All Questions

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Finding the change of variables to transform $u_{tt} - u_{xx} = 0$ into $u_{rs} = 0$

I'm just beginning to introduce myself to partial differential equations and one of the first problems presented in the textbook I have literally no idea how to do. I think the author intended the ...
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Looking for help with this elementary method of finding integer solutions on an elliptic curve.

In the post Finding all solutions to $y^3 = x^2 + x + 1$ with $x,y$ integers larger than $1$, the single positive integer solution $(x,y)=(18,7)$ is found using algebraic integers. In one of the ...
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Is this symmetric, block-diagonal matrix positive semi-definite?

I have a matrix of the following form, where $a,b,c>0$ \begin{align*} A = \left[ \begin{array}{cccccc} aM_{12}^2 & aM_{12}M_{13} & 0 & 0 & 0 & 0 & 0 \\ aM_{13}M_{12} ...
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If $\operatorname{rank}(A)=n$ then $\operatorname{rank}(AB)=\operatorname{rank}(B)$

I have looked here, but still I cannot understand how to get to equality. Let assume that the matrices are squared $\operatorname{rank}(AB) \leq \operatorname{rank}(B)$ is easy to show, but how can I ...
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Solving for $a^2+b^2+c^2$ given $\sqrt{a^2+1}+\sqrt{b^2+4}+\sqrt{c^2+9}$ and $a+b+c$. [on hold]

I am trying to solve the following problem involving three terms: $$\sqrt{a^2+1} + \sqrt{b^2+4} + \sqrt{c^2+9} = 8$$ $$a + b + c = 6$$ $$a^2 + b^2 + c^2 =?$$
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Arithmetic and Geometric Mean Inequalities [on hold]

Can someone help me to understand the logic of: $$\sqrt{ab} \le \frac{a+b}{2}$$ Proof: ?
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Sum/diff of matrix units

I understand what the product of matrix units means, but I don't understand what the sum/difference of two different matrix units represents. For example, what does ${e_{2,2}}-{e_{5,5}}$ equal? ...
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Form-invariant solution to PDEs

I'm trying to understand how to create form-invariant solutions to PDEs: $$\hat{L}u(x,t)=0$$ with the constraint $|u\big(x,t\big)|^2=|u\big(f(t)\cdot x + g(t),0\big)|^2$. During evolution, the ...
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Group of even order must contain $a:a=a^{-1}$ $(a\not = e)$ [duplicate]

Let $G$ be a finite group. If the order of $G$ is even, prove that there is at least one element $a$ in $G$ such that $a\not= e$ and $a=a^{-1}$. Here's my idea: Suppose $\{x_1,\cdots,x_n\}$ is ...
So say I have a differential equation of the form: $$(\alpha \frac{d^2}{dx^2}+fx^2)y(x)=\lambda y(x)$$ Whose solutions are known (a Gaussian multiplying a Hermite polynomial.) I am now curious how ...