Math question thread

[Surlethe]

Okay, I'm bored and putting off studying for my exams. Post questions you have about math, math problems, the philosophy of math, and I (or those of us who are math/science-literate, if y'all want) will try to answer. If I/we have time.

[Molyneux]

How the heck, conceptually, can you have a number raised to the big-O of something?

I can understand how it works, mathematically - I know how to juggle the numbers and get the right answer - but I just plain cannot understand what that represents in the real world.

If X=O(Y), that means that there exists some constant k for which, after a certain point, Y times k will always be greater than or equal to X. It means that the fastest X can grow is Y.

What the hell does that mean when you get to something like X = 2^O(Y)?

[Beowulf]

How the heck, conceptually, can you have a number raised to the big-O of something?

I can understand how it works, mathematically - I know how to juggle the numbers and get the right answer - but I just plain cannot understand what that represents in the real world.

If X=O(Y), that means that there exists some constant k for which, after a certain point, Y times k will always be greater than or equal to X. It means that the fastest X can grow is Y.

What the hell does that mean when you get to something like X = 2^O(Y)?

Do you have a context for when you saw it being used like that? Because it conceptually doesn't make any sense to me.

[Number Theoretic]

I have seen that use of O-Notation rarely but as far as i understood it, the O(X) can be thought of as a template for any function which fits into the given complexity class. But it appears to me as sloppy notation.

[Kuroneko]

I shouldn't mean anything but log X = O(Y). It's a different statement from any X = O(f) to say that X is bounded by 2[sup]kY[/sup] for some k, so there's some use for it.

[Singular Intellect]

If energy source 'S' currently provides for .5% of all energy consumption and that figure is doubling every two years, how many years until it provides all energy needs and what is the annual growth rate of 'S'? Assume total energy consumption is growing at 1% annually.

[Surlethe]

Doubling every two years means it can be modeled by S(t) = 0.005T_0 exp(t sqrt(2)). If total energy consumption T grows 1% annually, it can be modeled as T(t) = T_0 exp(0.01t). Then S(t) = T(t) iff 0.005exp(t sqrt(2)) = exp(0.01t), so ln(0.005) + tsqrt(2) = 0.01t, or t = ln(0.005)/(0.01-sqrt(2)). (Note that t>0 because ln(0.005) < 0 and 0.001 < sqrt(2).)

Exercise 1: How quickly is S increasing when it passes T? Is this realistic?

Exercise 2: Model S as a modified logistic function with carrying capacity T(t).

[Aniron]

I am lost on this question from my caclulus text book.

Consider the function, f(x)=6sin(3x). Calculate the total area trapped between the x-axis and the function, from x= -pi/3 to x=pi/3.

I know just calculating that using the fundamental theorem results in 0, but that's the net area, not the total area, if I recall correctly. Any tips?

I believe the integral goes from 0 to pi/3. Use substitution, where u = 3x and du=3dx. Then compensate the 3dx with 1/3.

[Surlethe]

I'm not going to do your calculus homework for you. However, here's a vague hint: you want to calculate the unsigned area. How can you get rid of minus signs without distorting the area? Compute accordingly.

Exercise: Come up with a second method for computing unsigned area. Explain why the two methods are equivalent.

[Aniron]

I'm not going to do your calculus homework for you. However, here's a vague hint: you want to calculate the unsigned area. How can you get rid of minus signs without distorting the area? Compute accordingly.

Exercise: Come up with a second method for computing unsigned area. Explain why the two methods are equivalent.

I think I figured it out. I used absolute value.

Thanks fro the tips.

[Feil]

Do the neat mathematical definitions of self-information, shannon entropy, etc, that one sees in information theory and computing, have application to real world things? Could one talk meaningfully about the shannon entropy of an atom? A cat?

[starslayer]

Doubling every two years means it can be modeled by S(t) = 0.005T_0 exp(t sqrt(2)). If total energy consumption T grows 1% annually, it can be modeled as T(t) = T_0 exp(0.01t). Then S(t) = T(t) iff 0.005exp(t sqrt(2)) = exp(0.01t), so ln(0.005) + tsqrt(2) = 0.01t, or t = ln(0.005)/(0.01-sqrt(2)). (Note that t>0 because ln(0.005) < 0 and 0.001 < sqrt(2).

