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comment What is the mixed strategy equilibrium bid, if any, for complete information auction games with minimum bid?
Does "complete-information" mean that both players know the values of $a$ and $\bar b$? If so, why are the bids not expressed as $b_i\in[\bar b, a)$? If not, what do they actually know?
Jul
30
comment Representing all pairs shortest path in a graph with a matrix
You will, at the very least, need to add an assumption that there are no negative-weight cycles, since otherwise the shortest paths are not well defined.
Jul
26
comment Generate all De Bruijn sequences
@qwr, DFS as typically described in algorithms text visits each edge only once and each node only twice. I can see how tracing an Eulerian path could be thought of as both depth-first and a search, but to call it DFS is to invite confusion. And to generate all Eulerian cycles you need to backtrack.
Jul
18
comment Generate all De Bruijn sequences
en.wikipedia.org/wiki/De_Bruijn_sequence#Construction en.wikipedia.org/wiki/De_Bruijn_graph en.wikipedia.org/wiki/…
Jul
13
comment Game on simple finite graphs
I'm glad that I now know why our calculations disagreed.
Jul
13
comment Game on simple finite graphs
@hardmath, "the smallest non-negative integer that is not already assigned to its neighbours". Since the only integers assigned to the neighbours are 2 and 2, the smallest unassigned non-negative one is 0.
Jul
13
comment Game on simple finite graphs
@hardmath, with both path(A,-1,K) and path(A,2,k) the value which will be played into the gap next to B can only be 0 or 1, and in particular is never -1 or 2 in either case; in fact, we could go further and say that for any $B \not\in \{0,1\}$, path(A,-1,K) = path(A,B,K). path(2,2,1) has only one possible move, which is to play 0 into the gap and lose, exactly as with path(-1,-1,1). Other discrepancies are (1,0,3), (1,0,5), (1,0,7), (2,0,7), (1,1,2), (1,1,7), (2,1,1), (2,1,5), (2,1,7).
Jul
2
comment Game on simple finite graphs
When working with paths, $a$ and $b$ are only going to be $0$ or $1$ for the most part: the exceptions are that filling the hole in $0?1$ requires a $2$ (although that's then a finished segment), and the $-1$ endpoint special case. One thing I don't understand from your table is that I think that $-1$ should be equivalent to $2$: in both cases, the node next to it will be either $0$ or $1$, with $1$ occurring only when the node at distance $2$ is $0$.
Jun
30
comment Game on simple finite graphs
Ah, hang on. That's not true for large values of $k$, because then a $0$ can be inserted in the middle of the path and the value of $b$ matters. I apologise for sending you down a blind alley.
Jun
30
comment Game on simple finite graphs
The $p(a,b,k)$ notation can be improved slightly by observing that the actual values of $a$ and $b$ don't matter: all that matters is $a - b$ and $b\mod 2$.
Apr
27
comment Computing efficiently a small base to the power a large number
Are you after exact results or results to some level of relative error?
Apr
24
comment Is an empty parenthesis a valid mathematical expression?
The meaning of mathematical notation is highly dependent on context. Which subfield(s) of mathematics are relevant to the case which interests you? And what is your motivation for the question: trying to interpret someone else's use of them, trying to justify using them yourself, trying to win a random bet, ...?
Apr
18
comment Conditional distribution of two binomials which both depend on a third
You have $C\sim\text{B}(N,c)$ and $N\sim\text{B}(C,\alpha)$. Is that the same $N$ in both cases??
Apr
17
comment Coupon collector variation (with deleterious coupons and tolerance)
It is better than it was, but it would be better still if you outlined what progress you have made with solving the problem.
Apr
17
comment Prove $\lim_\limits{x\to\infty}\dfrac{P_k(x)}{P_{k+1}(x)}=0$
possible duplicate of Finding the limit of $\frac{Q(n)}{P(n)}$ where $Q,P$ are polynomials
Apr
15
comment Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$
Could you explain where the 5 and 30 come from?
Apr
2
comment How can I find the closed form of this recursive relationship: $a_{n}=(a_{n-1})^2+a_{n-1},a_{0}=1$
Knuth's comment in OEIS that "Using the methods of Aho and Sloane, Fibonacci Quarterly 11 (1973), 429-437, it is easy to show that $a_n$ is the integer just a tiny bit below the real number $\theta^{2^n}-\frac12$, where $\theta \approx 1.597910218$ is the exponential of the rapidly convergent series $\ln\frac32+\sum_{n \ge 0} \ln(1+(2a_n+1)^{-2})$" gives you a starting point.
Mar
31
comment Prove that for all naturals $n \ge 6$ there is a set of $n$ positive naturals, $a_1$ to $a_n$ such that $\sum_{i=1}^n \left(\frac{1}{a_i}\right)^2 =1$
$\{2, 2, 2, 2\}$ is not a set. You might want to check that you're solving the right problem.
Mar
29
comment $\sin(\pi - a) = \sin (a)$. How/why?
What's your definition of $\sin$? Is it in terms of triangles or complex exponentials?
Mar
25
comment Math of password cracking
Spaces are \, or \;. Your formula looks correct, so if it overestimates then the obvious question is: why do you think it does 4 billion calculations per second? Edit: wait, I may have it. Does your program exit early when it finds the password? Your calculation is the worst case: if the password is generated uniformly and randomly then on average it will take half that time.