# All Questions

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### Velleman - How to prove it - Do these two statements really mean the same thing?

Hello and thanks in advance for reading! In How to Prove it P29 Velleman writes: " In general, the statement y ∈ { x | P(x)} means the same thing as P(y), ... " In my understanding the first ...
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### Differentiation optimisation

I have a question stating that an insulation strip is to be sealed completely around three edges of a rectangular solar panel. The length of this strip is 200cm. It is asking what dimension of the ...
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### Normed vector space and uniform continuity

Let $V, V'$ be normed vector spaces and $f: V\to V'$ a linear transformation. a. Prove that if $f$ is continuous at one point it is continuous everywhere and in fact uniformly continuous. ...
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### Is there a friendly pair $\{a, b\}$ where both $a$ and $b$ are odd?

Let $\sigma(x)$ denote the sum of the divisors of a positive integer $x$. $\{y, z\}$ is said to be a friendly pair if $$I(y) = I(z),$$ where $I(x) = \sigma(x)/x$ is the abundancy index of $x$. As ...
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### “Problem of points (POP)” explanation

Quite recently, there was a question related to "Problem of Points" at MSE. I did some literature survey on POP in the internet and found explanations using an example of coin toss. I have been ...
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### Show a subring contains certain elements.

Show that the set of all real numbers of the form $a_0 + a_1\pi + a_2\pi^2 +\cdots+ a_n\pi^n$ with $n≥0$ and $a_i ∈ \mathbb{Z}$ is a subring of $R$ that contains $\mathbb{Z}$ and $\pi$. ...
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### how to solve $x^2 \equiv -1 \pmod{13}$

Knowing that $p$ is prime and if $p \equiv 1 \pmod 4$, then $\left(\left(\frac{p-1}{2}\right)!\right)^2 \equiv -1 \pmod p$; how do I solve $x^2 \equiv -1 \pmod{13}$?
### we need to find the points where $|f(z)|$ has maximum and minimum value
$f(z)=(z+1)^2$ and $R$ be the triangle with vertices $(0,0),(0,1),(2,0)$, we need to find the points where $|f(z)|$ has maximum and minimum value so here $z=2$ is the point of maximum and $z=0$ is ...
### the induced homomorphism $i_\#:\pi_1(P,x_0) \to \pi_1 (X,x_0)$ is an isomorphism.
If $P$ is a path component of $X$ and $X_0\in P$, then the inclusion map $i:P\to X$ can be regarded as a map of pointed spaces $P(X.x_0) \to (X,x_0)$. Prove that the induced homomorphism ...