# All Questions

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### What is a $\mathbb{Z}$-form of an algebra?

A homework problem I have is to describe the Lie algebra associated to a Kac-Moody root datum $\mathcal{K} =(I,A,\Lambda ,(c_i)_{i\in I},(h_i)_{i\in I})$ as well as to describe the universal ...
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### Bound on expectation of function of standard normal, $\mathbb{E}[\exp(Z^a)]$

I'm trying to find the maximum (or sup) of the value of $a$ such that $$\mathbb{E}[\exp(Z^a)]<+\infty$$ where $Z\sim \mathcal{N}(0,1)$. Obviously for $a=1$ the expectation is finite since it is the ...
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### Show that $3^n-2^n\cdot 5$ is composite for infinitely many $n$

I came across this problem: Show that $3^n-2^n\cdot 5$ is composite for infinitely many $n$ and do not know how to solve it. I only know that it is true for $n=7$, since then $1547=17\cdot 91$.
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### Ellipse equation. What does it need to be in order for $b > a$?

We have the quadratic equation: $$ax^2 + bx + cy^2 + dy + e$$ $a$ and $c$ are both negative or both positive. How can I, by looking at that only, determine whether $b$ (the length of the semi-minor ...
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### Showing uniform convergence of series

Show that $\displaystyle \sum_{j=1}^{\infty} \frac{-2j}{(x^2 + j^2)^2}$ converges uniformly. Don't know how to do this problem since $x$ and $j$ are in the expression together. Is there a convergence ...
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### My proof regarding composition of permutations came to the same conclusion as the answer sheet, but through different methods. Is it valid?

Let $S_3$ be a set of all permutations of elements in $\{1,2,3\}$. Prove that there doesn't exist f $\in S_3$ where $\{f,f^2,f^3,f^4,f^5,f^6\} = S_3$. Where $f^n = f \circ f \circ \:... \circ \:f$ ...
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### Let $f:S\rightarrow\mathbb{R}$ be a continious function and $S$ be a closed subset of $\mathbb{R}$ does it imply that $f(S)$ is a closed set? [duplicate]

Let $f:S\rightarrow\mathbb{R}$ be a continious function and $S$ be a closed subset of $\mathbb{R}$ does it imply that $f(S)$ is a closed set? Can somebobly please explain me this.thanks for your kind ...
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### A tough inequality problem with condition $a+b+c+abc=4$

If, $a+b+c+abc=4$, with $a,b,c$ being positive reals, then prove or disprove the following inequality: $$\frac{a}{\sqrt{b+c}}+\frac{b}{\sqrt{a+c}}+\frac{c}{\sqrt{a+b}}\geq\frac{a+b+c}{\sqrt2}$$ I ...
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### Arc length in polar coordinates: Why isn't $dS=r×d\theta$

As a sort of exercise, I tried to derive the formula for arc length in polar coordinates, using the following logic: $$dS = r(\theta)d\theta\\ \implies S=\int r(\theta)d\theta$$ However, it turns ...
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### Functional Choice for p in a Bernoulli Distribution

Why is the functional choice $p = \exp(x)/(1+\exp(x))$ to model $p$ a good one in a Bernoulli distribution? Is it because it is limited at $0$ as $x$ approaches $0$ and $1$ as $x$ approaches ...
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### given x>0 and n contained in N show show that there exists a unique positive real number $r$ such that $x = r^{n}$

I have proved half of this proof but I'm stuck on the other half because it is a little harder than the first due to negatives. I have considered that $(r+h)$ so that $(r+h)^n$ is less than $x$ and ...
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### determine if $G = \{ f : \mathbb{R_+} \to \mathbb{R_+} \}$ is a group

I'm confused about how the identity was formed - if $e(x) = x$, then how does one get from $f(x)e(x) = f(x)\cdot 1$
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### Good Textbooks for Real Analysis and Topology.

I'm currently in my 3rd year of my undergrad in Mathematics and moving onto my 4th year next year. I took a course in Real Analysis I, but the professor was very confusing and we didn't use a textbook ...
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### Find the limit of the sequence if it converges, otherwise state divergence.

Find the limit of the sequence given by $$\frac{10+12n+20n^4}{7n^4 + 5n^3 - 20}$$ I think the answer is $\frac{20}{7}$ after dividing, but is that right?
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### Rational vs irrational

If two points on a number line is shown, are rational numbers between the two points is more or irrational number is more ? I have tried using probability , my collegue who was like my teacher also ...
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### Inequality, $\left(\frac{2}{x}+2\right)^{n}-\left(\frac{2}{x}-2\right)^{n}\leq \left(\frac 4 x \right)^n$

How do I show that $$\left(\frac{2}{x}+2\right)^{n}-\left(\frac{2}{x}-2\right)^{n}\leq \left(\frac 4 x \right)^n$$ for $x\in\left(0,1\right]$ and $n\in\mathbb N$?
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### A Criterion for being Sylow p-group

Show that if $H$ is a $p$-group of finite group $G$ and $N_G(H)=H$ then $H$ is a Sylow $p$-group of $G$? Or prove the following more general property,$$[G:H]\equiv1\ (\mod\ p)$$
Given a pair $[M_1,M_2]$ there is an easy encoding $\lambda x.x M_1 M_2$. For the n-tuple we have two options. First encoding: $$\lambda x.x M_1, M_2, \ldots , M_n$$ Second encoding: [M_0, [M_1 , ...
I'm trying to find the for for forward equations for a birth and death processes when all $\lambda$ coefficients are zero. The forward equation for a Birth and Death Process has the form: ...