# All Questions

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### How can we represent $10$ in a decimal system?

This may sound like a silly question to begin with but I'm having problems finding a proper answer. The question is generally targeting numeral systems of any base, but for simplicity, I will ...
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### Properties of the inverse of unit (lower) triangular matrix

Is there any special properties about the inverse of a unit lower triangular matrix? I'm trying to prove this: $$L^{-1}=I_n + N + N^2 + ... + N^{n-1}$$ where $L$ is a unit lower triangular matrix ...
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### how to prove the following recursive sequence produce relatively prime numbers

Sequence an is defined recursively: $a_1=2$ $a_{n+1}=a_n^{2}-a_n+1$ Prove that $a_i$ and $a_j$, $i\ne j$ are relatively prime. Hint: Prove that $a_{n+1} = a_1a_2…a_n + 1$ and use Euclid’s theorem. ...
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### Finding an angle on a triangle inscribed in a circle

How is angle $\angle OBW_2$ equal to $\theta$? $\angle AOB$ is given as $2\theta$. Is there a theorem? or simple geometry I'm not using. Would appreciate if someone drew out how $\angle OBW_1$ is ...
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### Probability related question, Permutations, combinations

Im doing a practice problem for an upcoming test, I had a hard time figuring out this question, could anyone walk me through it?
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### Complex analysis problem (entire functions)

How to solve following problems: Find all entire functions $f$ for which $|e^z-f(z)|\ge 2$, for every $z\in\mathbb{C}$ and $f(0)=-1$. If $f$ is entire function and $|f(z)|\le |z|^{2013}-|z|+2013$ ...
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### Can anybody provide some steps on how to do this?

How do I find a basis for the given plane $x+y+z=1$ in $\mathbb{R}^3$?
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### What is an intuitive way to think of Cauchy's theorem?

I am looking at a problem which involves an understanding of why a finite group $G$ has an element with order $p$ if $p$ is a prime factor of $|G|$. I have looked at several resources and proofs ...
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### Understanding averaging of symplectic matrices via Haar measure

In McDuff and Salamon's Intro. to Symplectic Topology (2nd edition), there's a proof that $U(n)$ is a maximal compact subgroup of $Sp(2n)$ which I'm trying to understand. The proof uses the Haar ...
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### Product Spaces and axioms of countability

Prove that a countable product of second countable spaces is second countable
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### Birational map between singular variety and smooth variety

$A$ is singular and $B$ are smooth algebraic varieties. Is it possible that $A$ is birationally equivalent to $B$? (over $\mathbb{C}$)
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### Tensor Algebra: Symmetrization & Antisymmetrization

Problem Given the tensor algebra: $$TV:=\sum_{k=0}^\infty{\bigotimes}^k V$$ Regard the symmetrization and antisymmetrization: ...
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### Find the order of $GL_2(Z_{p^{n}})$ for each prime ${p}$ and positive integer ${n}$.

Let $GL_2(Z_m)$ denote the multiplicative group of invertible $2 * 2$ matrices over the ring of integers modulo m. Find the order of $GL_2(Z_{p^{n}})$ for each prime ${p}$ and positive integer ${n}$.
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### reference needed for a property of the Riemann sphere

I need a reference citation for this fact, which I think is a common sense The bijective conformal mappings from the Riemann sphere to itself are Möbius transformations.