# All Questions

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### Definition of an inverse-powerseries

Let $t(q)=\sum_{n=0}^{\infty}t_n q^n$ be a complex powerseries convergent for all $|q|<1$. Assume $t_0=0$ and $t_1\neq0$. Not it says Let $q(t)$ be the local inverse of $t(q)$ with $q(0)=0$. ...
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### Homotopy type of mapping space

In Ralph Cohen's notes on the topology of fiber bundles (pp.63) he claims that the space of all $G$-equivariant maps from $P$ to $EG$ denoted by Map$^G(P,EG)$ is aspherical, where $EG$ is the total ...
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### Proof of unicity of decomposition of a representation

I'm studying representation theory and in the book the author makes the following proposition with the following proof: Proposition: For any representation $V$ of a finite group $G$, there is a ...
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### Probability density function for product and minimum of i.i.d. $U(0,1)$ random variables

If $U$ and $Y$ and $Z$ are i.i.d. $U(0,1)$ random variables, find the pdf for $A= U \times Y$ and $B = \min \{ U,Y,Z\}$.
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### Can anyone solve this geometric construction problem?

I remember when I was in high school, one of my all-time favorite books was College Geometry by Nathan Altshiller-Court. Some of its problems kept me wondering for days and even weeks. Now after about ...
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### Can $\mathbb{Z}/6\mathbb{Z}$ act freely and properly discontinuously on $\Sigma_4$?

Let $\Sigma_m$ denote the closed connected orientable surface of genus $m$. Let $N_m$ denote the closed connected non-orientable surface of genus $m$. I was wondering which cyclic groups could act ...
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### $f$ is factored into many same degree irreducible polynomials.

I met a problem when I study about Galois field and do this exercise. Hopefully, someone can help me. Suppose that $L/K$ is normal extension and $f$ is an irreducible polynomial in $K[X]$. Prove that ...
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### Proof verification: Every Euclidean space is complete

To prove this, I would like to use induction. For $n=1$ it is easy to prove that $\mathbb{R}$ is complete. For $n=k$ we assume it is true. For $n=k+1$, we have to show that $\mathbb {R}^{k+1}$ is ...
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### Tangent space to the intersection of two manifolds

Let $M, N \subset \mathbb{R}^n$ be two manifolds such that, for every $p \in M \cap N$, $(T_pM)^\bot \cap (T_pN)^\bot = \{0\}$. How do I determine the tangent space of $M \cap N$? I found some places ...
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### How to reverse a certain formula

I've got this formula which gives me the experience required for a specific level. $25X^2-25$ Which gives me a nice experience table like this Experience from level X to X+1 From level 4 to 5, 600 ...
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### $S_4/H \simeq S_3$ where $H$ is a normal subgroup

Prove that the group of permutations of four symbols $S_4$ contains a normal subgroup H such that the quotient group $S_4/H$ is isomorphic to the group of permutations of three symbols $S_3$. ...
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### Preferential Attachment and salton similarity in directed networks

Preferential Attachment similarity between two nodes in an undirected graph is the degree of the first node multiplied by the degree of the second node. But what about directed graphs? Which degree ...
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### On a linear non-homogeneous system of differential equations.

I rewrite my attempt at solving this system \begin{cases} x'(t) = 3x(t) + y(t) + e^{2t} \\y'(t) = - x(t) + y(t) + e^t\\ x(0) = 1 \\ y(0) = 0 \end{cases} I notice that the eigenvalue of the matrix ...
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### Generalized function in term of the Dirac-$\delta$-function

Personal question : Could it possible for the distribution distribution $<tr(e^{it \sqrt{\Delta}},\varphi)>=<-i \ln(1-e^{it}), \varphi'>$ to be expressed in term of the ...
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### Volume of intersection of the $n$-ball with a hyperplane

Let $\mathcal{B}_n$ be the $n$-ball of radius $r>0$ and centre $\mathbf{x}_0$, i.e., $\mathcal{B}_n=\{\mathbf{x}\in\mathbb{R}^n\colon \|\mathbf{x}-\mathbf{x}_0\| \leq r\}$. The volume of ...
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### Elliptic Curve Problem (finding a factor of n by using elliptic curve)

Let n be a a composite integer such that q is a factor of it. Consider the elliptic curve E defined by $y^2=x^3+3x+36$. The point $P=(0,6)$ is on the curve. Suppose it is given that the order of P mod ...
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### Integral inequality :$\int_0^1(f'(x))^2dx\geq 32\int_0^1(f(x))^2dx + 16\left(\int_0^{\frac{1}{2}}f(x)dx-\int_{\frac{1}{2}}^1f(x)dx\right)^2$
Assume $f:[0,1]\to \mathbb{R}$ is differentiable and $f'$ is integrable. Given $f\left(\frac{1}{4}\right)=f\left(\frac{3}{4}\right)=f(1)-f(0)=0$, then prove that \int_0^1(f'(x))^2dx\geq ...