Peter Taylor
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 Feb3 reviewed Leave Closed Does performance in math competitions accurately reflect natural aptitude in mathematics? Feb3 reviewed Leave Closed Optimization, travel by land and sea. Feb1 comment $f(xy)=\frac{f(x)+f(y)}{x+y}$ Prove that $f$ is identically equal to $0$ I think the missing step is that since $f: \mathbb{R} \to \mathbb{R}$, $f(0)$ must be defined. Then $\forall x \ne 0: f(0) = \frac{f(x)+f(0)}{x}$, and in particular $f(0) = f(1)+f(0)$ whence $f(1) = 0$. Jan31 comment $f(xy)=\frac{f(x)+f(y)}{x+y}$ Prove that $f$ is identically equal to $0$ $f$ identically equal to $0$ satisfies the functional equation, so you seem to be asking the impossible. Did you mean "Prove that there exists an $f$ satisfying this functional equation which isn't identically equal to $0$"? Also, what do you mean by $f : R ->$? Jan31 comment What is the number $p(n)$ of partitions of an abundant number $n$ into distinct, proper divisors of $n$? @Travis, yes, it does. $f(n, m, k)$ counts the number of partitions of $m$ into distinct divisors of $n$ each of which is at least $k$, so the first term on the RHS counts the ones where $k$ is not in the partition, and the second term counts the ones where $k$ is in the partition - but obviously it can only be in the partition if it's a divisor of $n$. Jan30 answered A digraph is a graph where every edge is directed. How many digraphs on $n$ vertices are there? Jan30 comment Why the space of all permutations of a vector (n!) is smaller than the space of all possible permutations of a sorting network? Ah, I've just seen your earlier question. If this question is also about abusing sorting networks to generate permutations, it would help a lot to edit the question and make that clear. I've been assuming that you were trying to use the sorting network to sort. You may find this thread on another site in the StackExchange network useful. Jan30 comment Why the space of all permutations of a vector (n!) is smaller than the space of all possible permutations of a sorting network? If you have a 3-gate sorting network for 3 elements, that's 6 possible inputs and (if I understand you correctly) 8 "solutions". That's small enough that you can draw the whole lot on a piece of paper and identify the interesting ones. Jan29 comment Why the space of all permutations of a vector (n!) is smaller than the space of all possible permutations of a sorting network? I'm not quite sure what the question is, but perhaps you would find it helpful to work through the 6 permutations of 3 elements with a 3-gate sorting network. Jan28 answered Street Fighter: is the game balanced? Jan27 comment Placing $4n$ non-attaking queens of in a $4n \times 4n$ chessboard. @tone, the Wikipedia article linked by hardmath shows solutions which do place a queen in a corner of the board. Jan19 comment Counting rings of order $p^3$ R. Raghavendran, Finite associative rings, Compositio Mathematica vol 21 no 2 (1969) p 195-229 (referenced from the OEIS page you link) claims that there are 11 rings with identity of order 8 and 12 rings with identity of order $p^3$ for odd prime $p$, so the discrepancy of $3(p-1)$ rings of order $p^3$ between the two sources presumably relates to rings which don't have an identity. Jan17 comment Simplification of recursive polynomials Is any of the stuff in this paper useful? Jan16 answered 6 lamp at the circumcircle Jan15 comment how to count possible planar bipartitions? Ah. At present the question says "any bipartition at all is a valid solution", so if you really want each half of the partition to be connected you should edit the text to clarify that. Jan14 comment how to count possible planar bipartitions? It seems to me that the number of ways a graph can be bipartitioned is $2^{|V|-1}$ unless you add some constraints. Have I missed a constraint? Jan12 awarded Yearling Jan3 comment How to denote sum over partitions? @ruadan, it depends. If you're just referring to frequencies and total then you can use the standard notation for the frequency representation and subscript the sum with $(1^{a_1}2^{a_2}\ldots)\vdash n$. If you're referring to both parts and frequencies you need to consider whether to subscript as $\lambda = (1^{a_1}2^{a_2}\ldots)\vdash n$ or just to explain in the accompanying text how the $\lambda_i$ relate to the $a_j$. Jan2 answered How to denote sum over partitions? Dec21 comment A combinatorial game theory problem What do you mean by "vicinal squares"? Does that cover diagonally adjacent, orthogonally adjacent, or both? And why do you call this a combinatorial game theory problem? It seems to be a single combinatorial question.