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Monthly Archives: November 2011
RT Part 7; Van Der Waerden’s Theorem
Last time we saw how we can use the technique of focusing to show that however one 2-colors [330], there always exists a 3-term, monochromatic arithmetic progression. As with last time, let’s abbreviate “-term monochromatic arithmetic progression” as “-TMAP”. We … Continue reading
Posted in Combinatorics, Ramsey Theory
3 Comments
Differentiation of Measures – The Radon-Nikodym Theorem Part 2
Hello everyone! My apologies for the long delay between posts in this series, these last few weeks have been a bit crazy with the NSF fellowship application due as well as an exam and a boatload of homework. Anyways, I’m … Continue reading
Posted in Analysis
3 Comments
Dear all: The other day I was reflecting on what material we covered in our undergrad courses, and I wondered what the most difficult proof we encountered in courses there was. Of course, we should keep in mind that every … Continue reading
Hi everyone! This is my first post and it is more of an aside than a true post. In this post I will briefly discuss a few methods on how one creates a ring out of a set . Forgive … Continue reading
RT Part 6; The Technique of Focusing
Last time I mentioned that we’d now move from Ramsey theory on graphs to Ramsey theory on the naturals. By popular demand, and to Zach‘s dismay, as decided by a Nebraska Math Department poll last year we will declare that … Continue reading
Posted in Combinatorics, Ramsey Theory
7 Comments
Differentiation of Measures – The Radon-Nikodym Theorem Part 1
In my last post, I ended by saying we need to discuss derivatives of measures. Now, at first you might think this sounds kind of strange, since usually to discuss derivatives we need the notion of a limit as, say … Continue reading
Determinants – Part 4
Ah we’ve finally arrived! We have defined, played with, and even proved the existence of alternating n-linear forms on a vector space . Even better we have defined the determinant of a linear transformation, a concept not usually seen in … Continue reading
Posted in Analysis, Linear Algebra
1 Comment
Basic AC, part 2: Choice in Algebra
The outline I (apparently) wrote on the previous post in this series says this post should talk about the Axiom of Choice in algebra, particularly how it affects vector spaces and groups. Before I talk about vector spaces and groups, … Continue reading
Posted in AC, Linear Algebra, Logic
11 Comments
RT Part 5; Intro to RT on the Naturals
We have talked about Ramsey theory on graphs. But what other structures can we do Ramsey theory on? Well, how about numbers? Remember that the basic idea behind Ramsey theory is to try to guarantee the existence of some sort … Continue reading
Posted in Combinatorics, Ramsey Theory, Uncategorized
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