All Questions

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Polynomial long division properties

So I have a basic question. I needed to simplify this expression $$\frac{x^{3}}{x^{2} +3x +2}$$ I was to do a polynomial long division in order to simplify it, but I still do not understand how they ...
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Determinants of matrices; Does the equation $\det M=\det(I_n-M)$ have a solution?

Suppose $M$ is some $n\times n$ real matrix. Then is it always true that $\det M\neq\det(I_n-M)$, where $I_n$ is the $n\times n$ identity matrix? I have a feeling that it is yes, but I am not sure... ...
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Problem on Borel real line

We are given a distribution with mean u, and variance sigma square. An unknown random variable X exists whose values take on belong to a range of numbers. This range of numbers from which X takes ...
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May be a foolish question, but why is it always being said that "Two non-parallel edges are said to be adjacent if they are incident on a common vertex." What about a situation where there exist ...
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Eliminate removable discontinuity

I have two rational functions which I have to examine for discontinuity and try to remove their domain gaps if possible. $$f(x)=\frac{|x|-1}{x^3-x}$$ and $$g(x)=\frac{sin (x)}{sin (2x)}$$ I ...
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How to show that product of a Lindelöf and a Compact space is Lindelöf?

How to show that if $X$ is Lindelöf and $Y$ is compact , then $X \times Y$ is Lindelöf?
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Proving the Formula for the Determinant of the Adjacency Matrix of a Complete Graph

A complete graph of $n$ nodes has an $n$x$n$ adjacency matrix $A_{ij}$ such that $$a_{ij} = 0 \text{, if } i = j \\ a_{ij} = 1 \text{, if } i \ne j$$ i.e. there are 0s down the diagonal and 1s ...
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Using natural deduction, show that: ∀x.(P (x) → Q(x)), ∃x.P (x) ⊢N ∃x.Q(x) [on hold]

How I would show: ∀x.(P (x) → Q(x)), ∃x.P (x) ⊢N ∃x.Q(x) using Natural Deduction? Would I not need to know what P(x) and Q(x) are to prove soundness and completeness?
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Is there a mistake in the solution given to this exercise?(compact linear operators)

The exewrcise is: Let $\mathcal{H}$ be an infinite-dimensional Hilbert space, with an orthonormal basis $\{e_n\}$ and let $T \in \mathcal{B}(\mathcal{H})$. Show that if T is compact then ...
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Automorphism group of a covering space $Aut(\tilde{X},p)\cong NH/H$

I want to show $Aut(\tilde{X},p)\cong NH/H$ where $H=p_*\pi_1(\tilde{X},\tilde{x_0})$ and $NH$ is the normalizer of $H$. In my text book, the author sketches the proof of the above by using the ...
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Expected number of rolls when repeatedly rolling an $n$-sided die

Suppose I roll an $n$-sided die once. Now you repeatedly roll the die until you roll a number at least as large as I rolled. What is the expected number of rolls you have to make? I know the answer ...
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Mandelbrot set, inequality proof

If I have the relation $z_{n+1} = z_{n}^2 + c$. How can I show that $|z_{n+1}| > k |z_n|$ for some $k>1$, if $|z_n| > |c| > 2$? I have no idea how to proof this, any help will be good.
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How to solve 2-nd order differential equation

How to find the general solution of the following 2nd order ODE: $f''(x)-3f'(x)-4f(x)=e^x +6e^{5x}$ ?
Stirling Numbers of the First Kind and $S_n$.
I know that, on the one hand, if $s(n, p)$ denotes the unsigned Stirling Numbers of the First Kind, then $(x)_n=\displaystyle\sum_{p=0}^n s(n, p)x^p$, where $(x)_n=x(x-1)\cdots(x-n+1)$. It follows ...
I am interesting in bounding the arithmetic sum $$\sum_{n \leq x} \frac{\mu(n)^2}{\varphi(n)}$$ (The motivation is that this is a sum that comes up a lot in sieving primes, in particular in the ...