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visits member for 2 years
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Oct
4
awarded  Yearling
Sep
15
comment Results true in a dimension and false for higher dimensions
Is this related to the accepted answer on Brownian motion?
Aug
22
comment Continuous relations?
@EricWofsey: Presumably the idea is that continuous relations should generalize continuous functions in the same way as ordinary relations generalize functions.
Jul
24
comment Both NP-hard but different
Are you considering the 0-1 knapsack problem? And how do you compare the size of a knapsack instance to the size of a TSP instance? By the number of variables (=edges for TSP)?
Jun
13
comment Why are polynomials so useful in mathematics?
@GerryMyerson: another way of saying that is that $K[x]$ is the free unital $K$-algebra on one generator. Similarly, $K[x_1,\ldots,x_n]$ is the free $K$-algebra on $n$ generators.
May
21
awarded  Excavator
May
21
revised Are there any books that take a 'theorems as problems' approach?
edited body
May
7
comment Why are smooth numbers called “smooth”?
@WillJagy: en.wikipedia.org/wiki/Mouse_%28set_theory%29
Apr
22
comment Entropy for Haar measure on $O(n)$
@Asaf: okay, good, then we perfectly agree! I just wanted to make sure that the basic concepts are not forgotten.
Apr
21
comment Entropy for Haar measure on $O(n)$
Kolmogorov-Sinai entropy is then a derived quantity defined in terms of the Shannon entropy of a partition, and it measures the scaling of this Shannon entropy with time. As far as I can see, the OP's question is about "entropy" in the sense of Shannon entropy, not Kolmogorov-Sinai entropy.
Apr
21
comment Entropy for Haar measure on $O(n)$
@Asaf: "entropy is defined by a dynamical system, not just a space". I disagree with this statement. Arguably, the most basic notion of entropy is Shannon entropy, which assigns to every finite probability space a number measuring its effective size: en.wikipedia.org/wiki/Shannon_entropy
Apr
1
comment Numbers of distinct products obtained by permuting the factors
What about generalizing the problem: given any equivalence relation on $S_n$, can one find $G$ and the $g_i$ such that the map $S_n\to G,\: \sigma \mapsto g_{\sigma(1)}\cdot\ldots\cdot g_{\sigma(n)}$ identifies two permutations if and only if they are equivalent? Or is this trivially false?
Mar
25
awarded  Suffrage
Feb
28
comment Does every separated measurable space embed into a power of $\{0,1\}$?
That's precisely what I had tried initially (see the original version of my question). If you know about Gelfand duality and Stone duality, it seems like a natural thing to try and indeed the resulting injective function is measurable (with respect to the Baire, and hence also the Borel $\sigma$-algebra on $\{0,1\}^\Sigma$). However, I don't see why this function should also take measurable sets to measurable sets, and I think this fails in general. In fact, I suspect that its image may not even be measurable, but I don't have a concrete example of that.
Feb
28
accepted Does every separated measurable space embed into a power of $\{0,1\}$?
Feb
28
comment Does every separated measurable space embed into a power of $\{0,1\}$?
Neat answer! I can't accept both, so I will go with Joseph's which covers the Borel case, since it also contains a sufficient condition which is close to his necessary one.
Feb
27
comment Does every separated measurable space embed into a power of $\{0,1\}$?
@Joseph: I updated the question to reflect your correction. A more detailed answer would be great!
Feb
27
revised Does every separated measurable space embed into a power of $\{0,1\}$?
added 838 characters in body
Feb
27
comment Does every separated measurable space embed into a power of $\{0,1\}$?
@Joseph: you're right. I will have to think about it a bit more. Sorry for the confusion...
Feb
27
revised Does every separated measurable space embed into a power of $\{0,1\}$?
clarified the question after a request in the comments