You might want to check that math. This gives a time of only 3.8 years from .5% to meeting all energy needs, which is obviously absurd unless I'm missing something here.

[Surlethe]

Whoops, I think S should be S(t) = 0.005T_0 exp(t ln(sqrt(2))).

And I don't know shit about Shannon entropy or information theory

[Surlethe]

Wait, there's something a little subtler going on with that. Bonus points to the person who figures out my latest mistake.

[Eternal_Freedom]

If the differentional of a value represents the rate of change of said value, what does the integral represent?

[Surlethe]

The integral represents a total quantity. Think of the value as a density; integrating the density over a region represents the total amount of mass in that region.

(Nitpick: the differential of a value is not exactly the same as the rate of change of the value.)

[Eternal_Freedom]

Thanks, I could never quite wrap my head around what the integral actually was in the real world rather than graphland.

So what does the differential represent if not the rate of change?

[Surlethe]

I am distinguishing a function's derivative from its differential. The derivative is the instantaneous rate of change. The differential is its instantaneous change. If f is the function we're talking about, think about an infinitesimal triangle with two sides parallel to the axes and hypotenuse tangent to f at x. The differential df is the length of the side parallel to the y axis. The differential dx is the length of the side parallel to the x axis. The derivative df/dx is the slope of the tangent line. (Kind of crappy picture: http://upload.wikimedia.org/wikipedia/c ... uncion.png)

In a more general setting, the two are less closely related. Over the real numbers, there is only one degree of freedom: forward-backward. When you have more, "derivative" loses meaning -- you have to ask which direction, leading to the (creatively-named) directional derivative. However, the differential retains its meaning as the "total derivative." In even more general settings, the differential of a function is a one-form which encodes information about all of the directional derivatives.

[Eternal_Freedom]

Oooooook...

Thanks, I got it now. Took me three readings to get it, but I'm there.

[Feil]

I'm not going to do your calculus homework for you. However, here's a vague hint: you want to calculate the unsigned area. How can you get rid of minus signs without distorting the area? Compute accordingly.

Exercise: Come up with a second method for computing unsigned area. Explain why the two methods are equivalent.

I think I figured it out. I used absolute value.

Thanks fro the tips.

Unfortunately, you can't integrate smoothly over an absolute value sign: you have to integrate it stepwise between the zeros and add. For a cleaner function, look at the resultant graph of y=|a*sin(bx)|. It has the form of a new sine function, y=c*sin(dx)+e. Derive the new function graphically to obtain a function that you can integrate over the entire length.

For any limits of integration which give you a whole number of periods in your function and for which your sine function has constant amplitude, you can also use root-mean-square (RMS) for your integration. Look it up. Prove that the integral of the RMS line is equivalent to the integral of the absolute value over a whole number of periods.

[fnord]

Given an unknown target, x, on the real interval [0..1], and an oracle truthfully answering questions of the form "is x <= c", bisecting the interval n times will suffice to localise x to within an interval of size 2 ^ -n.

How does your approach change to localise x given that the oracle can return false positive answers with conditional probability alpha, and false negative answers with conditional probability beta, 0 < alpha < 1, 0 < beta < 1?

[Surlethe]

Ooh, that's interesting.

Let me rephrase the question. "Suppose c is fixed in the real line (or really any metric space). Given x, an oracle answers "Yes" or "No" to the question "Is x < c?" such that Pr(Yes | x < c) = a and Pr(Yes | x > c) = b, a, b between 0, 1. Is there a strategy to 'localize' to c, by which I mean, a strategy to construct a sequence x_i converging to c? If so, how quickly does it converge?"

Let me think about this.

[fnord]

I maybe wasn't too clear

P(Oracle says x > c | x > c) = 1 - b
P(Oracle says x <= c | x > c) = b

P(Oracle says x > c | x <= c) = a
P(Oracle says x <= c | x <= c) = 1 - a

The original x is uniformly distributed over the interval.

[Bottlestein]

^ Bayes Theorem.

Partition the sample space on x > c, and x < = c (Due to X being distributed an uniform pdf over compact [0,1] the event probabilities are 1-c and c respectively)

[Bottlestein]

@ Surly:

Not math per se, but interesting nonetheless:

So with negative index of refraction (meta)materials being made - could we see Cerenkov in air?
You know, accelerate an electron out of a metamaterial and with c_m > c_air ...?