# All Questions

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### Local maxima when multiplying two functions

I have two functions, $f(x)$ and $g(x)$, where: $f(x) = \frac{1}{x^2 + a}, a>0$ and the only things I know about $g(x)$ are that: $g(x) > 0, \forall x \in \mathbb R$ $g(x)$ is a polynomial ...
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### Coming up with an alternative proof by induction

Kindly refer to Q4 of this handout. "$2n$ dots are placed around the outside of the circle. $n$ of them are colored red and the remaining $n$ are colored blue. Going around the circle clockwise, you ...
316 views

### complex numbers find greatest value of z

I've to sketch the complex number $z$ such that it satisfy both the inequality $|(z-2i)|\le2$ and $0\le \arg(z+2)\le 45\deg$ I was able to sketch and shade the region that satisfies both ...
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### Proving that if $\sum\|f_n-e_n\|^2< 1$, $\{f_n\}$ is a complete sequence

Let $\{e_n\}$ be a complete orthonormal sequence in an Hilbert space $H$ and let $\{f_n\}$ be an arbitrary sequence of elements in $H$ s.t $$\sum_{n=1}^\infty\|f_n-e_n\|^2<1$$Show that ...
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### Computational cost of solving $Ax_i = b_i$ for $i=1,…,m$

$A$ is an invertible matrix square $n$ matrix. The exercise is about 3 different ways you can solve this and I have to determine its efficiency. It's always the same matrix $A$ but a different right ...
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### Determining a consistent estimator/asymptotic relative efficiency

Question: Let $X_1,\ldots,X_n$ be i.i.d. as $N(0,\sigma^2)$. a) Show that $\delta_1 = k \sum_{i=1}^n |X_i|/n$ is a consistent estimator of $\sigma$ if and only if $k = \sqrt{\pi/2}$. b) Determine ...
581 views

### Extract real and imaginary parts of $\operatorname{Li}_2\left(i\left(2\pm\sqrt3\right)\right)$

We know that polylogarithms of complex argument sometimes have simple real and imaginary parts, e.g. ...
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### How do you solve this equation?

$$\sqrt{x + \sqrt{x + \sqrt{x + \cdots}}}= 5$$
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### Limit definition question

Just curious, the definition of a limit is: For every $\epsilon\gt0$, there exists a $\delta\gt0$, such that for every $x$, the expression $0\lt|x-c|\lt\delta$ implies $|f(x)-L|\lt\epsilon$. Is ...
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### Exact values of error function

The error function is defined as $$\operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} \, dt.$$ We know that the Gaussian integral is $$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}.$$ ...
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### If $P$ is a projection operator, is $1-P$ also a projection operator?

Show that if $P$ is a (hermitian) projection operator, so are (a) $1-P$ and (b) $$U^{+}PU$$ for any operator $U$
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My question is this: Why does Rudin use $\delta$ in this proof? Would it not work just as well if $\forall i \ge1,$ $$x_{i}\in N_{2^{-(i+1)}}(q) \cap E^*$$ $$p_{n_i}\in N_{2^{-(i+1)}}(x_i) ... 1answer 113 views ### Calculate winner of soccer match I am writing a program that simulates a soccer tournament between countries using their FIFA rankings. I am looking for a function that takes two country rankings and outputs a number between (about) ... 2answers 105 views ### A fair coin is flipped until the first tail appears, in general we win \ 2^k . St. Petersburg problem. For the St.Peterburg problem (Example 3.5.5), find the expected payoff if (a) the amounts won are c^k instead of 2^k, where 0 < c < 2. (b) the amounts won are \log(2^k). The original ... 1answer 25 views ### Maxima and Minima, If s = 60, what should the side of the cut out be… A square piece of steel, s cm on a side, is to made into an equipment chassis by cutting equal squares out of the corners, folding up the sides , and welding the seam to form a pan. A) If s=60, ... 2answers 212 views ### Notation involving the Lebesgue integral. I have a measurable function f : \mathbb{R}^d \to \mathbb{R}. Let E be a measurable subset of \mathbb{R}^d. Then then$$\int_{E} f(x) \, dx = \int f(x) \chi_E (x) \, dx. If we are taking an ...
Say I have a set $H$ and I want to describe the union of all elements in $H$. How would I write that? I believe I've seen a big U with a subscript used before.