H-H
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Proof that the largest eigenvalue of a stochastic matrix is $1$
7 votes

If we can show that $A$ doesn't increase the 1-norm, i.e., $$\|Ax\|_1\leq\|x\|_1$$ Then $$\|Ax\|_1=\|\lambda x\|_1=|\lambda|\|x\|_1\leq\|x\|_1$$ which is $|\lambda|\leq 1$, we are done, but how to ...

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Nice identity: the norm of the second fundamental form and scalar, Ricci, mean, Gauß curvatures together
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4 votes

Let's do calculation following your idea (some different notations). $$\operatorname{Ric}(\nu,\nu)=\sum_{i=1}^3<\tilde{R}(e_i,e_3)e_3,e_i>=\sum_{i=1}^2<\tilde{R}(e_i,e_3)e_3,e_i>=\tilde{R}...

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How does one calculate $A^n$ when $A$ can't be diagonalized?
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3 votes

For $3\times 3$ matrix, if it can't be diagonalized, it will have Jordan forms $A=PJP^{-1}$ for following two cases $$J=\begin{pmatrix}\lambda&1&0\\ 0&\lambda&0\\ 0&0&\mu\end{...

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Metric on unit circle
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2 votes

For the first question, I think the metric for circle depends how you parametrize the circle, usually we choose $(\cos\theta,\sin\theta)$, then the metric induced from $\mathbb{R^2}$ is $g_{circle}=d\...

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Theorem $2.10$ - A course in minimal surfaces by Colding and Minicozzi
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2 votes

I think the first inequality in the red box should be equality because it is just the decomposition of ring into finite smaller rings. The second inequality is because for $x\in B_{e^{l+1}r_0}\...

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Harmonic coordinate functions in minimal surfaces
2 votes

I am not sure if you have background of Riemannian geometry, following is what I do. Suppose this surface $(\Sigma^2,g)$ which is parametrized by $$F=(f_1,f_2): \ \ (x^1,x^2)\in D\subset{R}^2\...

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Geodesic curvature change under conformal metrics
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2 votes

I am not sure if it is right but following is my calculation. First, the geodesic curvature is defined to be $$k_g=g(\nabla_{v}v,N)$$ where $v,\ N$ respectively are the unit tangent vector and inner ...

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Isometries preserve geodesics
2 votes

It is actually an exercise in the Lee's book, I try to do it by following the hint. First, you have to understand the naturality of Riemannian connection, then everything will be clear. I like using $\...

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Laplacian on minimal surface
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1 votes

It is a local calculation. Suppose $p\in \Sigma$, and $\{e_1,\cdots,e_{n-1},N\}$ is local O.N. basis of $R^n$ such that $\{e_1,\cdots,e_{n-1}\}$ is local O.N. basis of $\Sigma$, $N$ is unit normal ...

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Using the Gauss-Bonnet Theorem
1 votes

Ted Shifrin already gave the explanation. You can deduce it from definition. Choose a local O.N. frame $\{e_1,e_2\}$ on $M$, with a normal vector $N$, it gives a O.N. frame for $S^3$. By definition \...

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Derivative of boundary volume element
1 votes

It is just $$\frac{dA(g_t)}{dt}\big|_{t=0}=\frac{1}{2}\int_{\partial M}h(T,T)dV_{\bar{g}}$$ where $T$ is the tangent vector field on the boundary. Since for orthonormal basis $\{e_i\}$ we have $$...

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Product of inverse matrices $ (AB)^{-1}$
1 votes

I have some personal opinions which might perfect it (many students made mistakes about this), in your case it works fine $$(AB)^{-1}=B^{-1}A^{-1}$$ Because you already have the fact that $A,B$ are ...

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Constraints $X^TBX = 1$, what's the maximum $X^TAX$?
0 votes

It is equivalent to $\max f(x)=x^tAx$ with constraint $g(x)=x^tBx-1=0$. By Lagrange multiplier, to find critical point we solve following equation for $(\lambda,x)$ $$\nabla f=\lambda \nabla g,\ \ \ \ ...

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How do I know if it's a subspace of $\mathbb{P}^2$?
0 votes

I think you mean to prove $\operatorname{Span}(S)$ is a subspace of $P^2$, in fact, any element $p$ in $\operatorname{Span}(S)$ is $$p=a+bx^2+c(2+x^2)=(a+2c)+(b+c)x^2$$ that is why only vectors of the ...

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