Michael Smith
Reputation
633
Top tag
Next privilege 1,000 Rep.
Create new tags
 Oct 3 comment Permutating dance partners with least distance moved Thanks that is useful link and solve all the parts except minimizing how far the people have to move to swap partners and what this problem is called in graph theory Oct 3 comment Permutating dance partners with least distance moved I looked up Maximal independent set here en.wikipedia.org/wiki/Maximal_independent_set but it doesn't seem to capture the same idea Oct 3 comment Permutating dance partners with least distance moved @Ross - that does seem to solve the first part and the rotating lines part. It does not address what this is called in graph theory (Qiaochu Yuan commented: I think he means counting the number of maximal sets of disjoint edges in a complete graph on n vertices but did not give a reference at the time). And it also does not address how we minimize the amount of movement required by the people. Aug 28 comment On the Math Mindset I got "A survey of modern algebra" by Birkhoff and Mclane while I was in grade 12 and while I enjoy it now I did find it a bit abstract at the time. When I am getting into a subject I like to have more motivation and history in a math book. If you are into Algebra then "Rings, Fields and Groups" by RBJT Allenby is a much easier read that still covers most of the same ground. And the author's enthusiasm for math really shines through. Given that you are not finding your current teaching method motivating I think finding books that explain why the material is interesting are useful. Aug 11 comment How to tell $i$ from $-i$? That is an interesting question - is there an algebraic definition of orientation in the (complex) plane. For the question that was asked for an elementary explanation of difference between i and -i and I think it is reasonable to use the complex plane and the natural notions of geometry that students usually already know. Here is my attempt at describing clockwise in words - if you multiply a complex number by another complex number of length 1 that is close to 1 then if the y coordinate increases it is an anti-clockwise rotation and if it decreases it is clockwise. Aug 11 comment How to tell $i$ from $-i$? Yes there is an automorphism (a mirror reflection of the complex plane) that swaps i to -i and changes clockwise to anti-clockwise. I don't think that affects the above way to explain the difference between i and -i Jun 3 comment How can we find and categorize the subgroups of R? Yes that is true. And the point I am trying to make is that I think there is a infinite hierarchy of dense subgroups between Q or these other non-Q subgroups and the full group R. And I am wondering how to classify those... My intuition says that this is somehow related to the diagonal proof of the uncountablity of the reals and the point that @rschwieb made on the related question math.stackexchange.com/questions/152263/… about stripping away leading digits doesn't matter. Jun 3 comment How can we find and categorize the subgroups of R? @benmachine - thanks for correction of definition of dyadic rationals - I updated the question text to reflect this. Jun 3 comment How can we find and categorize the subgroups of R? I am thinking that the structure of the subgroups of R might be clearer if we look at the generators of them. For example Z = [1], nZ = [n], aZ = [a], Dyadic =[1/2^n n in N], Q = [1/p p prime] etc. Then it is clearer which subgroups are subgroups or supergroups of others. Perhaps this idea can be extended to infinite (uncountable?) sets of irrational generating elements... Jun 3 comment How can we find and categorize the subgroups of R? @benmachine Oh, thanks for that example! That is interesting - we have a set that spans the subgroup but no subset is linearly independent! In fact we can remove any number of elements (finite or infinite) from this spanning set and so long as an infinite number of elements remain the subset still spans the subgroup... However I am not clear how to prove that no basis exists for dyadic subgroups of Q. I tried to get a contradiction from expressing 1/2^n as Z linear combination of basis elements but don't see that yet. Jun 3 comment How can we find and categorize the subgroups of R? I am probably missing something here but is not the infinite set { 1/2n where n in N} a basis for the dyadic rationals of binary form ( 1/2a )? Thanks for the idea on Z-modules too! Jun 3 comment How can we find and categorize the subgroups of R? @benmachine - thanks for clarifying, I will look into those more and I found a proof of the classification of subgroups of Q at answers.yahoo.com/question/index?qid=20080425160119AA19f8E which shows that they are all either finitely generated or of the type of the dyadic rationals. Jun 2 comment How can we find and categorize the subgroups of R? That is a useful way to look at the problem. I know that R has a Hamel basis over Q and also any subgroup H that contains Q would also have a Hamel basis. Then we can make other subgroups by picking arbitrary subsets of these bases. I did some google searching on submodules of infinite power modules without any luck so you have any links for articles on that I would appreciate it. Jun 2 comment How can we find and categorize the subgroups of R? That is useful because it shows that there are no subgroups between the Z ones and the Q ones. However it doesn't say anything about the structure beyond Q once the subgroups have gotten dense in R. I am wondering how "many" subgroups there are between Q and the full group of R... Jun 2 comment How can we find and categorize the subgroups of R? @Norbert thanks for that article - looks interesting, though I hope the task is not completely hopeless! Jun 2 comment How can we find and categorize the subgroups of R? @benmachine - thanks - I added in dyadic rationals and link to article on them Jun 2 comment What do the cosets of $\mathbb{R} / \mathbb{Q}$ look like? To help get a better intuitive handle on R / Q I thought to look at R / Z. In this case we get representatives as the interval [0,1) or equivalently as the circle Lie group. This is where we only look at the digits of the real number after the decimal point. Now look at the subgroup Yn = 1/n Z. R / Yn = [0, 1/n). We are looking at digits to the right of a certain point. We can consider R / Q as a limit of R / Yn - the circles get infinitesimally small. This reminds me of some modern physical models of space-time with 11 dimensions where the non-space time dimensions curl in on themselves Jun 1 comment What do the cosets of $\mathbb{R} / \mathbb{Q}$ look like? @Leo - thanks for your edits - I wasn't quiet sure how to add in Tex. Your edits are correct and I added a few more clean ups too Jun 1 comment What do the cosets of $\mathbb{R} / \mathbb{Q}$ look like? @Arturo - thanks for your insights. I will think about the Q-linear function approach as I have been reading about functional analysis and distributions recently. Jun 1 comment What do the cosets of $\mathbb{R} / \mathbb{Q}$ look like? that idea of stripping out all leading digits in the decimal representation of a coset representative reminds me of infinitesimals in non-standard analysis. Just a thought here - perhaps the costs of R / Q are homeomorphic to the set of infintestimal numbers that have zero real part.