Joel Cohen
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 Nov 19 comment Determine maximal addend in Newton Binomial Expansion. I think the expansion should be $$\left(2n+\frac{1}{2n}\right)^{4n+1} = \sum_{k = 0}^{4n+1} \binom{4n+1}{k} (2n)^k \left(\frac{1}{2n}\right)^{4n+1-k} = \sum_{k = 0}^{4n+1} \binom{4n+1}{k} (2n)^{2k-4n-1}$$ Nov 16 comment What is this dynamics question asking me to do? I think the question is asking you what the limit of this expression is as $u$ goes to infinity. Nov 16 comment Why is $GL_2(\mathbb C)$ connected? @learnmore : Well, you don't need to know about Lie groups for this proof (you just need to know about exponential of matrices, which are defined with the usual series $\exp(A) = \sum_{n = 0}^{+\infty} \frac{A^n}{n!}$). I was just mentioning where this idea came from. Nov 16 answered Why is $GL_2(\mathbb C)$ connected? Nov 16 revised Finding all group homomorphisms $(\mathbb{Q},+)\to (\mathbb{Q}-\lbrace 0\rbrace,\cdot)$ added 901 characters in body Nov 15 answered Finding all group homomorphisms $(\mathbb{Q},+)\to (\mathbb{Q}-\lbrace 0\rbrace,\cdot)$ Nov 11 comment How to prove that this ring homomorphism is isomorphic? Remember the notation $F[x]/(m(x))$ is for the space of polynomials $F[x]$ modulo the ideal $(m(x))$. This not a fraction in the usual sense (which would make no sense here), but a space in which $m(x)$ and all its multiples are zero. Nov 11 reviewed Approve How to get the geometric shape of an amount with complex numbers? Nov 11 reviewed Approve finding the integral with substitution Nov 11 reviewed Approve How would you determine $\sin(x) = -\cos(x)$ Nov 11 reviewed Approve Proving Vectors are linearly dependent in certain fields Nov 11 comment Proving Vectors are linearly dependent in certain fields Yes, you can think of $F$ as a number line, regardless of the field $F$ (of course this is just a mental picture). Nov 11 comment How would you determine $\sin(x) = -\cos(x)$ Note that your two cases can be condensed into just one : $\frac{3\pi}{4} + k \pi$. Nov 11 comment Is the function $\alpha$ an isomorphism? Check that $\alpha$ is linear, so showing it's injective reduces to showing that $\delta_v = 0$ implies $v = 0$. Nov 10 comment Is there a way to approximate a polynomial as another, binary-coefficient polynomial? In which sense do you want $p$ to be approximated (maybe minimizing the maximal distance on $[0,1]$, or the integral of the square of the difference ...) ? For example, how would you approximate $p(x) = 1000$ ? Nov 10 reviewed Reject Helmoltz equation on the torus Nov 10 reviewed Approve Prove $\frac{\partial ln|X|}{\partial X_{ij}}=tr[X^{-1} \frac{\partial X}{\partial X_{ij}}]$ not using adjoint. Nov 10 comment Why doesn't $\sqrt {x^3} = x \sqrt {x}$ Note that $\sqrt{ab} = \sqrt{a}\sqrt{b}$ only works for $a, b \ge 0$, and so does $\sqrt{a^2} = a$. So yes, if $a \ge 0$, we have $\sqrt{a^3} = a \sqrt{a}$. And otherwise, if $a < 0$, then the square root is complex. Nov 9 comment $f(x)$: Nth derivative function of $\frac{1}{100!}x^{100}(1-x)^{100}$ Show that $\int^{1}_{0}f(x)dx=0$ You might want to use the formula for the $n^{\textrm{th}}$ derivative of a product, namely $$(f g)^{(n)} = \sum_{k=0}^n \binom nk\ f^{(k)}\ g^{(n-k)}$$ Nov 9 comment Additivity & one-point Continuity of $f \in \mathbb{R}^\mathbb{R}$ imply there is $\alpha \in \mathbb{R}$ s.t. $f(\alpha x) = \alpha x$ @Kolmin : sorry my comment was not clear, I was just pointing a small typo :) Yes proving $f(x) = \alpha x$ is not directly obvious. To do that, start with $x$ and integer, then extend it to rational numbers, and then to any real number using continuity.