# All Questions

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### Does Euler totient function gives exactly one value(answer) or LEAST calculated value(answer is NOT below this value)?

I was studying RSA when came across Euler totient function. The definition states that- it gives the number of positive values less than n which are relatively prime to n. I thought I had it, untill I ...
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Is it true that one complex addition requires 2 real additions. Could you show a proof that one complex addition requires 2 real additions?
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### $f_n(z)={z^n\over n}$, $z\in D$ open unit disk then

$f_n(z)={z^n\over n}$, $z\in D$ open unit disk then 1.$\sum f_n$ converges uniformly on $D$? 2.$f_n$ and $f'_n$ converges uniformly on $D$? 3.$\sum f'_n$ converges on $D$ pointwise? 4.$f_n''(z)$ ...
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### Statistics Question

This is probably super simple to most of you on here, but I was chatted by a friend earlier with a question. It reads just like this: ...
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### FFT: Why does 4log(4)/log(2) result into 8 complex adds and 4 complex multiplications?

FFT supposed to take O(N log N) operations. Take an radix-2 FFT with N=4. Does 4log(4)/log(2) really result into 8 complex adds and 4 complex multiplications? If so, why doesn't it result into 8 ...
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### Can an odd perfect number be divisible by $165$?

I know that an odd perfect number cannot be divisible by $105$ or $825$. I wonder if that's also the case for $165$ (this is actually a stronger statement).
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### Does $P(A\cap B) + P(A\cap B^c) = P(A)$?

Based purely on intuition, it would seem that the following statement is true, when thinking of the events as sets: $$P(A\cap B) + P(A\cap B^c) = P(A)$$ However, I am not sure if this is true, and ...
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### Analogy between prime numbers and singleton sets?

While trying -- in vain -- to write an alternative answer for another question (If $\cup \mathcal{F}=A$ then $A \in \mathcal{F}$. Prove that $A$ has exactly one element.), I discovered the following ...
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### Prove that if $n\geq\text{lcm}(a,b)$ and $\gcd(a,b)|n$ then $n=xa+yb$ for some integers $x,y\geq 0$

I thought I had it, but then I realized I didn't. Even just a hint—am I going in the right direction or should I try something completely different? We know that $\gcd(a,b)=wa+zb$ for some integers ...
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### Help with school solid geometry

In the triangular pyramid $MABC$ all side edges equals $1$, $\angle AMB = \angle BMC = 60 ^\circ$, $\angle AMC = 45 ^\circ$. Find: 1) square of the $\triangle ABC$; 2) dihedral angle on the $AB$ ...
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### Prove that $[L,Rad(L)] \subseteq N$ for finite-dimensional Lie algebra $L$

I need to prove the following fact: if $L$ is a finite-dimensional Lie algebra over field of characteristic $0$, $Rad(L)$ is its radical, and $N$ is the maximal nilpotent ideal in $L$, then ...
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### If $d(x,A)=0\forall x\in X$ for some subset $A$ of $X$, does it follow that $A$ is dense?

If $d(x,A)=0\forall x\in X$ for some subset $A$ of $X$ then $A$ is dense in $X$, right? Once I did one problem which says $d(x,A)=0\Leftrightarrow x\in \bar{A}$ so by the condition here we get ...
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### probability to cover straight line with circles

suppose sensors of homogeneous radius r are dropped by Poisson distribution on a straight line of length L. how to calculate that the straight line is covered by sensors with probability P
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### $\int_{0}^{\pi/2} \text{arctanh}(\sin x) \text{arctan}(a \tan(x)) \cos(x) \ dx$

Are you kind to let me know the way? By the way, don't you have a "curiosity" tag? $$\int_{0}^{\pi/2} \text{arctanh}(\sin x) \text{arctan}(a \tan(x)) \cos(x) \ dx, \quad a>0$$
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### Finding Area of a shape

I'm doing some revision and I'm a bit stuck. How do you find out the area of this shape? I know you have to do 10cm x 2cm = 20cm devided by 2 = 10cm - 2cm x ? (something) = area. I'm not sure what ...
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### Is there a continuous version of $tan^{-1}(\frac{y}{x})$ for the entire unit circle?

The fact that $tan^{-1}(\frac{y}{x})$ only "works" for the upper-right quadrant makes some calculations (for a physics simulator) impossible. I of course use $atan2(y,x)$ in the code, that's not what ...
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### Size and Order of a Line Graph of Another Graph G

This is an extension of the problem at this thread: Graphs, line graph and complement of graph. Just to reiterate definitions: The line graph $L(G)$ of a graph $G$ is defined in the following ...
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### A is a matrix of integers , prove that A+I is invertible

Question: $2 \le d \in \Bbb Z$ Let $A \in M_n(\Bbb Q) s.t$ All of it's elements are integers divisible by d. Prove that $I+A$ is invertible. What I thought: I thought of using the determinant of ...
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### Proving an identity regarding the Cauchy problem (using convolutions)

Given $u_0 \in C_c(\mathbb{R}^n)$, consider the solution of the Cauchy problem $$u(x,t) = \int_{\mathbb{R}^n} \Gamma (x - y,t)u_0(y) dy \qquad x \in \mathbb{R}^n,t>0\, \, .$$ Given $0<s<t$ , ...
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### $p(z,w)=a_0(z)+a_1(z)w+\dots +a_k(z)w^k$

$p(z,w)=a_0(z)+a_1(z)w+\dots +a_k(z)w^k$ where $a_i(z)$ are non constant poly in complex variable, $k\ge 1$, I need know $$\{(z,w):p(z,w)=0\}$$ is 1.bounded with empty interior 2.unbounded with ...
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### numerical approximation to logarithm

we know that $$\ln(x)=\frac{1}{x} \int_{0}^{1}dt \frac{1}{1+xt}$$ then given a cuadrature formula inside $(0,1)$ is that true $$\ln(x)= \frac{1}{x}\sum_{i}\frac{w_{i}}{1+xt_{i}}$$ wht other ...
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### Find the limit without use of L'Hôpital or Taylor series: $\lim \limits_{x\rightarrow 0} \left(\frac{1}{x^2}-\frac{1}{\sin^2 x}\right)$

Find the limit without the use of L'Hôpital's rule or Taylor series $$\lim \limits_{x\rightarrow 0} \left(\frac{1}{x^2}-\frac{1}{\sin^2x}\right)$$
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### Can we get just $3$ from $\pi$?

Today, a friend and I solved a question and one point came up where we were discussing whether we should write $(-1)^{n-1}$ or $(-1)^{n+1}$ and quickly we remembered that it was the same thing (for ...
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### Questions about epimorphisms and projectives in functor categories

Suppose $I$ is a small category, $R$ is a ring and $_R\mathrm{Mod}$ is the category of left $R$-modules. How do I show that the category $[I,~_R\mathrm{Mod}]$ of all functors from $I$ to ...
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### Unity in the rings of matrices

Suppose we are given an arbitrary ring $R$. Then the set $M_n(R)$ of all square matrices with elements from $R$, together with usual matrix addition and multiplication forms a ring. If R is a unitary ...