# All Questions

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### How to solve an equation involving euclidean norm operation?

On page 3 of Scalable, Versatile and Simple Constrained Graph Layout it describes the equation: ||(p-r)-(q+r)||=d Where p and q are points q, r is a vector and d is a scalar. It then goes on to ...
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### $f$ is monotone on D and $f(D)$ is an interval

$f$ is monotone on D and $f(D)$ is an interval then $f$ is continuous Is my proof right? pf) First, suppose it is monotone increasing Since $f(D)$ is an interval there is $[c,d]$ such that ...
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### hypothesis testing ; F-testing

lnQ=1.37+0.632lnK+0.452lnL (0.257). (0.219) cov(bk,bl)=0.055 R^2=0.98 H0: bk+bl=1 F=(Rb(hat)-r)'[R(X'X)^-1R']^-1(Rb(hat)-r)/e'e/n-k-1 I found the numerator value but i ...
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Is the radius of convergence of $$\frac{n(x+3)^n}{4^n}$$ equals 4? I got $|x+3|\lt 4$ as the final result. How do you know, what is the radius from here?
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### Interesting summation question: If $a$ and $b$ are the roots of $x^2+x+1$, then what is the below expression equal to?

Question: If $a$ and $b$ are the roots of $x^2+x+1$, then what is the below expression equal to? $$\sum_{n=1}^{1729} \left[(-1)^n\cdot V(n)\right]$$ Where $$V(n)=a^n+b^n$$ My effort: I think I ...
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### Finding this Probability Density Function

I would much appreciate if you help me out with this problem Let $X \sim Unif(0,1)$ Find the density of $Y = -\lambda^{-1} \log(1-X)$ with $\lambda > 0$ Then calculate $P(Y>t+s|Y>t)$ for ...
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### Find the population

Every year, the emigration rate from country A to B is 𝛼 (0 < 𝛼 < 1), whereas the emigration rate from country B to A is 𝛽 (0 < 𝛽 < 1). Note that the fluctuation in population of both ...
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### Wave Operators: Cook

Problem Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}H_\#\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote for shorthand: ...
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### The sum of the multiples of 2 and 17 under 767 [on hold]

What is the sum of the multiples of 2 or 17 under 767
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### Why does symmetry happen in reset-based random walks?

I am studying the basic concepts about random walks / brownian motion, and based on the idea of a Möbius-based walk in Wolfram's website, I wanted to try my own version of it in Python to compare it ...
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### Show that the operator sequence $A_n = 1/2(A_{n-1} + A^{-1}_{n-1})$ converges strongly, $A_0 = I+T$, where $T$ is compact and $||T|| \le 1/2$.

I'm studying for an analysis prelim and am stumped on an old exam problem for which there are no solutions given. The full question is as follows: Let $X$ denote a Hilbert space, and $T$ a ...