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1500 questions
77
votes
3 answers
Characterizing units in polynomial rings
I am trying to prove a result, for which I have got one part, but I am not able to get the converse part. Theorem. Let $R$ be a commutative ring with $1$. Then $f(X)=a_{0}+a_{1}X+a_{2}X^{2} + \cdots + a_{n}X^{n}$ is a unit in $R[X]$ if and only if…
anonymous
77
votes
4 answers
Graph theory: adjacency vs incident
Okay, so I think if 2 vertices are adjacent to each other, they are incident to each other....or do I have it wrong? Is this just different terminology. I thought I was totally clear on this for my class, but now I am doubting myself reading the…
pqsk
- 875
77
votes
1 answer
Gerrymandering on a high-genus surface/can I use my powers for evil?
Somewhat in contrast to this question . Let's say the Supreme Court has just issued a ruling that the upper and lower roads of an overpass need not be in the same congressional district. This makes states with lots of overpasses into high-genus…
Alex Becker
- 62,289
77
votes
8 answers
What is the chance to get a parking ticket in half an hour if the chance to get a ticket is 80% in 1 hour?
This sounds more like a brain teaser, but I had some kink to think it through :( Suppose you're parking at a non-parking zone, the probability to get a parking ticket is 80% in 1 hour, what is the probability to get a ticket in half an hour? Please…
Rock
- 816
77
votes
4 answers
Teenager solves Newton dynamics problem - where is the paper?
From Ottawa Citizen (and all over , really): An Indian-born teenager has won a research award for solving a mathematical problem first posed by Sir Isaac Newton more than 300 years ago that has baffled mathematicians ever since. The solution devised…
jnm2
- 3,260
77
votes
6 answers
How do I tell if matrices are similar?
I have two $2\times 2$ matrices, $A$ and $B$, with the same determinant. I want to know if they are similar or not. I solved this by using a matrix called $S$: $$\left(\begin{array}{cc} a& b\\ c& d \end{array}\right)$$ and its inverse in terms of…
user4681
- 835
77
votes
3 answers
how to read a mathematical paper?
I hope that this question is on-topic, though it is not quite technical. I am curious to hear from people how they approach reading a mathematical paper. I am not asking specific questions on purpose, though at first I had a few. But I want to keep…
normvector
- 713
76
votes
3 answers
What's an example of a discontinuous linear functional from $\ell^2$ to $\mathbb{R}$?
I'm trying to find a discontinuous linear functional into $\mathbb{R}$ as a prep question for a test. I know that I need an infinite-dimensional Vector Space. Since $\ell_2$ is infinite-dimensional, there must exist a linear functional from $\ell_2$…
FPP
- 2,193
76
votes
3 answers
Path-connected and locally connected space that is not locally path-connected
I'm trying to classify the various topological concepts about connectedness. According to 3 assertions ((Locally) path-connectedness implies (locally) connectedness. Connectedness together with locally path-connectedness implies…
qyy
- 1,187
76
votes
4 answers
Check if a point is within an ellipse
I have an ellipse centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane. How do I determine if a point $(x,y)$ is within the area bounded by the ellipse?
76
votes
14 answers
Dividing 100% by 3 without any left
In mathematics, as far as I know, you can't divide 100% by 3 without having 0,1...% left. Imagine an apple which was cloned two times, so the other 2 are completely equal in 'quality'. The totality of the 3 apples is 100%. Now, you can divide those…
RAO
- 1,027
76
votes
2 answers
Why is $\mathbb{Z}[\sqrt{-n}], n\ge 3$ not a UFD?
I'm considering the ring $\mathbb{Z}[\sqrt{-n}]$, where $n\ge 3$ and square free. I want to see why it's not a UFD. I defined a norm for the ring by $|a+b\sqrt{-n}|=a^2+nb^2$. Using this I was able to show that $2$, $\sqrt{-n}$ and $1+\sqrt{-n}$ are…
Danielle Intal
- 1,382
76
votes
6 answers
Functions which are Continuous, but not Bicontinuous
What are some examples of functions which are continuous, but whose inverse is not continuous? nb: I changed the question after a few comments, so some of the below no longer make sense. Sorry.
isomorphismes
- 4,768
76
votes
5 answers
What does "curly (curved) less than" sign $\succcurlyeq$ mean?
I am reading Boyd & Vandenberghe's Convex Optimization . The authors use curved greater than or equal to ( \succcurlyeq ) $$f(x^*) \succcurlyeq \alpha$$ and curved less than or equal to ( \preccurlyeq ) $$f(x^*) \preccurlyeq \alpha$$ Can someone…
Dinesh K.
- 861
76
votes
20 answers
How can you prove that the square root of two is irrational?
I have read a few proofs that $\sqrt{2}$ is irrational. I have never, however, been able to really grasp what they were talking about. Is there a simplified proof that $\sqrt{2}$ is irrational?
John Gietzen
- 3,631