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A few musings.
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https://indelibled.wordpress.com/2014/03/13/64
The Chinese Remainder theorem. The system of congruences:. Has a unique solution modulo. We first show, by induction, that the given system has a solution. The result is obvious when. As we can set. So, let us consider the case. If we also require that. We see that this congruence, with. As the unknown has a unique solution modulo. Then we see that. So substituting back into the previous equation for. Satisfies both of the initial two congruences as well. Now suppose that the initial system of. Address n...
Sperner’s Lemma | indelibled
https://indelibled.wordpress.com/2014/04/06/sperners-lemma
In this post we prove Sperner’s Lemma. Be a triangle in the plane. Let. Be a triangulation of this triangle, i.e. every face excepting the exterior face is surrounded by three edges. Such a triangulation is depicted in the picture below, which I took from Wikipedia. Now sperner color the vertices of the triangulation with the colors. Such that all vertices on the edge between. And $A,C$ is 1 and 3, and CB is 2,3, and. Then Sperner’s lemma states:. First we create a graph,. From this triangulation,. Shows...
A Case of Dirichlet’s Theorem | indelibled
https://indelibled.wordpress.com/2014/03/09/24
A Case of Dirichlet’s Theorem. Contains infinitely many prime numbers. Suppose, for the sake of contradiction, that. Contains finitely many primes. Then call them. There are two cases. Is not prime, as it is congruent to. Yet is larger than every of the. If every prime divisor of. Has at least one prime divisor,. Which is congruent to. As none of the. Yet is congruent to. And thus of the form. We have a contradiction. So, as before there is some divisor. Thus there are infinitely many primes of the form.
March | 2014 | indelibled
https://indelibled.wordpress.com/2014/03
Month: March, 2014. March 13, 2014. The Chinese Remainder theorem. The system of congruences:. Has a unique solution modulo. We first show, by induction, that the given system has a solution. The result is obvious when. As we can set. So, let us consider the case. If we also require that. We see that this congruence, with. As the unknown has a unique solution modulo. Then we see that. So substituting back into the previous equation for. Equations has a solution,. Then we show that it has a solution modulo.
Schroder-Bernstein Theorem | indelibled
https://indelibled.wordpress.com/2014/04/11/schroder-bernstein-theorem
Be distinct sets sets such that. And suppose we have a bijection. Then there is a bijection. Is clearly a bijection between. Now consider the map. We aim to show that this map is a bijection between. We see that this map is injective, so it will suffice to show that. Is a bijection, this implies. A contradiction. Thus. For the reverse inclusion, suppose. A contradiction. Thus we see that $latex f (A 0 backslash A 1)$ is a bijection. Similarly, we can let. And repeat this process creating. April 11, 2014.
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Tuesday, May 19, 2009. If I use brushes I did not create myself when designing a layout, they will have been created by a brilliant artist at Obsidian Dawn. She creates her high quality brushes in Photo Shop using her own photography and artistic talents. If I use her brushes, I will include her button, along with my own, in a credited HTML gadget on your blog. Posted by Lex the mom. At 5/19/2009 05:23:00 PM. Subscribe to: Posts (Atom). Don't forget to drop by my Etsy shop for one of a kind jewelry!
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indelibled | A few musings.
July 5, 2014. This is a guest post by my girlfriend, Chris, who upon reading this blog decided that she would contribute. Something about “egotistical”. You can just read the first sentence, and if she asks about it just say it was great. Anyway, here it is:. Yeah it was fine like I told you it would be. You always worry too much about these things. It was just a birthday cake. April 11, 2014. Be distinct sets sets such that. And suppose we have a bijection. Then there is a bijection. Now consider the map.
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Thursday, February 11, 2010. Is when Clairbear gets very very happy? Is just Clairbear's birthday month! I had a great birthday week. Packed, fun week. First present was from Mummy. First cake was from 老爷. Dinner with Mummy and dy. It just made me smell like a dork. It was on the 27 itself, and when i got home to open the cake it just made me too damn happy. It was my first cake this year, and a cake to pass into the 12am mark for 28. I prep the cake. Also called up (he was in some jungle). Then i got mo...
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