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The Art of Problem Solving
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| The Axiom of Infinity |
[May. 9th, 2012|09:25 pm] |
What am I missing here? These definitions are taken from Smullyan's Set Theory and the Continuum Hypothesis.
A class W is called inductive if(f) ∅ is an element of W and for any set x in W, x ∪ {x} is also an element of W. We call a set n a natural number if(f) it belongs to every inductive set.
From this alone, we don't seem to be guaranteed that there are even any natural numbers, for we don't seem to be guaranteed that there exists an inductive set. Clearly the class N = {∅, {∅}, {{∅}}, ...} exists as an inductive class, but we aren't guaranteed it's a set. Indeed, we need the axiom of infinity to deem it a set.
So if we define ω to be the class of all natural numbers (i.e. the class of all sets which belong to every inductive set), then I don't see how we've actually defined anything except the empty class, ω = ∅. It seems like we need an axiom asserting that there exists an inductive set, which I've seen some authors do.
So is Smullyan's text lacking in some way? Is there a better introduction to set theory? Or is there something I'm just missing about this whole setup? |
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| Harmonic functions |
[May. 6th, 2012|06:04 pm] |
Let G in C be an open set and take a in G. Suppose that u(z) is harmonic in G-{a} such that lim_{z-> a} u(z) exists and equals A. Show that U:G-->R given by U(z)=u(z) for z=/=a and U(a)=A is harmonic in G.
Somehow this relates to the Dirichlet problem, but I only know how to solve it for a disk and can't figure out quite how it applies to this problem. Any help is appreciated! |
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| The Lottery |
[Mar. 30th, 2012|11:40 pm] |
1) What are your thoughts on lotteries in general,
2) When the expected winnings are greater than the cost of the ticket (as in tonight's Mega Millions game*, $640 million jackpot at odds of 1 in ~176 million; $1 to play), would that make you substantially more likely to buy a ticket?
Note: It is not physically feasible to buy one of each combination. (If it takes 5 seconds to fill out a ticket, it would take 28 years to get them all, and then there's the sheer weight/volume of the paper, ink quantities etc.)
*Edit: I neglected to account for taxes, the annuity's present value (or lump sum), and splitting the jackpot among winners. According to AP, the actual jackpot (present value after taxes) is around $347m, and nearly 1.5b tickets were sold (was not expecting it to be that high!), making the expected winnings about 23 cents on the dollar (about 40 cents including the smaller prizes). So yeah, bad bet. |
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| Breech of Contract Incentives |
[Mar. 29th, 2012|10:08 pm] |
The question is this: If a Client has a certain amount of capital (let’s assume 100) and wants to to pay someone (the Agent) to complete a task, how much should the Client offer up front and how much should be left for after the completion of the project? We’ll make the simplifying assumption that the Client has 100 discrete units of currency, a positive integer k of which may be given as payment. We must also assume that the Client will have perfect knowledge of the completion of the project. There is the possibility for two uncertainties in this problem and thus two opportunities for a probabilistic model. The first is whether the Agent will take the down payment and jet off to Schmuck Island without completing the task, in which case the Client has lost k units of currency and has an unfinished project. We will ignore, for now, the second risk: that the Client will not pay the Agent the remainder of the 100 after completing the task. In some situations this is perfectly valid: if the Client wants the Agent to retrieve a certain object, he cannot obtain that object until the remainder is paid.
It may help to elucidate precisely what the contract is. The Agent and Client agree to the following:
The Client pays k units of currency to the Agent prior to the beginning of the task.
The Agent will perform the agreed upon task and report back to the Client with proof that the task is complete.
The Client will pay the Agent the remaining 100 – k units of currency.
Considered as the expected outcome of an uncertain event, we condition on the possible breeches of contract (including none).
Since the only risk under our assumptions is from the Client’s perspective, we denote the degree of trust the Client has for the Agent by p. We assume 0 < p < 1. In fact, we may assume p > 1/2 (if the Client felt it was more likely that the Agent would renege than complete the assignment, there would be greater concerns about the Client’s judgment apparatus than how much money he lost).
From the Client’s perspective, there are two possible outcomes of the contract: the Agent completes it or does not. The relative probability the Agent does not complete the task is given by 1 – p. So we can condition on whether the assignment is completed. The unknown event in this situation is whether the Agent completes the assignment. It is in the Client’s best interest that the Agent completes it.
Now we introduce the second layer of uncertainty: whether the Client will pay the remaining 100 – k or jet himself off to Schmuck Island. It’s of course not possible for the Client to renege on the contract if the Agent already has. We’ll denote the level of trust the Agent has for the Client by q.
We have to consider the random variables X and Y that denote what the Agent and Client, respectively, will obtain from the contract from their own perspectives. We are assuming that neither party has any intention of reneging, only that they think there is a possibility the other party might. In particular, to ensure an equal level of trust between both parties, we must require that E(X) = E(Y), i.e. that both parties can expect to gain the same amount from engaging in the contract. We denote the value that the Client gains from the Agent’s completion of the assignment by v. We assume that the agent gains nothing from the completion of the project, i.e. that v = 0 for the Agent.
Possible outcomes of X and their probabilities:
X = k with probability 1 – q. [The Agent makes at least k from agreeing to the contract, but doesn't obtain the remainder because the Client has reneged.]
