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1d
comment Would this proof strategy work for proving the lonely runner conjecture?
@DavidvonRudisill, following r.e.s.' reasoning, we can consider the general case for three runners of speeds $0$, $a$, $b$. Then wlog the equispacing would have $at = \frac{3\alpha + 1}{3}$ and $bt = \frac{3\beta + 2}{3}$ with $\alpha, \beta \in \mathbb{N}$. But then $\frac{a}{b} = \frac{3\alpha + 1}{3\beta + 2} \in \mathbb{Q}$, so your strategy can at best work for a subset of runner speeds which has measure $0$ in the space of all possible runner speeds. This is (related to) the point James Hines was trying to make with the comment on irrational speeds.
Aug
23
comment How many legal states of chess exists?
web.archive.org/web/20140601124247/http://homepages.cwi.nl/…
Aug
10
comment Complexity of subset-generation algorithm
At a high level, $2^n$. At a low level, it depends on implementation details. There are probably some questions about it on the sister site stackoverflow.com
Aug
9
comment A combinatorial identity
1. Not a typo: the $!!$ is a double factorial. 2. You're quite right. I've edited.
Aug
9
comment Complexity of subset-generation algorithm
It's not very clear to me what most of the operations are, or what the purpose of k is (it seems to be unused). If what you want is a way to iterate through a powerset efficiently then you should look into Gray codes.
Aug
9
comment Complexity of subset-generation algorithm
$\binom{i}{i} = 1$, so the sum is equal to $N + 1$. I think you've probably made a mistake in your analysis.
Aug
4
comment What is the mixed strategy equilibrium bid, if any, for complete information auction games with minimum bid?
I think you might be misunderstanding the extreme equilbria. They're the vertices of a polytope (or, in this case, a triangle) such that any point in the polytope is a Nash equilibrium. So in the discrete case there are mixed strategies which are equilibria, being any convex sum of the three extrema, but it just happens that the extrema are pure.
Aug
3
comment What is the mixed strategy equilibrium bid, if any, for complete information auction games with minimum bid?
That depends on whether you consider pure strategies to be a subset of mixed strategies or whether you consider the two sets to be disjoint.
Jul
31
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, ...?