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Dec
20
comment Understanding the random variable definition of Markov chains
FWIW I found some lecture notes that answer this question in the fashion I had envisioned (hamilton.ie/ollie/Downloads/ProbMain.pdf) -- see section 3.1.
Dec
20
comment Understanding the random variable definition of Markov chains
Well thanks for all your input -- much appreciated.
Dec
20
comment Understanding the random variable definition of Markov chains
Isn't it also valid to think of the chain as a dynamical system exploring a space of states (not necessarily pmf's) with a dynamical rule that is probabilistic so that at any given time, the state is determined by a pmf associated with that time, but is not the pmf itself? This (in addition so my impressions from some literature) is my motivation for wanting to associate the state of a possible trajectory of the chain at each time $k$ with a possible value in the range of a random variable.
Dec
19
comment Understanding the random variable definition of Markov chains
(contd.) sense of a simple way of constructing an appropriate $\Omega$ by thinking of the chain as a probabilistic experiment in which one generates a sequence of states (elements of the ranges of the random variables), and in that sense an outcome of the experiment would be such a sequence, leading to the sample space being the set of all such sequences. Imagine flipping one of two biased coins depending on the outcome of the last flip, then I want to try to take $\Omega$ to be the set of infinite sequences of $H$ or $T$, and then $X_k$ is $1$ if the $k$th flip is heads, and $0$ otherwise.
Dec
19
comment Understanding the random variable definition of Markov chains
Thanks, but I don't think this answers the question. I know that in a practical sense, knowing $\Omega$ isn't important, but I still want to know if the way I attempted to construct $\Omega$ works. I understand why one might want to view the trajectory of the chain as a sequence of pmf's, but is that necessarily the way it's commonly viewed? If so, why would the range of the random variables often (as far as I can tell) be called the "state space" of the chain? Lastly, I wasn't thinking of $X(s_0, s_1\dots)$ as the probability of various sequences of states, I was trying to make
Dec
19
comment Understanding the random variable definition of Markov chains
@A.S. Thank you. I will update the question to reflect my misuse of the terminology.
Sep
17
comment Is there more than one way to divide the “L”-shaped tromino into four congruent, connected pieces?
I don't suppose you have an idea of how to do this if boundaries are excluded or an opinion on whether a solution besides the one I originally came up with is possible?
Sep
17
comment Is there more than one way to divide the “L”-shaped tromino into four congruent, connected pieces?
+1 This is clever. If it also answered the original polygons version, then I'd award your dad the bounty. Maybe I should anyway...I don't anticipate anyone proving it can't be done in another way with four polygons or coming up with another solution.
Sep
14
comment Is there more than one way to divide the “L”-shaped tromino into four congruent, connected pieces?
@MikePierce Oh gotcha. Yeah I would consider Jon Mark Perry's answer cheating. I'd be interested in a solution using something like the bowtie, but I don't think such a solution would be in line with the spirit of the original question.
Sep
14
comment Is there more than one way to divide the “L”-shaped tromino into four congruent, connected pieces?
@Rick To be fair to Mike, I did mention "parts" in my addendum.
Sep
14
comment Is there more than one way to divide the “L”-shaped tromino into four congruent, connected pieces?
@MikePierce I'll make the appropriate edit. Thanks.
Sep
14
comment Is there more than one way to divide the “L”-shaped tromino into four congruent, connected pieces?
@MikePierce If I understand your suggestion correctly, I'd consider that cheating. You should be able to cut along the boundaries between the subdivisions with scissors and obtain three congruent shapes without any extra left over. Am I interpreting your suggestion correctly?
Sep
14
comment Is there more than one way to divide the “L”-shaped tromino into four congruent, connected pieces?
@jakemckenzie Would you happen to know if anyone has studied tilings of polyominoes themselves as opposed to using polyominoes to tile other regions?
Sep
9
comment Is there more than one way to divide the “L”-shaped tromino into four congruent, connected pieces?
This is nice if one wants an analysis of the tiling of the tromino by trominos, but that's not ultimately what the question is about. I'm asking if there is a tiling by another shape.
Sep
4
comment Is there more than one way to divide the “L”-shaped tromino into four congruent, connected pieces?
@StevenStadnicki I'm asking the second question. Perhaps I'll make an edit to make that more clear, thanks.
Sep
3
comment Is there more than one way to divide the “L”-shaped tromino into four congruent, connected pieces?
@Blue Very cool! Thanks for the terminology. I hope someone on math.SE is a rep-tile master.
Apr
8
comment what's the difference between a rational number and an irrational number?
@gideon Well thanks, especially since this caused me to look back at the answer and correct an error.
Mar
7
comment Complex distributions - what are the appropriate test functions?
@StephenMontgomery-Smith I wouldn't be surprised in the least, but I'd prefer to defer final judgement until I can be a bit more certain there's a mistake. Perhaps someone more familiar with CFT will someday have something definitive to say about this.
Mar
7
comment Complex distributions - what are the appropriate test functions?
@StephenMontgomery-Smith Yeah I'm strongly inclined to agree -- I have never heard of analytic test functions either. Nonetheless, the calculation I wrote down appears in at least the most well-known, standard conformal field theory text, so I'd like to make sense of it somehow.
Feb
20
comment Proving that a family of functions limits to the Dirac delta.
+1: This is great, thank you. I'm still hoping, however, that someone will be able to tell me how to make the approach that uses complex analysis work as a matter of interest.