X = 100 with probability q. [The Agent completes the assignment and the Client fulfills his end of the contract.]
Possible outcomes of Y and their probabilities:
Y = 100 – k with probability 1 – p. [The Agent reneges, so the Client is left with his remainder.]
Y = v with probability p. [The Agent fulfills her end of the contract. The Client has paid his remainder and has the value v of a completed project.]
Now we impose the requirement that E(X) = E(Y). In particular,
k*(1 – q) + 100*q = (100 – k)*(1 – p) + v*p
k = [100(1 - p - q) + vp]/[2 - p - q].
Note that from the beginning we assumed p and q exceeded 1/2, so that their sum exceeded 1 but was less than 2, so we have no risk of dividing by 0 or a negative number. Let’s assume for simplicity that v = 100. In that case,
k = [100(1 - q)]/[2 - p - q].
As a specific example, let’s assume that the Client and the Agent have equal trust for each other p = q = 0.8. Then k = 50. In fact, whenever p = q, k = [100(1 - q)]/[2 - 2q] = 50*(1 – q)/(1 – q) = 50. Since p only appears in the denominator with a negative sign, increasing it will cause k to increase. In English, the more the Client trusts that the Agent will complete the task, the more k should be. This makes perfect sense. If the Client had complete trust of the Agent, he could just pay her all 100 at the beginning. Note that k also must increase when the Agent trusts the Client less, for in that case the Agent wants to ensure that she is well compensated for efforts that may not be reimbursed.
For different values of v the situation might look different. The reader can investigate situations where v = 0, 50, 150, 200, 777. |
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| Finite fields, vector spaces, and linear dependence |
[Feb. 25th, 2012|03:30 pm] |
Hey all.
I'm working on a proof for a theorem that I know to be true, and I'm hoping to derive an easy, elegant proof for the theorem. It relies on vector spaces over finite fields, which isn't a strong area of mine, so I was hoping someone here could help me out.
Essentially, say we have GF(q), for q any prime power, and the vector space / field V=GF(q^3) over GF(q). Let f be a degree 3 primitive polynomial over GF(q) with root a so that GF(q^3) = GF(q)[a]/(f).
Now, say for 0 <= i < j < (q^3-1)/(q-1), we know that {a^i, a^j} is linearly independent.
What I would like to show is that there is no s in GF(q) such that for all b in GF(q^3),
Tr(b * a^i) = s Tr(b * a^j), where Tr is the trace function from GF(q^3) to GF(q).
If I try to prove via contradiction, i.e. assume s exists, and then rewrite as:
Tr(b * (a^i - s * a^j)) = 0 for all b in GF(q^3)
(or equivalently, Tr(a^(i+k) - s a^(j+k)) = 0 for all k)
is there any way that I can conclude that this implies a^i - s * a^j = 0, which would give me the contradiction that I need? I'm hoping since it's true for all b, somehow this fact has to be easily shown to be true.
Help, anyone? Pretty please? |
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| (no subject) |
[Jan. 27th, 2012|01:04 pm] |
I would like to simulate the chaotic flow along the geodesics of a compact Riemann Surface of constant positive negative curvature (cf. Hadamard Billard). However, I could not find out how to represent the manifold numerically. Any ideas? |
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| Number of functions |
[Jan. 17th, 2012|01:35 pm] |
Assuming the generalized continuum hypothesis, how many functions are there from R^n to R?
Is it correct that there are |R|^{|R^n|}=|R|^{|R|}= \aleph_2? |
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| Arg Sup vs Arg Max |
[Jan. 15th, 2012|12:44 pm] |
Let X be the space of functions from R -> R.
Let f be a function from X->R
Is arg max_{x \in X} f(x) = arg sup_{x \in X} f(x)?
I think this is true because, if the supremum exists, but the maximum doesn't, then there is no x that creates the supremum's value, so the "arg supremum" does not exist. I'm well outside of my comfort zone though, so I wanted to check with you guys. |
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| Conformal mappings |
[Nov. 6th, 2011|10:17 pm] |
I have two questions:
1. Find a bijective conformal mapping from G={z in C : |z|<2 and |z-1|>1} onto the open unit disc D.
I'm not really sure where to begin. This set G is more complicated than any of the examples we have of this sort of process; all of our examples have been simply connected.
2. Show that f(z)=(z2 - 1)/(z2 + 1) is a bijective conformal mapping from the half-plane H={z : Re(z)>0} to C\{x in R: |x|>= 1}, and use this to find an analytic function such that tan(f(z))=z, where f: C-{iy : |y|>=1} to H.
I thought at first f would just be the composition of w=z^2 with (w-1)/(w+1), but this doesn't seem to work. The map w=z^2 sends H to C\{x in R : x=<0}, and I know that (w-1)/(w+1) maps the upper half plane to the upper half plane, but I'm not sure what it does to C\{x in R : x=<0}...
For the second part, it seems like I want to consider the function if-1, yes?
Thanks for the help |
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| Convolution |
[Oct. 28th, 2011|10:20 am] |
Trying to teach myself the black art, I attempted to convolute (convolve?) t2 and et.
I get 2et - t2 - 2t - 2.
Have I got it right, or do I need to go bash my head against the brick walls of algebra and integration-by-parts again? |
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