# All Questions

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### Show every subgroup of D4 can be regarded as an isotropy group for a suitable action of D4

Show every subgroup of D4 can be regarded as an isotropy group for a suitable action of D4 I know that D4={1,R,R2,R3,D1,D2,M1,M2} and the subgroups are {1,R,R2,R3} {1,D1} {1,D2} {1,M1} {1,M2} ...
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### Interscholastic Mathematics League Senior B #12

Compute the product of the nonreal roots of the equation $x^4+4x^3+6x^2+1004x+1001=0$. So here is what I have done so far. I got two of the roots to be zero and 4 since ...
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### Another Question in Hatcher

First of all, I apologize for asking yet another question about the hypotheses of a problem in Hatcher, but the statement of one of his problems has stumped me again. The problem is 1.3.15. It reads ...
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### What does unique “minimal” partition mean (Context: Partitioning of Vertex-Sets)?

I am studying R. Diestel's Book Graph Theory and I encountered a formulation which I don't quiet understand. Mr. Diestel speaks in this proof on page 180 (Google Books Link) in the second last line of ...
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### This is the most difficult question I could get without using mass point geometry

In triangle ABC, points D and E are on sides BC and CA respectively, and points F and G are on side AB with G between F and B. BE intersects CF at point O_1 and BE intersects DG at point O_2. If FG ...
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### Pi approximation

If $d(a,b)=$ largest $n$ such that $a$ and $b$ agree on all digits upto $n$. Eg. $d(\pi,3.14)=3$, $d(0.1234667,0.1234669)=7$. What is the asymptotics of $d(\pi/4,1-1/3+1/5-1/7+\cdots(\pm)1/m)$ as ...
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### Why can't $\mathbb{Z}/(p^k)$ for $k>1$ be the direct sum of two submodules?

If you mod out $\mathbb{Z}$ be a nontrivial prime power $p^k$, $k>1$, then why can't $\mathbb{Z}/(p^k)=\mathbb{Z}/(n)\oplus\mathbb{Z}/(m)$ for some such submodules? If that where the case, then ...
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### how to find correlation between 2 arrays of 1's and 0's?

For my case, I have 2 arrays or sets of data, 100 elements, and the values are only 0 and 1. What test or procedure would measure the correlation or independence of the 2 sets? To give an example of ...
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### How to “explain” Szemerédi's Regularity Lemma so that classmates may understand its value?

I am a student, preparing myself for a talk in which I want to present and prove Szemerédi's Regularity Lemma. I understand the proof and I am able to reproduce it - that is no problem. But I am ...
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### functions $f=g$ $\lambda$-a.e. for continuous real-valued functions are then $f=g$ everywhere

I am supposed to show that if $f$ and $g$ are continuous, real-valued functions on $\mathbb{R}$, then if $f=g\;\;$, $\lambda$-a.e., then $f=g$ everywhere. So I have been reading and I think that this ...
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### Is this a well known determinant identity? Are there any generalizations?

Let $A$ be a $3\times3$ matrix and for any $i,j\subseteq\{1,2,3\}$, let $A^{i,j}$ denote the $2\times2$ matrix resulting from removing row $i$ and column $j$ from $A$. Then: ...
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### Interpretation of a question: “group of all p-power roots of unity”

I have a homework problem I'm trying to do, but I'm not sure what it's asking. The problem is as follows: Recall that $\mathbb{Q}/\mathbb{Z}$ is isomorphic to the group of all roots of unity in ...
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### Apostol Section 13.25 #13 - Conic Sections

Question: Prove that a similarity transformation (replacing $x$ by $tx$ and $y$ by $ty$) carries an ellipse with center at the origin into another ellipse with the same eccentricity. (The next ...
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### How can I convert lines intersecting a plane into a focused image?

I am writing a particle transport code. I would like to be able to obtain an image of my geometry when transporting photons given the following information: The photons are incident on a plane. For ...