# Tagged Questions

If you already have a proof for some result, but want to ask for a different proof (using different methods).

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### Seeking Additional Solutions for the Number of Network Links

The Problem Show that the number of possible links in a computer network of $n$ computers ($n \in Z \land n \geq 1$) is $\frac{n(n-1)}{2}$ in as many ways as you can. My Work Solution 1 Given $n$ ...
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### Pentomino Tessellation Explanation

I need to explain why this pentomino tessellates in a mathematically coherent way. Here is the pentomino and the tessellation I have made. This pentomino can be translated to form a diagonal ...
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### How can i proof that every high order derivative of $\frac{1}{1+x}$ is equal to$(-1)^kk!$ at point $0$.

In order to calculate the Taylor-Maclaurin polynomial $\frac{1}{1+x}$ of order $n$ at point $0$, i used the identity: $$\sum_{i=0}^n x^i + \frac{x^{n+1}}{1+x} = \frac{1}{1+x}\qquad (I)$$ and i ...
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### Why are an even number of flips required to get back to the original list?

Consider the list of numbers $[1, \cdots, n]$ for some positive integer $n$. Two distinct elements $i$ and $j$ of the list can be switched in a so-called flip. For example, let $f$ be a flip that ...
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### $A = \sum_{n=0}^\infty a_n$ and $b_n \to B$ implies $\sum_{k=0}^n a_k b_{n-k} \to AB$

This question is motivated by the answer With $y_n$ a sequence of real numbers, prove that if $y_n=x_{n-1}+2x_{n}$ converges then $x_n$ also converges, where essentially the following fact is used: ...
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### Alternative proof : Group algebra contains all irreducible G-modules.

It is well-known that the group algebra $F[G]$ is a direct sum of irreducible $G$-modules. The proof in my text book is as follows Write $F[G] = \oplus_{i=1}^n V_i$, where $V_i$ is a set of ...
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### For all real numbers x and y there is a real number $z$ such that $x + z = y − z$.

To Prove: For all real numbers $x$ and $y$ there is a real number $z$ such that $x + z = y − z$. Proof: $x+z=y-z \Rightarrow y-x=2z$. Since $y$ and $x$ are real numbers, $2z$ is also real. Therefore ...
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### Proving three asymptotic identities (Murray (1984)'s Exercise 1.1.4)

(Context: I'm self-studying Murray (1984). I learned (and have forgotten quite a lot of) real and complex analysis. I'm willing to relearn and to look up references.) Problem: if $f=O(g)$, show that ...
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### How to improve my proof and whether or not one condition in the statement is important in writing the proof?

A simple graph $G$ is connected iff for every partition of the vertices into two non-empty sets $X$ and $Y$, there is a vertex $x\in X$ and a vertex $y\in Y$ such that $xy$ is an edge of $G$. My ...
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### $f$ is bijective, show that $h(x)=\left(f(x); g(x)\right) \rm{\ is\ bijective\ } \iff G$ is Singleton

Let E, F and G be three sets ($E\neq 0;F\neq 0,G\neq 0 )$ Let $h$ defined by : \begin{align} h \ \colon\ E & \to F\times G\\ x & \mapsto h(x)=\biggl(f(x);g(x)\biggr). \end{align} with ...
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### Check my proof on showing a graph with each vertex's degree at least $e$ has every tree with $e$ edges a subgraph
Let $T$ be a tree with $e$ edges and $G$ be a simple graph such that ech vertex has degree at least $e$. We need to show that $T$ is a subgraph of $G$. I tried to prove this by induction. The base ...
### Constructing topology on $\Bbb{Z}$
Fix an infinite subset $A$ of $\mathbb Z$ whose complement $\mathbb{Z}\setminus A$ is also infinite. Construct a topology on $\mathbb{Z}$ in which: (a) $A$ is open (b) Singletons are never open (i.e ...