root-finding algorithm. Bo Jacoby (talk) 09:24, 15 September 2009 (UTC).

Irrational perfect set

How can I find a nonempty perfect set in $\mathbb {R}$ which has no rational numbers in it. The book in which I found this problem has not discussed measure theory as yet, and so I do not want to use it in any way. Thanks--Shahab (talk) 07:45, 11 September 2009 (UTC)

for instance play with the binary expansions: fix an irrational number c, and consider all numbers 0<x<1 whose binary expansion has x_2k=c_k for all k, the other digits being free.--pma (talk) 08:03, 11 September 2009 (UTC)

Re OP: what do you mean by "perfect"? If you mean perfect set, the set of all irrationals is an example. If you mean perfect closed set, you can do it by a Cantor-set like construction, where you add an extra technique that uses an enumeration of the rationals. I don't want to say more in case this is a homework problem; once you see how the hint works you will be fine. — Carl (CBM · talk) 13:34, 11 September 2009 (UTC)

Perfect sets are always closed. The set of all irrationals is not perfect. — Emil J. 13:50, 11 September 2009 (UTC)

You're right, of course; I am so used to saying "perfect closed set" that I have to think about whether the terms are redundant. In any case, we have three solution techniques now. — Carl (CBM · talk) 13:53, 11 September 2009 (UTC)

More abstractly, you could begin with the inclusion of the Cantor space 2^ω into the Baire space ω^ω and compose this with a homeomorphism between Baire space and the irrational numbers. — Carl (CBM · talk) 13:40, 11 September 2009 (UTC)

No Carl this isn't homework. (Unfortunately I don't go to school). Anyway I thought about pma's hint and came up with the following reasoning. I want to confirm whether this is correct. Let c be an irrational number in [0,1] and $E:=\{x\in [0,1]:x_{2k}=c_{k}\}$. Clearly as the 2kth digit of any $x\in E$ doesn't follow a pattern so x is irrational. Moreover any neighborhood of such an x is bound to contain a $y(\neq x)\in E$ as other then the even placed digits all others may be freely chosen. So all points of E are its limit points. Also if y is any limit point outside E then let 2t be the first position from the left in the decimal expansion of y such that y_2t doesn't coincide with c_t, and let e be the rational number in (0,1) with 1 at the 2t position and 0's elsewhere. Then (y-e,y+e) doesn't contain any element of E. Hence E is closed. --Shahab (talk) 19:05, 11 September 2009 (UTC)

Correct! (Alternatively, you can prove that E is homeomorphic to the Cantor space 2^ω quoted by Carl, certainly a compact space with no isolated points.) --pma (talk) 20:41, 11 September 2009 (UTC)

Life expectancy of the oldest

Today's passing away of Gertrude Baines made me think of the kind of question my old physics prof liked to ask: A general question that requires an understanding of what is relevant, which assumptions to make, and how to apply the correct mathematical methods. Maybe I shouldn't post this here, because what I'm looking for is not a mathematically precise result, but rather an estimate of the order of magnitude. I wanted to figure this out myself, but I need to do other things now; and because it is connected to a current event I rather posted it here for others who might enjoy it as long as it's fresh:

What is the life expectancy of someone who just became the oldest person? Or, asked differently, how long is the expectation value for the time a person holds the title "oldest person"? Of course, you could just go to Oldest people#Oldest living people since 1955 and divide the observed time by the number of people, but how would you go about it if you didn't have that information? If you don't like ignoring what you already know, you might go the other way and ask: Given the observed life expectancy of the oldest, what conclusions can we draw about Theories of aging? — Sebastian 20:23, 11 September 2009 (UTC)

The fact that they are the oldest person isn't relevant. You would calculate their life expectancy in the same way as with anyone else, using a life table. Life tables aren't particularly accurate at the upper end due to the very small sample size. If someone is the older person ever, then there aren't any data points to base the expectation on. You could try and extrapolate based on the last few years you do have data for, but that wouldn't be very reliable at all. --Tango (talk) 20:35, 11 September 2009 (UTC)

What the conditional expected lifetime is, given that the person is the oldest person, but not given the age, is a perfectly valid question in the right context. But I think the "right context" for me might involve knowing more than we're given here. Michael Hardy (talk) 20:57, 11 September 2009 (UTC)

I think the events are independent. How does someone dying 1000 miles away affect the chances of another person dying? Even if they were both in some kind of cohort initially. An insurance company might revise it's estimate of the probability based on the new evidence, but I think the actual probability remains the same.--RDBury (talk) 21:23, 11 September 2009 (UTC)

Well, yes, you could work out the probability distribution of the age of the current oldest person and use that, combined with the life tables, to get the life expectancy of a randomly selected oldest person, but I'm not sure why you would want to. --Tango (talk) 21:41, 11 September 2009 (UTC)

Jeanne Calment's answer at age 120 was "A very short one!".John Z (talk) 22:41, 11 September 2009 (UTC)

I guess she already said that at age 90, when she negotiated the reverse mortgage.  ;-) — Sebastian 07:45, 12 September 2009 (UTC)

The general statistical methods for such problems are described in extreme value theory. Robinh (talk) 06:23, 12 September 2009 (UTC)

Yup, that's basically what I've been looking for. I'm a bit surprised that it takes its own theory and that it was discovered so late - I expected it to be rather simple exercise. — Sebastian 07:45, 12 September 2009 (UTC)

Using discrete crime data points to estimate relative rates by geography

I have a set of datapoints that represent individual murders-- each datapoint has a longitude and latitude. How can I produce an estimate for any geographic point, such that the map could be painted with an overlay representing the estimated murder rate for a given area.

