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Mathematical Notes | On some mathematical topics I found interesting. | chiasme.wordpress.com Reviews
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On some mathematical topics I found interesting.
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Almost all continuous functions are nowhere differentiable | Mathematical Notes
https://chiasme.wordpress.com/2013/05/18/almost-all-continuous-functions-are-nowhere-differentiable
On some mathematical topics I found interesting. Almost all continuous functions are nowhere differentiable. Is a Baire space if a countable intersection of open dense sets is still dense in. Then Baire category theorem consists in:. Baire) A complete metric space is a Baire space. It is a surprising consequence of Baire category theorem that almost all continuous functions are nowhere differentiable in the following sense: Let. Be the set of continuous functions. Endowed with the sup norm. Then, for all.
Free groups acting on the circle | Mathematical Notes
https://chiasme.wordpress.com/2014/02/27/free-groups-acting-on-the-circle
On some mathematical topics I found interesting. Free groups acting on the circle. We present here an unusual application of Baire category theorem between topology, group theory and dynamical systems. Roughly speaking, we prove that almost all pairs of homeomorphisms of the circle are unrelated; in particular, it leads to natural occurrences of free groups. More precisely, if. Denotes the set of orientation-preserving homeomorphisms of the circle, endowed with the distance. The set of pairs. And show th...
Van Kampen diagrams: an application to tiling problems | Mathematical Notes
https://chiasme.wordpress.com/2015/07/29/van-kampen-diagrams-an-application-to-tiling-problems
On some mathematical topics I found interesting. Van Kampen diagrams: an application to tiling problems. In this note, we are interested in the following problem: given a chessboard and a set of dominoes, is it possible to tile our chessboard using the dominoes we have? For example, let. Be the following chessboard. Denote the set of the “true” dominoes of size 2, that is. Is it possible to tile. Surprisingly, this problem can be solved thanks to the van Kampen diagrams used in group theory. By our induc...
A prime as a sum of two squares | Mathematical Notes
https://chiasme.wordpress.com/2015/09/26/a-prime-as-a-sum-of-two-squares
On some mathematical topics I found interesting. A prime as a sum of two squares. In this note, we are interested in the following well-known result:. Can be written as the sum of two squares if and only if. There exist many different proofs of this theorem, but a surprising one is exposed by D. Zagier in his article. A one-sentence proof that every prime. Is a sum of two squares. Our note is dedicated to this proof. First of all, it is easy to prove that the sum of two squares is either congruent to.
Not all finitely-presented groups are fundamental groups of closed 3-manifolds | Mathematical Notes
https://chiasme.wordpress.com/2015/04/12/not-all-finitely-presented-groups-are-fundamental-groups-of-closed-3-manifolds
On some mathematical topics I found interesting. Not all finitely-presented groups are fundamental groups of closed 3-manifolds. In our previous note Amalgamated products and HNN extensions (IV): Markov properties. We saw that for every. And for every finitely-presented group. Dimensional closed manifold whose fundamental group is. However, such a result does not hold for. See for example the note On subgroups of surface groups. The present note is devoted to the 3-dimensional case. We say that the triple.
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Mathematical Notes | On some mathematical topics I found interesting.
On some mathematical topics I found interesting. A prime as a sum of two squares. In this note, we are interested in the following well-known result:. Can be written as the sum of two squares if and only if. There exist many different proofs of this theorem, but a surprising one is exposed by D. Zagier in his article. A one-sentence proof that every prime. Is a sum of two squares. Our note is dedicated to this proof. First of all, it is easy to prove that the sum of two squares is either congruent to.
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