# All Questions

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### How can I find an upper bound for the radius of an arc, given arc length and chord length?

This is similar to this question, except I am finding the radius now using the Bisection method and then Newton's method for finding a zero. This is a computer science for a Numerical Methods course. ...
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### Different ways to simplify the structure of a graph

Let $G = (V,E)$ be an undirected finite simple graph with no loops. Definition Let's call a subset $U\subset V$ autonomous if every vertex of $V\ \setminus U$ is either adiacent to every vertex of ...
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### Correlation Coefficient between these two random variables

Suppose that $X$ is real-valued normal random variable with mean $\mu$ and variance $\sigma^2$. What is the correlation coefficient between $X$ and $X^2$?
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### Cylindrical Shells problem (can't find region)

"Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis." I have two equations : x= 2y^(2)-y^(3) and x= 0. ...
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### Are there disjoint closed sets which contain sequences that get arbitrarily close to each other?

I had a question of curiosity. Take the interval $(0,1)$ with the usual metric in $\mathbb{R}$. Is it possible to find closed sets $X$ and $Y$ with $X\cap Y=\varnothing$ such that there is a sequence ...
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### Limit involving the exponential of an expression plus a small-o

I'm trying to formalize the following limit: $$\lim_{x \to \infty} \left[1+\frac{a}{x}+o\left(\frac{1}{x}\right) \right]^x= e^a$$ however I can't see how to rigorously show this identity. Any ...
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### Finding an $f(x)$ that satisfies $f(f(x)) = 4 - 3x$

I need to find $f(f(x)) = 4 - 3x$ In other examples, such as $f(2)$, I can see that the result equates to $-2$ or $f(x^2)$ becomes $-3x^2 + 4$. Do I really just substitute $f(x)$ for $x$ and ...
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### Let $f(t) = \frac{t - x}{t + y}$

I have the problem. Let $$f(t) = \frac{t - x}{t + y}.$$ Show that $$f(x + y) + f(x - y) = \frac{-2y^2}{x^2 + 2xy}.$$ I know that this is just some substitution followed by simplification, ...
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### Logical NOT of an implication

I was looking through my notes but I was unable to find the answer to this, which I need to start am assignment question. What would the following be, in terms on moving the negation inside the ...
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### Best method for this example to get from transfer function to state space

I have this system here: In this example the state space representation $\frac{dx}{dt} = Ax + bu$ and the corresponding transition matrix $\Phi(t)$ is asked for. So to get the state space, I ...
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### Mathematical explanation behind a picture posted (lifted from facebook)

In this image given below, there is an actor's (famous south Indian actor Rajinikanth) image which can be seen only if you shake your head ! I had lifted this from Facebook. I am just curious to ...
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### Motivation behind the definition of Prime Ideal

Can someone explain what's the motivation behind the definition of a prime ideal? Or why is it exactly called a prime ideal? Has it anything to do it prime numbers?
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### Polynomial equations in 2 variables with symmetry

Suppose $P(x,y)$ is a polynomial with real coefficients. Is it true that any solution $(x_0,y_0)$ of the system $P(x,y)=P(y,x)=0$ has the property that $y_0 = \overline{x_0}$ (i.e. they are ...
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### Using Fourier series to calculate an infinite sum

Given the Fourier series of the $2\pi$-periodic function defined for $$-\pi\leqslant x \leqslant \pi$$ by $$f(x) = |x|$$ is  \frac{\pi}{2} -\frac{4}{\pi} \sum_{k\geq 1, k\ odd}^{\infty} ...
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### What is this special type of factor called?

I'm wondering if there's a special term for the following: The (special factor) of a number $x$ is a pair of numbers that multiply to give $x$ but has the smallest difference compared to other ...
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### How do I prove that $\cos{\frac{2\pi}{7}}\notin\mathbb{Q}$?

How do I prove that $\cos{\frac{2\pi}{7}}\notin\mathbb{Q}$? Should I use some geometrical approach or apagoge?
### Show that rays of the form $(-\infty, a)$ and $(b, \infty)$ ; $a,b \in \mathbb R$, are a sub-basis for the standard topology on $\mathbb R$?
Show that rays of the form $(-\infty, a)$ and $(b, \infty)$ ; $a,b \in \mathbb R$, are a sub-basis for the topology generated by open intervals of $\mathbb R$ on $\mathbb R$? I'd just like to know if ...