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www.kuniga.me | ||
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mattbaker.blog
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| | | | | In my last blog post, I discussed a simple proof of the fact that pi is irrational. That pi is in fact transcendental was first proved in 1882 by Ferdinand von Lindemann, who showed that if $latex \alpha$ is a nonzero complex number and $latex e^\alpha$ is algebraic, then $latex \alpha$ must be transcendental. Since... | |
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www.jeremykun.com
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| | | | | This proof assumes knowledge of complex analysis, specifically the notions of analytic functions and Liouville's Theorem (which we will state below). The fundamental theorem of algebra has quite a few number of proofs (enough to fill a book!). In fact, it seems a new tool in mathematics can prove its worth by being able to prove the fundamental theorem in a different way. This series of proofs of the fundamental theorem also highlights how in mathematics there are many many ways to prove a single theorem... | |
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mikespivey.wordpress.com
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| | | | | The Riemann zeta function $latex \zeta(s)$ can be expressed as $latex \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$, for complex numbers s whose real part is greater than 1. By analytic continuation, $latex \zeta(s)$ can be extended to all complex numbers except where $latex s = 1$. The power sum $latex S_a(M)$ is given by $latex S_a(M) =... | |
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pfzhang.wordpress.com
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| | | Consider a smooth one-parameter family $latex {f_t}$ of diffeomorphisms on a manifold $latex M$. It is a flow if $latex f_0(x)=x$ and $latex f_{t}\circ f_{s}(x) = f_{s+t}(x)$ for every $latex x\in M$, , $latex t ,s \in \mathbb{R}$. Set $latex X(x)=\lim\limits_{h\to 0} \frac{1}{h}(f_h(x) - x)$. This generates a vector field $latex X: M \to TM$.... | ||