Obviously, there will be limits to how accurate the estimates can be, just trying to find a good ballpark sort of measure.

I had thought about doing something along the lines of:

Estimated rate at a point ~= Sum over all data points(1/distance to murder)

But this seems sort of arbitrary, in that, it's something I just made up. Perhaps murder rate should use an inverse square rule instead. Or maybe there is some more advanced procedure for estimating a location-based-rate using a random sampling of data points.

Advice greatly welcomed.

(also know, murder rate isn't the actual issue being explored, it's just easier to describe) --Alecmconroy (talk) 20:47, 11 September 2009 (UTC)

Since when did this forum get so morbid? Anyway, you probably want to find some sort of data analysis/data mining software. This kind of problem occurs in business all the time, though more people who might want to by a product than people who get murdered, and there is commercial software out there to help with it. I don't know if I'm allowed to mention a specific product but try Googling "mapping software" as a first step. It's an interesting problem but it's already being worked by people who are actually getting paid to do it.--RDBury (talk) 21:37, 11 September 2009 (UTC)

Yes, that is an interesting Rorschach isn't it-- when asked to come up with a meaning for a binary discrete samples, I just think murder. Murder has a certain binariness to it in a way that "crime" or "disease" or "profit" just lack. lol I guess I am morbid.

Any way, thanks to the commenters, I think I'm on the right trail. RDBury (or others)-- do we know offhand of any such software that would be easily available to do this? I actually already have the data in google maps, so if it can do it, that would be extra awesome-- but I can't find any mention it in the documentation right now. --Alecmconroy (talk) 09:20, 12 September 2009 (UTC)

Sounds like you want some kind of 2D histogram. Or, more sophisticatedly, density estimation. Yaris678 (talk) 21:47, 11 September 2009 (UTC)

Differential equation

Just wondering if anyone could give a hint (or can solve)

d2[fn(x)]/dx2 = -k/[fn(x)]2

Starts at f(0)=real number, f'(x)=0. Any ideas? (not advanced student)83.100.250.79 (talk) 22:45, 11 September 2009 (UTC)

It's a one dimensional motion with an inverse square law force. Multiply by d[fn(x)]/dx and integrate. You'll get the conservation of energy, that gives a first order autonomous equation, that you can integrate again. --78.13.143.41 (talk) 23:18, 11 September 2009 (UTC)

Got stuck at "Multiply by d[fn(x)]/dx and integrate." - left hand side integration by parts is giving me more dⁿ[f(x)]/dxⁿ ...83.100.250.79 (talk) 23:38, 11 September 2009 (UTC)

Use substitution. --COVIZAPIBETEFOKY (talk) 02:52, 12 September 2009 (UTC)

Substitute what? There's not a lot there.83.100.250.79 (talk) 11:00, 12 September 2009 (UTC)

I'm not going to do all your homework for you. Both integrals can be evaluated using the technique of substitution. Figure out what u should be in each integral, such that the expression inside the integral is of the form f(u)du. --COVIZAPIBETEFOKY (talk) 15:23, 12 September 2009 (UTC)

It's not homework, it's curiosity. And I'm looking for a link to an answer, or more helpful answer, I'm familiar with integration by substitution, but I'm stuck. Is this a well known differential. Can someone link to a method to solve it. Thanks.83.100.250.79 (talk) 15:34, 12 September 2009 (UTC)

If you do the multiplication suggested then on the LHS you will have a function (d(fn(x))/dx) and its derivative - that kind of integral is generally done by substituting u=(the function) (in this case, u=d(fn(x))/dx). On the RHS you have an expression involving a function (fn(x)) and the derivative of that function, so you should substitute u=fn(x). I haven't actually tried it, but that ought to work. --Tango (talk) 16:24, 12 September 2009 (UTC)

I still don't understand the multiplication method... I found a substitution: I found this [1] which gives the answer - I suppose that's what user:78... meant when they said "its one dimensional motion with inverse square law" - I found it by using that as a search term.

It'll probably make sense in a few moments once I've worked it through...83.100.250.79 (talk) 16:51, 12 September 2009 (UTC)

You just multiply both sides by d[fn(x)]/dx before you integrate, it makes for easier integrals (which you can do using the substitutions I described). --Tango (talk) 02:09, 13 September 2009 (UTC)