Since I am trying to get the bounty room going, perhaps a reasonable thing could be to promote it here. So here is link to the relevant meta post and here is the room.
Feel free to tell me off if mentioning that room here is getting too repetivitive and everybody of the chat regulars already saw it - if that's the case I will try to space it out a bit.
@TobiasKildetoft That is on purpose, so that the first thing people see when they click the link is: "Before posting here, please, read the rules."
@mick If that bounty expires (or if you have other bounties from the past with no answer you are still interested in, the above room might be of interest for you).
But if I want to get it going, I have to try somehow spread the word. Mentioning it in chat is one possibility.
Basically, if enough users (both bounty offerers and potential answerers) become aware of the room, then there is chance that it eventually becomes useful.
What information do we get from the graph of the scalar function $\psi (\lambda )=f(\lambda x_1, \lambda x_2)$ at the point $(x_1, x_2)=(1,1)$ for a function $f$ ?
I'd say that by looking only on one direction, you cannot say anything about global behavior of the function of two variables. But you might be able to refute some things.
For example, if the section is not convex, the whole function is not convex. If a point is non a minimum of the section, then it is not a minimum of two-variables functions.
@MaryStar Well, you said you are looking at values at $(\lambda x_1,\lambda x_2)=(\lambda,\lambda)$ (for $x_1=x_2=1$). Those are precisely the points where the two coordinates are equal.
Anyway, I will have to leave. But I don't think I am able to provide some more inside into your question from chat than what I posted above.
Consider the system of equations
$$
\begin{align*}
x^2+(1-y)^2&=a\\
y^2+(1-z)^2&=b\\
z^2+(1-x)^2&=c\\
\end{align*}
$$
Compute $x(1-x)$ in terms of $a,b,c$.
Edit:
The question should say
Compute all possible values of $x(1-x)$ in terms of $a,b,c$
Oh man. I really don't understand how students dare to claim they didn't copy answers from eachother because their constants are named differently, when they have exactly the same steps, same solutions and same mistakes.
a fun problem I came across: $x$ and $y$ are uniformly independently distributed random variables on $(0,1)$. Find the probability that $\lfloor\frac{1}{xy}\rfloor$ is even
@DHMO ~0.25, first, half of the intervall will yield 1, which is odd, and half of the other half will yield odd numbers. But that isn't to be taken serious haha
A curious puzzle for which I would appreciate an explanation.
For $x$ and $y$ both uniformly and independently distributed in $[0,1]$,
the value of $\lfloor 1/(x y) \rfloor$ has a bias toward odd numbers.
Here are $10$ random trials:
$$51, 34, 1, 239, 9, 4, 2, 1, 1, 1 $$
with $7$ odd numbers.
H...
my question got 5 votes and the answer was perfect and I gave it one upvote and no one else upvoted the answer. :( I feel bad for the guy. I gave him my bounty though
hey guys, have a question with method, "Find the sum of the first $n$ terms of the series : $(2*3)+(3*4)+(4*5)+...+(n+1)(n+2)$" I can be awarded 5 marks for this however it seems really simple; $\sum_{r=1}^n (r+1)(r+2)$. Where are the 5 marks allocated?
"For first order logic, note that the first order logic is undecidable, i.e. given a formula there is no algorithm to check if it is a valid formula. Any computable upperbound on the proof size would give an algorithm for deciding first order logic, check all possible proofs up to that size.
(Note that an upper bound on proof length in natural deduction/sequent calculus will give an upper bound on the size because of normalization/cut elimination). Since there is no such algorithm, there cannot be any computable upperbound."
Regarding the "This statement…10^100 steps" thing I mentioned earlier, the reason that we know it's provable at all is that there are finitely many theorems of length less than 10^100, so we can just check them all.
None of the proofs of length less than 10^100 can prove it, because proving a statement means it's true (assuming consistency), and if it's true the proof can't be less than 10^100 long, contradiction.
So we check all proofs of length less than 10^100, see that none of them work, and conclude that the theorem is true.
The metamathematical proof that I just gave doesn't count because it can't be written in the language of PA.
What's weird about that hydra game is that, on the one hand, by its design (normal + dire heads) it definitely seems worse than the standard hydra. but it's also got a probability aspect i.e. it doesn't always grow new heads, and when it does it's not always the same number of copies
I was just mentioning it because it's a similar idea (in terms of proof)
The hydra game itself came up because metamathematics came up
@MikeMiller Because the hydra in the applet is constrained by the fact that it can't clutter the screen too much, there's probably a (relatively) reasonable bound on how many hydras could possibly appear
which means that the worst case scenario, which goes through every possible hydra once, would be relatively reasonable and not at all Goldstein-like
@MikeMiller Someone was asking why the question "Does this language prove statement T" was unsolvable — specifically, why it's not NP. I think the idea is that the length of the proof is not a polynomial in the length of the theorem, or even a computable function
@Akiva Right. The only pseudo-algorithm involves enumerating all strings and checking if one is a proof. If the theorem is not provable, you're in bad shape!
You could also enumerate strings to see if one is a proof of falsity but if it's independent you're toast
In regards to my previous message of ""Find the sum of the first $n$ terms of the series : $(2*3)+(3*4)+(4*5)+...+(n+1)(n+2)$" I can be awarded 5 marks for this however it seems really simple; $\sum_{r=1}^n (r+1)(r+2)$. Where are the 5 marks allocated?"
I have found the sum of the first n terms to be $\frac{n^3+6n^2+11}{3}$
oh, btw: I just realized that the hydra game's page is an xhtml script, so you can actually view the source code in-browser [here](view-source:madore.org/~david/math/hydra.xhtml)
I don't have a specific question. @Semiclassical I wanted to know if it is a good topic for a presentation and if so, how the structure of the presentation could look like.
To be honest, I haven't occupied so far with boolean functions. I found the definition in wikipedia: In mathematics and logic, a (finitary) Boolean function (or switching function) is a function of the form ƒ : B^k → B, where B = {0, 1} is a Boolean domain and k is a non-negative integer called the arity of the function. In the case where k = 0, the "function" is essentially a constant element of B. @TobiasKildetoft
@TobiasKildetoft I thought that it would be interesting. @Semiclassical The prof said that we should find the topic, without him since he has no time right now. Because of the fact that we gave no restrictions, I don't know how to choose one...
Consider the system of equations
$$
\begin{align*}
x^2+(1-y)^2&=a\\
y^2+(1-z)^2&=b\\
z^2+(1-x)^2&=c\\
\end{align*}
$$
Compute $x(1-x)$ in terms of $a,b,c$.
Edit:
The question should say
Compute all possible values of $x(1-x)$ in terms of $a,b,c$
